diff --git a/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/2301.11406v1.pdf.txt b/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/2301.11406v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..68b7cd9ed8b4221aa840581a0e48ffb5ac840148 --- /dev/null +++ b/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/2301.11406v1.pdf.txt @@ -0,0 +1,944 @@ +Preprint to appear in the Proceedings of the 7th Arabic Natural Language Processing Workshop (WANLP), 2022. +EMNLP, Abu Dhabi, United Arab Emirates, December 7–11, 2022. +Beyond Arabic: Software for Perso-Arabic Script Manipulation +Alexander Gutkin† Cibu Johny† Raiomond Doctor‡∗ Brian Roark◦ Richard Sproat⊛ +Google Research +†United Kingdom +‡India +◦United States +⊛Japan +{agutkin,cibu,raiomond,roark,rws}@google.com +Abstract +This paper presents an open-source software +library that provides a set of finite-state trans- +ducer (FST) components and corresponding +utilities for manipulating the writing sys- +tems of languages that use the Perso-Arabic +script. +The operations include various lev- +els of script normalization, including visual +invariance-preserving operations that subsume +and go beyond the standard Unicode normal- +ization forms, as well as transformations that +modify the visual appearance of characters in +accordance with the regional orthographies for +eleven contemporary languages from diverse +language families. The library also provides +simple FST-based romanization and transliter- +ation. We additionally attempt to formalize the +typology of Perso-Arabic characters by provid- +ing one-to-many mappings from Unicode code +points to the languages that use them. While +our work focuses on the Arabic script diaspora +rather than Arabic itself, this approach could +be adopted for any language that uses the Ara- +bic script, thus providing a unified framework +for treating a script family used by close to a +billion people. +1 +Introduction +While originally developed for recording Arabic, +the Perso-Arabic script has gradually become one +of the most widely used modern scripts. Through- +out history the script was adapted to record many +languages from diverse language families, with +scores of adaptations still active today. This flexi- +bility is partly due to the core features of the script +itself which over the time evolved from a purely +consonantal script to include a productive system +of diacritics for representing long vowels and op- +tional marking of short vowels and phonologi- +cal processes such as gemination (Bauer, 1996; +Kurzon, 2013). +Consequently, many languages +productively evolved their own adaptation of the +∗ On contract from Optimum Solutions, Inc. +Perso-Arabic script to better suit their phonology +by not only augmenting the set of diacritics but +also introducing new consonant shapes. +This paper presents an open-source software li- +brary designed to deal with the ambiguities and +inconsistencies that result from representing var- +ious regional Perso-Arabic adaptations in digital +media. Some of these issues are due to the Uni- +code standard itself, where a Perso-Arabic char- +acter can often be represented in more than one +way (Unicode Consortium, 2021). Others are due +to the lack or inadequacies of input methods and +the instability of modern orthographies for the lan- +guages in question (Aazim et al., 2009; Liljegren, +2018). +Such issues percolate through the data +available online, such as Wikipedia and Common +Crawl (Patel, 2020), negatively impacting the qual- +ity of NLP models built with such data. The script +normalization software described below goes be- +yond the standard language-agnostic Unicode ap- +proach for Perso-Arabic to help alleviate some of +these issues. +The library design is inspired by and consis- +tent with prior work by Johny et al. (2021), in- +troduced in §2, who provided a suite of finite- +state grammars for various normalization and (re- +versible) romanization operations for the Brah- +mic family of scripts.1 +While the Perso-Arabic +script and the respective set of regional orthogra- +phies we support – Balochi, Kashmiri, Kurdish +(Sorani), Malay (Jawi), Pashto, Persian, Punjabi +(Shahmukhi), Sindhi, South Azerbaijani, Urdu +and Uyghur – is significantly different from those +Brahmic scripts, we pursue a similar finite-state in- +terpretation,2 as described in §3. Implementation +details and simple validation are provided in §4. +1https://github.com/google-research/nisaba +2https://github.com/google-research/nisaba/ +tree/main/nisaba/scripts/abjad alphabet + +2 +Related Work +The approach we take in this paper follows in +spirit the work of Johny et al. (2021) and Gutkin +et al. (2022), who developed a finite-state script +normalization framework for Brahmic scripts. We +adopt their taxonomy and terminology of low- +level script normalization operations, which con- +sist of three types: Unicode-endorsed schemes, +such as NFC; further visually-invariant transfor- +mations (visual normalization); and transforma- +tions that modify a character’s shape but preserve +pronunciation and the overall word identity (read- +ing normalization). +The literature on Perso-Arabic script normal- +ization for languages we cover in this paper is +scarce. The most relevant work was carried out +by Ahmadi (2020) for Kurdish, who provides +a detailed analysis of orthographic issues pecu- +liar to Sorani Kurdish along with corresponding +open-source script normalization software used +in downstream NLP applications, such as neu- +ral machine translation (Ahmadi and Masoud, +2020). In the context of machine transliteration +and spell checking, Lehal and Saini (2014) in- +cluded language-agnostic minimal script normal- +ization as a preprocessing step in their open-source +n-gram-based transliterator from Perso-Arabic to +Brahmic scripts. Bhatti et al. (2014) introduced +a taxonomy of spelling errors for Sindhi, includ- +ing an analysis of mistakes due to visually confus- +able characters. Razak et al. (2018) provide a good +overview of confusable characters for Malay Jawi +orthography. +For other languages the regional +writing system ambiguities are sometimes men- +tioned in passing, but do not constitute the main +focus of work, as is the case with Punjabi Shah- +mukhi (Lehal and Saini, 2012) and Urdu (Humay- +oun et al., 2022). The specific Perso-Arabic script +ambiguities that abound in the online data are of- +ten not exhaustively documented, particularly in +work focused on multilingual modeling (N. C., +2022; Bapna et al., 2022). As one moves towards +lesser-resourced languages, such as Kashmiri and +Uyghur, the NLP literature provides no treatment +of script normalization issues and the only reli- +able sources of information are the proposal and +discussion documents from the Unicode Techni- +cal Committee (e.g., Bashir et al., 2006; Aazim +et al., 2009; Pournader, 2014). A forthcoming pa- +per by Doctor et al. (2022) covers the writing sys- +tem differences between these languages in more +Op. Type +FST +Language-dep. +Includes +NFC +N +no +− +Common Visual +Vc +no +N +Visual +V +yes +Vc +Reading +R +yes +− +Romanization +M +no +Vc +Transliteration +T +no +− +Table 1: Summary of script transformation operations. +detail than we can include in this short paper. +One area particularly relevant to this study is +the work by the Internet Corporation for Assigned +Names and Numbers (ICANN) towards develop- +ing a robust set of standards for representing vari- +ous Internet entities in Perso-Arabic script, such as +domain names in URLs. Their particular focus is +on variants, which are characters that are visually +confusable due to identical appearance but differ- +ent encoding, due to similarity in shape or due to +common alternate spellings (ICANN, 2011). In +addition, they developed the first proposal to sys- +tematize the available Perso-Arabic Unicode code +points along the regional lines (ICANN, 2015). +These studies are particularly important for cyber- +security (Hussain et al., 2016; Ginsberg and Yu, +2018; Ahmad and Erdodi, 2021), but also inform +this work. +This software library is, to the best our knowl- +edge, the first attempt to provide a principled ap- +proach to Perso-Arabic script normalization for +multiple languages, for downstream NLP applica- +tions and beyond. +3 +Design Methodology +The core components are implemented as individ- +ual FSTs that can be efficiently combined together +in a single pipeline (Mohri, 2009). +These are +shown in Table 1 and described below.3 +Unicode Normalization +For the Perso-Arabic +string encodings which yield visually identical +text, the Unicode standard provides procedures +that normalize text to a conventionalized normal +form, such as the well-known Normalization Form +C (NFC), so that visually identical words are +mapped to a conventionalized representative of +their equivalence class (Whistler, 2021). We im- +plemented the NFC standard as an FST, denoted +N in Table 1, that handles three broad types of +transformations: compositions, re-orderings and +3When referring to names of Unicode characters we low- +ercase them and omit the common prefix arabic (letter). + +FST +Letter +Variant (source) +Canonical +V∗ +l +⟨ڑ⟩ +reh + small high tah +rreh +Vn +l +⟨ک⟩ +kaf +keheh +Vf +l +⟨ی⟩ +alef maksura +farsi yeh +Vi +l +⟨ہ⟩ +heh +heh goal +Table 2: Example FST components of Vl for Urdu. +combinations thereof. +As an example of a first type, consider the alef +with madda above letter ⟨آ⟩ that can be composed +in two ways: as a single character (U+0622) or +by adjoining maddah above to alef ({ U+0627, +U+0653 }). The FST N rewrites the adjoined form +into its equivalent composed form. The second +type of transformation involves the canonical re- +ordering of the Arabic combining marks, for exam- +ple, the sequence of shadda (U+0651) followed by +kasra (U+0650) is reversed by N. More complex +transformations that combine both compositions +and re-orderings are possible. For example, the se- +quence { alef (U+0627), superscript alef (U+0670), +maddah above (U+0653) } normalizes to its equiv- +alent form { alef with madda above (U+0622), su- +perscript alef (U+0670) }. +Crucially, N is language-agnostic because the +NFC standard it implements does not define any +transformations that violate the writing system +rules of respective languages. +Visual Normalization +As mentioned in §2, +Johny et al. (2021) introduced the term visual nor- +malization in the context of Brahmic scripts to +denote visually-invariant transformations that fall +outside the scope of NFC. We adopt their defini- +tion for Perso-Arabic, implementing it as a sin- +gle language-dependent FST V, shown in Table 1, +which is constructed by FST composition: V = +N ◦ Vc ◦ Vl, where ◦ denotes the composition op- +eration (Mohri, 2009).4 +The first FST after NFC, denoted Vc, +is +language-agnostic, constructed from a small set of +normalizations for visually ambiguous sequences +found online that apply to all languages in our li- +brary. +For example, we map the two-character +sequence waw (U+0648) followed by damma +(U+064F) or small damma (U+0619) to u (U+06C7). +The second set of visually-invariant transforma- +tions, denoted Vl, is language-specific and addi- +tionally depends on the position within the word. +Four special cases are distinguished that are rep- +4See Johny et al. (2021) for details on FST composition +and other operations used in this kind of script normalization. +Op. Type +FST +# states +# arcs +# Kb +NFC +N +156 +1557 +28.10 +Roman. +M +32 546 +52 257 +1487.10 +Translit. +T +340 +518 +15.15 +Table 3: Language-agnostic FSTs over UTF-8 strings. +resented as FSTs: position-independent rewrites +(V∗ +l ), isolated-letter rewrites (Vi +l), rewrites in the +word-final position (Vf +l), and finally, rewrites in +“non-final” word positions, which include visually- +identical word-initial and word-medial rewrites +(Vn +l ). The FST Vl is composed as Vi +l ◦Vf +l ◦Vn +l ◦V∗ +l . +Some examples of these transformations for Urdu +orthography are shown in Table 2, where the vari- +ants shown in the third column are rewritten to +their canonical Urdu form in the fourth column. +Reading Normalization +This type of normaliza- +tion was introduced for Brahmic scripts by Gutkin +et al. (2022), who noted that regional orthographic +conventions or lack thereof, which oftentimes con- +flict with each other, benefit from normalization +to some accepted form. Whenever such normal- +ization preserves visual invariance, it falls under +the rubric of visual normalization, but other cases +belong to reading normalization, denoted R in Ta- +ble 1. Similar to visual normalization, R is com- +piled from language-specific context-dependent +rewrite rules. One example of such a rewrite is +a mapping from yeh ⟨ي⟩(U+064A) to farsi yeh ⟨ی⟩ +(U+06CC) in Kashmiri, Persian, Punjabi, Sorani +Kurdish and Urdu. For Malay, Sindhi and Uyghur, +the inverse transformation is implemented as man- +dated by the respective orthographies. +For efficiency reasons R is stored independently +of visual normalization V. At run-time, the read- +ing normalization is applied to an input string s +as s′ = (s ◦ V) ◦ R, which is more efficient than +s′ = s ◦ R′, where R′ = V ◦ R. +Romanization and Transliteration +We also +provide language-agnostic romanization (M) and +transliteration (T ) FSTs. The FST M converts +Perso-Arabic strings to their respective Latin rep- +resentation in Unicode and is defined as M = +N ◦ Vc ◦ Mc, where N and Vc were described +above, and Mc implements a one-to-one mapping +from 198 Perso-Arabic characters to their respec- +tive romanizations using our custom romanization +scheme derived from language-specific Library of +Congress rules (LC, 2022) and various ISO stan- +dards (ISO, 1984, 1993, 1999). For example, in + +Language Information +Visual Normalization (V) +Reading Normalization (R) +Code +Name +# states +# arcs +# Mb +# states +# arcs +# Mb +azb +South Azerbaijani +315 933 +635 647 +16.49 +21 +735 +0.012 +bal +Balochi +620 226 +1 244 472 +32.31 +24 +738 +0.013 +ckb +Kurdish (Sorani) +1 097 937 +2 199 732 +57.15 +39 +753 +0.013 +fa +Persian +940 436 +1 884 347 +48.96 +36 +750 +0.013 +ks +Kashmiri +1 772 494 +3 547 448 +92.21 +44 +794 +0.014 +ms +Malay +199 777 +403 373 +10.45 +21 +735 +0.012 +pa +Punjabi +2 050 154 +4 105 465 +106.69 +24 +738 +0.013 +ps +Pashto +291 564 +587 552 +15.23 +24 +738 +0.013 +sd +Sindhi +1 703 726 +3 403 283 +88.53 +34 +748 +0.013 +ug +Uyghur +1 255 054 +2 513 231 +65.31 +24 +738 +0.013 +ur +Urdu +2 071 139 +4 138 950 +107.65 +31 +745 +0.013 +Table 4: Summary of FSTs over UTF-8 strings for visual and reading normalization. +our scheme the Uyghur yu ⟨ۈ⟩(U+06C8) maps +to ⟨¨u⟩. +The transliteration FST T converts the +strings from Unicode Latin into Perso-Arabic. It +is smaller than M and is defined as T = M−1 +c . +Character-Language Mapping +The geography +and scope of Perso-Arabic script adaptations is +vast. To document the typology of the characters +we developed an easy-to-parse mapping between +the characters and the respective languages and/or +macroareas that relate to a group of languages +building on prior work by ICANN (2015). For ex- +ample, using this mapping it is easy to find that +the letter beh with small v below ⟨ࢠ⟩(U+08A0) is +part of the orthography of Wolof, a language of +Senegal (Ngom, 2010), while gaf with ring ⟨ڰ⟩ +(U+06B0) belongs to Saraiki language spoken in +Pakistan (Bashir and Conners, 2019). This map- +ping can be used to auto-generate the orthographic +inventories for lesser-resourced languages. +4 +Software Details and Validation +Our software library is implemented using Pynini, +a Python library for constructing finite-state gram- +mars and for performing operations on FSTs (Gor- +man, 2016; Gorman and Sproat, 2021). +Each +FST is compiled from the collections of individ- +ual context-dependent letter rewrite rules (Mohri +and Sproat, 1996) and is available in two versions: +over an alphabet of UTF-8 encoded bytes and +over the integer Unicode code points. The FSTs +are stored uncompressed in binary FST archives +(FARs) in OpenFst format (Allauzen et al., 2007). +The +summaries +of +language-agnostic +and +language-dependent FSTs over UTF-8 strings are +shown in Table 3 and Table 4, respectively. As +can be seen from the tables, the language-agnostic +and reading normalization FSTs are relatively un- +complicated and small in terms of number of +Lang. +s′ = s ◦ V +s′ = (s ◦ V) ◦ R +% tokens +% types +% tokens +% types +ckb +18.27 +25.84 +30.07 +41.26 +sd +17.32 +14.83 +21.74 +17.31 +ur +0.09 +1.16 +0.10 +1.23 +Table 5: Percentage of tokens and types changed. +states, arcs and the overall (uncompressed) size on +disk. The visual normalization FSTs are signifi- +cantly larger, which is explained by the number +of composition operations used in their construc- +tion (see §3). The reading normalization FSTs for +South Azerbaijani and Malay shown in Table 4 im- +plement the identity mapping. This is because we +could not find enough examples requiring reading- +style normalization in online data (see the Limita- +tions section for more details). +As an informal sanity check we validate the +prevalence of normalization on word-frequency +lists for Sorani Kurdish (ckb), Sindhi (sd) and +Uyghur (ug) from project Cr´ubad´an (Scannell, +2007). Table 5 shows the percentages of tokens +and types changed (s′ ̸= s) by visual normaliza- +tion on one hand and the combined visual and +reading normalization on the other. Urdu has the +fewest number of modifications compared to So- +rani Kurdish and Sindhi, most likely due to a more +regular orthography and stable input methods man- +ifest in the crawled data. Significantly more ex- +tensive analysis and experiments in statistical lan- +guage modeling and neural machine translation for +the languages covered in this paper are presented +in a forthcoming study (Doctor et al., 2022). +Example +The use of the library is demonstrated +by the following Python example that implements +a simple command-line utility for performing read- +ing normalization on a single string using Pynini +APIs. The program requires two FAR files that + +Lang. +Input +Output +Correct Output +balٽﯿﺋدﺖﯿﺋد +teh +ckbﺮڪﺷەﻟﺮﮑﺷەﻟ +keheh +faﻪﺴﺳﺆﻣﻪﺴﺳﻮﻣ +waw +ksﮏﺗۍﮬﮏﺘؠﮬ +kashmiri yeh +paﻲﺌﮐﯽﺌﮐ +farsi yeh +sdﻪﻫﻮﮘﮧﮨﻮﮘ +heh goal +ugیﺎﺳيﺎﺳ +yeh +urةرﻮﺻۃرﻮﺻ +teh marbuta goal +Table 6: Some examples of reading normalization. +store compiled visual and reading normalization +grammars, the upper-case BCP-47 language code +for retrieving the FST for a given language, and an +input string:5 +example.py +from absl import app +from absl import flags +from collections.abc import Iterable, Sequence +import pynini as pyn +flags.DEFINE_string("input", None, "Input string.") +flags.DEFINE_string("lang", None, "Language code.") +flags.DEFINE_string("reading_grm", None, "Reading FAR.") +flags.DEFINE_string("visual_grm", None, "Visual FAR.") +FLAGS = flags.FLAGS +def load_fst(grammar_path: str, lang: str) -> pyn.Fst: +"""Loads FST for specified grammar and language.""" +return pyn.Far(grammar_path)[lang] +def apply(text: str, fsts: Iterable[pyn.Fst]) -> str: +"""Applies sequence of FSTs on an input string.""" +try: +composed = pyn.escape(text) +for fst in fsts: +composed = (composed @ fst).optimize() +return pyn.shortestpath(composed).string() +except pyn.FstOpError as error: +raise ValueError(f"Error for string `{text}`") +def main(argv: Sequence[str]) -> None: +# ... initializing FLAGS +visual_fst = load_fst(FLAGS.visual_grm, FLAGS.lang) +reading_fst = load_fst(FLAGS.reading_grm, FLAGS.lang) +out = apply(FLAGS.input, [visual_fst, reading_fst]) +print(f"=> {out}") +if __name__ == "__main__": +app.run(main) +The visual and reading FSTs for a given language +are retrieved from the relevant FAR files using +load_fst function. The input string is first con- +verted to a linear FST. The visual and reading nor- +malization FSTs are then sequentially composed +with the input FST and a shortest path algorithm is +applied on the result, which is then converted from +a linear FST back to a Python string in apply func- +tion to yield the final normalized output. +Some examples of reading normalization pro- +5The infrastructure for compiling the Pynini grammars is +described in Johny et al. (2021). +duced using the example.py utility above for +some of the supported languages are shown in Ta- +ble 6. For each language, the input string in the +second column of the table is normalized to a +string shown in the third column. The final col- +umn shows the name of a particular letter in the +output string that replaced the original letter from +the input string, e.g., for Sorani Kurdish (ckb) +the following rewrite occurs: swash kaf (U+06AA) +→ keheh (U+06A9), while for Punjabi (pa), yeh +(U+064A) → farsi yeh (U+06CC). +5 +Conclusion and Future Work +We have presented a flexible FST-based software +package for low-level processing of orthographies +based on Perso-Arabic script. We described the +main components of the architecture consisting +of various script normalization operations, roman- +ization/transliteration, and character-language in- +dex. +We expect to increase the current lan- +guage coverage of eleven languages to further rel- +atively well-documented orthographies, but also +provide treatment for resource-scarce orthogra- +phies, such as the Ajami orthographies of Sub- +Saharan Africa (Mumin, 2014). +Limitations +When developing the visual and reading normal- +ization rules for the eleven languages described in +this paper we made use of publicly available on- +line data consisting of the respective Wikipedias, +Wikipron (Lee et al., 2020), Cr´ubad´an (Scannell, +2007) and parts of Common Crawl (Patel, 2020). +The latter corpus is particularly noisy and requires +non-trivial filtering (Kreutzer et al., 2022). 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The COL- +ING 2012 Organizing Committee. +Gurpreet Singh Lehal and Tejinder Singh Saini. 2014. +Sangam: A perso-Arabic to indic script machine +transliteration model. +In Proceedings of the 11th +International Conference on Natural Language Pro- +cessing, pages 232–239, Goa, India. NLP Associa- +tion of India. +Henrik Liljegren. 2018. +Supporting and sustaining +language vitality in Northern Pakistan. In Leanne +Hinton, Leena Huss, and Gerald Roche, editors, +The Routledge Handbook of Language Revitaliza- +tion, pages 427–437. Routledge. +Mehryar Mohri. 2009. Weighted automata algorithms. +In Manfred Droste, Werner Kuich, and Heiko Vogler, +editors, Handbook of Weighted Automata, Mono- +graphs in Theoretical Computer Science, pages 213– +254. Springer. +Mehryar Mohri and Richard Sproat. 1996. An efficient +compiler for weighted rewrite rules. +In 34th An- +nual Meeting of the Association for Computational +Linguistics, pages 231–238, Santa Cruz, California, +USA. Association for Computational Linguistics. +Meikal Mumin. 2014. +The Arabic script in Africa: +Understudied literacy. In Meikal Mumin and Kees +Versteegh, editors, The Arabic Script in Africa, vol- +ume 71 of Studies in Semitic Languages and Linguis- +tics, pages 41–76. Brill, Leiden, The Netherlands. +Gokul N. C. 2022. Unified NMT models for the In- +dian subcontinent, transcending script-barriers. In +Proceedings of the Third Workshop on Deep Learn- +ing for Low-Resource Natural Language Processing, +pages 227–236, Hybrid. Association for Computa- +tional Linguistics. +Fallou Ngom. 2010. Ajami scripts in the Senegalese +speech community. Journal of Arabic and Islamic +Studies, 10:1–23. +Jay M. Patel. 2020. Introduction to Common Crawl +datasets. In Getting Structured Data from the Inter- +net, pages 277–324. Springer. +Roozbeh Pournader. 2014. The right HEHs for Arabic +script orthographies of Sorani Kurdish and Uighur. +Technical Report L2/14-136, Unicode Consortium. +Sitti Munirah Abdul Razak, Muhamad Sadry Abu Se- +man, Wan Ali Wan Yusoff Wan Mamat, and Noor +Hasrul Nizan Mohammad Noor. 2018. +Translit- +eration engine for union catalogue of Malay +manuscripts in Malaysia: E-Jawi Version 3. +In +2018 International Conference on Information and +Communication Technology for the Muslim World +(ICT4M), pages 58–63. IEEE. +Kevin P. Scannell. 2007. The Cr´ubad´an Project: Cor- +pus building for under-resourced languages. +In +Building and Exploring Web Corpora (WAC3-2007): +Proceedings of the 3rd Web as Corpus Workshop, +volume 4, pages 5–15. Presses universitaires de Lou- +vain. http://crubadan.org/. +Unicode Consortium. 2021. Arabic. In The Unicode +Standard (Version 14.0.0), chapter 9.2, pages 373– +398. Unicode Consortium, Mountain View, CA. +Ken Whistler. 2021. +Unicode normalization forms. +Technical Report TR15-51, Unicode Consortium. +Version 14.0.0. + diff --git a/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/load_file.txt b/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..61e21f0a05eed88370237a1e1510e41941177067 --- /dev/null +++ b/-9FIT4oBgHgl3EQf9SvZ/content/tmp_files/load_file.txt @@ -0,0 +1,447 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf,len=446 +page_content='Preprint to appear in the Proceedings of the 7th Arabic Natural Language Processing Workshop (WANLP), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' EMNLP, Abu Dhabi, United Arab Emirates, December 7–11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Beyond Arabic: Software for Perso-Arabic Script Manipulation Alexander Gutkin† Cibu Johny† Raiomond Doctor‡∗ Brian Roark◦ Richard Sproat⊛ Google Research †United Kingdom ‡India United States ⊛Japan {agutkin,cibu,raiomond,roark,rws}@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='com Abstract This paper presents an open-source software library that provides a set of finite-state trans- ducer (FST) components and corresponding utilities for manipulating the writing sys- tems of languages that use the Perso-Arabic script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The operations include various lev- els of script normalization, including visual invariance-preserving operations that subsume and go beyond the standard Unicode normal- ization forms, as well as transformations that modify the visual appearance of characters in accordance with the regional orthographies for eleven contemporary languages from diverse language families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The library also provides simple FST-based romanization and transliter- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We additionally attempt to formalize the typology of Perso-Arabic characters by provid- ing one-to-many mappings from Unicode code points to the languages that use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' While our work focuses on the Arabic script diaspora rather than Arabic itself, this approach could be adopted for any language that uses the Ara- bic script, thus providing a unified framework for treating a script family used by close to a billion people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' 1 Introduction While originally developed for recording Arabic, the Perso-Arabic script has gradually become one of the most widely used modern scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Through- out history the script was adapted to record many languages from diverse language families, with scores of adaptations still active today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' This flexi- bility is partly due to the core features of the script itself which over the time evolved from a purely consonantal script to include a productive system of diacritics for representing long vowels and op- tional marking of short vowels and phonologi- cal processes such as gemination (Bauer, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Kurzon, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Consequently, many languages productively evolved their own adaptation of the ∗ On contract from Optimum Solutions, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Perso-Arabic script to better suit their phonology by not only augmenting the set of diacritics but also introducing new consonant shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' This paper presents an open-source software li- brary designed to deal with the ambiguities and inconsistencies that result from representing var- ious regional Perso-Arabic adaptations in digital media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Some of these issues are due to the Uni- code standard itself, where a Perso-Arabic char- acter can often be represented in more than one way (Unicode Consortium, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Others are due to the lack or inadequacies of input methods and the instability of modern orthographies for the lan- guages in question (Aazim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Liljegren, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Such issues percolate through the data available online, such as Wikipedia and Common Crawl (Patel, 2020), negatively impacting the qual- ity of NLP models built with such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The script normalization software described below goes be- yond the standard language-agnostic Unicode ap- proach for Perso-Arabic to help alleviate some of these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The library design is inspired by and consis- tent with prior work by Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2021), in- troduced in §2, who provided a suite of finite- state grammars for various normalization and (re- versible) romanization operations for the Brah- mic family of scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='1 While the Perso-Arabic script and the respective set of regional orthogra- phies we support – Balochi, Kashmiri, Kurdish (Sorani), Malay (Jawi), Pashto, Persian, Punjabi (Shahmukhi), Sindhi, South Azerbaijani, Urdu and Uyghur – is significantly different from those Brahmic scripts, we pursue a similar finite-state in- terpretation,2 as described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Implementation details and simple validation are provided in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='com/google-research/nisaba 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='com/google-research/nisaba/ tree/main/nisaba/scripts/abjad alphabet 2 Related Work The approach we take in this paper follows in spirit the work of Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2021) and Gutkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2022), who developed a finite-state script normalization framework for Brahmic scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We adopt their taxonomy and terminology of low- level script normalization operations, which con- sist of three types: Unicode-endorsed schemes, such as NFC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' further visually-invariant transfor- mations (visual normalization);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' and transforma- tions that modify a character’s shape but preserve pronunciation and the overall word identity (read- ing normalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The literature on Perso-Arabic script normal- ization for languages we cover in this paper is scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The most relevant work was carried out by Ahmadi (2020) for Kurdish, who provides a detailed analysis of orthographic issues pecu- liar to Sorani Kurdish along with corresponding open-source script normalization software used in downstream NLP applications, such as neu- ral machine translation (Ahmadi and Masoud, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' In the context of machine transliteration and spell checking, Lehal and Saini (2014) in- cluded language-agnostic minimal script normal- ization as a preprocessing step in their open-source n-gram-based transliterator from Perso-Arabic to Brahmic scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Bhatti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2014) introduced a taxonomy of spelling errors for Sindhi, includ- ing an analysis of mistakes due to visually confus- able characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Razak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2018) provide a good overview of confusable characters for Malay Jawi orthography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For other languages the regional writing system ambiguities are sometimes men- tioned in passing, but do not constitute the main focus of work, as is the case with Punjabi Shah- mukhi (Lehal and Saini, 2012) and Urdu (Humay- oun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The specific Perso-Arabic script ambiguities that abound in the online data are of- ten not exhaustively documented, particularly in work focused on multilingual modeling (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Bapna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' As one moves towards lesser-resourced languages, such as Kashmiri and Uyghur, the NLP literature provides no treatment of script normalization issues and the only reli- able sources of information are the proposal and discussion documents from the Unicode Techni- cal Committee (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', Bashir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Aazim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Pournader, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' A forthcoming pa- per by Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2022) covers the writing sys- tem differences between these languages in more Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Type FST Language-dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Includes NFC N no − Common Visual Vc no N Visual V yes Vc Reading R yes − Romanization M no Vc Transliteration T no − Table 1: Summary of script transformation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' detail than we can include in this short paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' One area particularly relevant to this study is the work by the Internet Corporation for Assigned Names and Numbers (ICANN) towards develop- ing a robust set of standards for representing vari- ous Internet entities in Perso-Arabic script, such as domain names in URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Their particular focus is on variants, which are characters that are visually confusable due to identical appearance but differ- ent encoding, due to similarity in shape or due to common alternate spellings (ICANN, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' In addition, they developed the first proposal to sys- tematize the available Perso-Arabic Unicode code points along the regional lines (ICANN, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' These studies are particularly important for cyber- security (Hussain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Ginsberg and Yu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Ahmad and Erdodi, 2021), but also inform this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' This software library is, to the best our knowl- edge, the first attempt to provide a principled ap- proach to Perso-Arabic script normalization for multiple languages, for downstream NLP applica- tions and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' 3 Design Methodology The core components are implemented as individ- ual FSTs that can be efficiently combined together in a single pipeline (Mohri, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' These are shown in Table 1 and described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='3 Unicode Normalization For the Perso-Arabic string encodings which yield visually identical text, the Unicode standard provides procedures that normalize text to a conventionalized normal form, such as the well-known Normalization Form C (NFC), so that visually identical words are mapped to a conventionalized representative of their equivalence class (Whistler, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We im- plemented the NFC standard as an FST, denoted N in Table 1, that handles three broad types of transformations: compositions, re-orderings and 3When referring to names of Unicode characters we low- ercase them and omit the common prefix arabic (letter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' FST Letter Variant (source) Canonical V∗ l ⟨ڑ⟩ reh + small high tah rreh Vn l ⟨ک⟩ kaf keheh Vf l ⟨ی⟩ alef maksura farsi yeh Vi l ⟨ہ⟩ heh heh goal Table 2: Example FST components of Vl for Urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' combinations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' As an example of a first type, consider the alef with madda above letter ⟨آ⟩ that can be composed in two ways: as a single character (U+0622) or by adjoining maddah above to alef ({ U+0627, U+0653 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The FST N rewrites the adjoined form into its equivalent composed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The second type of transformation involves the canonical re- ordering of the Arabic combining marks, for exam- ple, the sequence of shadda (U+0651) followed by kasra (U+0650) is reversed by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' More complex transformations that combine both compositions and re-orderings are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For example, the se- quence { alef (U+0627), superscript alef (U+0670), maddah above (U+0653) } normalizes to its equiv- alent form { alef with madda above (U+0622), su- perscript alef (U+0670) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Crucially, N is language-agnostic because the NFC standard it implements does not define any transformations that violate the writing system rules of respective languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Visual Normalization As mentioned in §2, Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2021) introduced the term visual nor- malization in the context of Brahmic scripts to denote visually-invariant transformations that fall outside the scope of NFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We adopt their defini- tion for Perso-Arabic, implementing it as a sin- gle language-dependent FST V, shown in Table 1, which is constructed by FST composition: V = N ◦ Vc ◦ Vl, where ◦ denotes the composition op- eration (Mohri, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='4 The first FST after NFC, denoted Vc, is language-agnostic, constructed from a small set of normalizations for visually ambiguous sequences found online that apply to all languages in our li- brary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For example, we map the two-character sequence waw (U+0648) followed by damma (U+064F) or small damma (U+0619) to u (U+06C7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The second set of visually-invariant transforma- tions, denoted Vl, is language-specific and addi- tionally depends on the position within the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Four special cases are distinguished that are rep- 4See Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2021) for details on FST composition and other operations used in this kind of script normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Type FST # states # arcs # Kb NFC N 156 1557 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='10 Roman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' M 32 546 52 257 1487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='10 Translit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' T 340 518 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='15 Table 3: Language-agnostic FSTs over UTF-8 strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' resented as FSTs: position-independent rewrites (V∗ l ), isolated-letter rewrites (Vi l), rewrites in the word-final position (Vf l), and finally, rewrites in “non-final” word positions, which include visually- identical word-initial and word-medial rewrites (Vn l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The FST Vl is composed as Vi l ◦Vf l ◦Vn l ◦V∗ l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Some examples of these transformations for Urdu orthography are shown in Table 2, where the vari- ants shown in the third column are rewritten to their canonical Urdu form in the fourth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Reading Normalization This type of normaliza- tion was introduced for Brahmic scripts by Gutkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2022), who noted that regional orthographic conventions or lack thereof, which oftentimes con- flict with each other, benefit from normalization to some accepted form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Whenever such normal- ization preserves visual invariance, it falls under the rubric of visual normalization, but other cases belong to reading normalization, denoted R in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Similar to visual normalization, R is com- piled from language-specific context-dependent rewrite rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' One example of such a rewrite is a mapping from yeh ⟨ي⟩(U+064A) to farsi yeh ⟨ی⟩ (U+06CC) in Kashmiri, Persian, Punjabi, Sorani Kurdish and Urdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For Malay, Sindhi and Uyghur, the inverse transformation is implemented as man- dated by the respective orthographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For efficiency reasons R is stored independently of visual normalization V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' At run-time, the read- ing normalization is applied to an input string s as s′ = (s ◦ V) ◦ R, which is more efficient than s′ = s ◦ R′, where R′ = V ◦ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Romanization and Transliteration We also provide language-agnostic romanization (M) and transliteration (T ) FSTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The FST M converts Perso-Arabic strings to their respective Latin rep- resentation in Unicode and is defined as M = N ◦ Vc ◦ Mc, where N and Vc were described above, and Mc implements a one-to-one mapping from 198 Perso-Arabic characters to their respec- tive romanizations using our custom romanization scheme derived from language-specific Library of Congress rules (LC, 2022) and various ISO stan- dards (ISO, 1984, 1993, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For example, in Language Information Visual Normalization (V) Reading Normalization (R) Code Name # states # arcs # Mb # states # arcs # Mb azb South Azerbaijani 315 933 635 647 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='49 21 735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='012 bal Balochi 620 226 1 244 472 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='31 24 738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 ckb Kurdish (Sorani) 1 097 937 2 199 732 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='15 39 753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 fa Persian 940 436 1 884 347 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='96 36 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 ks Kashmiri 1 772 494 3 547 448 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='21 44 794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='014 ms Malay 199 777 403 373 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='45 21 735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='012 pa Punjabi 2 050 154 4 105 465 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='69 24 738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 ps Pashto 291 564 587 552 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='23 24 738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 sd Sindhi 1 703 726 3 403 283 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='53 34 748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 ug Uyghur 1 255 054 2 513 231 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='31 24 738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 ur Urdu 2 071 139 4 138 950 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='65 31 745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='013 Table 4: Summary of FSTs over UTF-8 strings for visual and reading normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' our scheme the Uyghur yu ⟨ۈ⟩(U+06C8) maps to ⟨¨u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The transliteration FST T converts the strings from Unicode Latin into Perso-Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' It is smaller than M and is defined as T = M−1 c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Character-Language Mapping The geography and scope of Perso-Arabic script adaptations is vast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' To document the typology of the characters we developed an easy-to-parse mapping between the characters and the respective languages and/or macroareas that relate to a group of languages building on prior work by ICANN (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For ex- ample, using this mapping it is easy to find that the letter beh with small v below ⟨ࢠ⟩(U+08A0) is part of the orthography of Wolof, a language of Senegal (Ngom, 2010), while gaf with ring ⟨ڰ⟩ (U+06B0) belongs to Saraiki language spoken in Pakistan (Bashir and Conners, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' This map- ping can be used to auto-generate the orthographic inventories for lesser-resourced languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' 4 Software Details and Validation Our software library is implemented using Pynini, a Python library for constructing finite-state gram- mars and for performing operations on FSTs (Gor- man, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Gorman and Sproat, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Each FST is compiled from the collections of individ- ual context-dependent letter rewrite rules (Mohri and Sproat, 1996) and is available in two versions: over an alphabet of UTF-8 encoded bytes and over the integer Unicode code points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The FSTs are stored uncompressed in binary FST archives (FARs) in OpenFst format (Allauzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The summaries of language-agnostic and language-dependent FSTs over UTF-8 strings are shown in Table 3 and Table 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' As can be seen from the tables, the language-agnostic and reading normalization FSTs are relatively un- complicated and small in terms of number of Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' s′ = s ◦ V s′ = (s ◦ V) ◦ R % tokens % types % tokens % types ckb 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='27 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='84 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='07 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='26 sd 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='32 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='83 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='74 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='31 ur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='23 Table 5: Percentage of tokens and types changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' states, arcs and the overall (uncompressed) size on disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The visual normalization FSTs are signifi- cantly larger, which is explained by the number of composition operations used in their construc- tion (see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The reading normalization FSTs for South Azerbaijani and Malay shown in Table 4 im- plement the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' This is because we could not find enough examples requiring reading- style normalization in online data (see the Limita- tions section for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' As an informal sanity check we validate the prevalence of normalization on word-frequency lists for Sorani Kurdish (ckb), Sindhi (sd) and Uyghur (ug) from project Cr´ubad´an (Scannell, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Table 5 shows the percentages of tokens and types changed (s′ ̸= s) by visual normaliza- tion on one hand and the combined visual and reading normalization on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Urdu has the fewest number of modifications compared to So- rani Kurdish and Sindhi, most likely due to a more regular orthography and stable input methods man- ifest in the crawled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Significantly more ex- tensive analysis and experiments in statistical lan- guage modeling and neural machine translation for the languages covered in this paper are presented in a forthcoming study (Doctor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Example The use of the library is demonstrated by the following Python example that implements a simple command-line utility for performing read- ing normalization on a single string using Pynini APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The program requires two FAR files that Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Input Output Correct Output balٽﯿﺋدﺖﯿﺋد teh ckbﺮڪﺷەﻟﺮﮑﺷەﻟ keheh faﻪﺴﺳﺆﻣﻪﺴﺳﻮﻣ waw ksﮏﺗۍﮬﮏﺘؠﮬ kashmiri yeh paﻲﺌﮐﯽﺌﮐ farsi yeh sdﻪﻫﻮﮘﮧﮨﻮﮘ heh goal ugیﺎﺳيﺎﺳ yeh urةرﻮﺻۃرﻮﺻ teh marbuta goal Table 6: Some examples of reading normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' store compiled visual and reading normalization grammars, the upper-case BCP-47 language code for retrieving the FST for a given language, and an input string:5 example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='py from absl import app from absl import flags from collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='abc import Iterable, Sequence import pynini as pyn flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='DEFINE_string("input", None, "Input string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='") flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='DEFINE_string("lang", None, "Language code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='") flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='DEFINE_string("reading_grm", None, "Reading FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='") flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='DEFINE_string("visual_grm", None, "Visual FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='") FLAGS = flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='FLAGS def load_fst(grammar_path: str, lang: str) -> pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='Fst: """Loads FST for specified grammar and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='""" return pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='Far(grammar_path)[lang] def apply(text: str, fsts: Iterable[pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='Fst]) -> str: """Applies sequence of FSTs on an input string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='""" try: composed = pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='escape(text) for fst in fsts: composed = (composed @ fst).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='optimize() return pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='shortestpath(composed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='string() except pyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='FstOpError as error: raise ValueError(f"Error for string `{text}`") def main(argv: Sequence[str]) -> None: # .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' initializing FLAGS visual_fst = load_fst(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='visual_grm, FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='lang) reading_fst = load_fst(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='reading_grm, FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='lang) out = apply(FLAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='input, [visual_fst, reading_fst]) print(f"=> {out}") if __name__ == "__main__": app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='run(main) The visual and reading FSTs for a given language are retrieved from the relevant FAR files using load_fst function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The input string is first con- verted to a linear FST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The visual and reading nor- malization FSTs are then sequentially composed with the input FST and a shortest path algorithm is applied on the result, which is then converted from a linear FST back to a Python string in apply func- tion to yield the final normalized output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Some examples of reading normalization pro- 5The infrastructure for compiling the Pynini grammars is described in Johny et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' duced using the example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='py utility above for some of the supported languages are shown in Ta- ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' For each language, the input string in the second column of the table is normalized to a string shown in the third column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The final col- umn shows the name of a particular letter in the output string that replaced the original letter from the input string, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', for Sorani Kurdish (ckb) the following rewrite occurs: swash kaf (U+06AA) → keheh (U+06A9), while for Punjabi (pa), yeh (U+064A) → farsi yeh (U+06CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' 5 Conclusion and Future Work We have presented a flexible FST-based software package for low-level processing of orthographies based on Perso-Arabic script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We described the main components of the architecture consisting of various script normalization operations, roman- ization/transliteration, and character-language in- dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' We expect to increase the current lan- guage coverage of eleven languages to further rel- atively well-documented orthographies, but also provide treatment for resource-scarce orthogra- phies, such as the Ajami orthographies of Sub- Saharan Africa (Mumin, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Limitations When developing the visual and reading normal- ization rules for the eleven languages described in this paper we made use of publicly available on- line data consisting of the respective Wikipedias, Wikipron (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2020), Cr´ubad´an (Scannell, 2007) and parts of Common Crawl (Patel, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' The latter corpus is particularly noisy and requires non-trivial filtering (Kreutzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Fur- thermore, many Wikipedia and Common Crawl documents contain code-switched text in several languages that are recorded in Perso-Arabic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Ro- bust language identification (LID) is required to distinguish between tokens in such sentences (for example, Kashmiri vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Pashto or Balochi) in or- der not to confuse between the respective orthogra- phies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FIT4oBgHgl3EQf9SvZ/content/2301.11406v1.pdf'} +page_content=' Since we did not have access to robust LID models for the languages under study, for lesser- resourced languages such as Kashmiri, Malay in Jawi orthography, South Azerbaijani and Uyghur, it is likely that some of the words we used as exam- ples requiring normalization may have been mis- classified resulting in normalizations that should not be there.' metadata={'source': 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R. China +2 School of Remote Sensing and Information Engineering, Wuhan University, P. R. China +{JingtaoLi, wangxinyu, whu_zhaohw, wangshaoyu, zhongyanfei}@whu.edu.cn + + +Abstract +Anomaly segmentation in high spatial resolution (HSR) re- +mote sensing imagery is aimed at segmenting anomaly pat- +terns of the earth deviating from normal patterns, which plays +an important role in various Earth vision applications. How- +ever, it is a challenging task due to the complex distribution +and the irregular shapes of objects, and the lack of abnormal +samples. To tackle these problems, an anomaly segmentation +model based on pixel descriptors (ASD) is proposed for +anomaly segmentation in HSR imagery. Specifically, deep +one-class classification is introduced for anomaly segmenta- +tion in the feature space with discriminative pixel descriptors. +The ASD model incorporates the data argument for generat- +ing virtual abnormal samples, which can force the pixel de- +scriptors to be compact for normal data and meanwhile to be +diverse to avoid the model collapse problems when only pos- +itive samples participated in the training. In addition, the +ASD introduced a multi-level and multi-scale feature extrac- +tion strategy for learning the low-level and semantic infor- +mation to make the pixel descriptors feature-rich. The pro- +posed ASD model was validated using four HSR datasets and +compared with the recent state-of-the-art models, showing its +potential value in Earth vision applications. +1. Introduction +Anomaly segmentation is aimed at segmenting the anomaly +patterns which deviate from the normal patterns (Pimentel +et al. 2014; Pang et al. 2021). Due to the lack of abnormal +samples, anomaly segmentation is a challenging task, but +plays an important role in many computer vision applica- +tions, including medical analysis (Fernando et al. 2021), in- +dustrial defect detection (Bergmann et al. 2019), video sur- +veillance (Liu, Li, and Poczos 2018), and environmental +monitoring (Miau and Hung 2020). + Anomaly segmentation in high spatial resolution (HSR) +remote sensing images (e.g., Figure 1) is a powerful tool for +environmental monitoring (Miau and Hung 2020; Wang et +al. 2019). Despite this, few related works have focused on +anomaly segmentation in HSR imagery because of the + +*Corresponding author + +unique characteristics when compared to the industrial and +medical images used in most anomaly segmentation tasks, +which have a regular structure. The objects in HSR images +typically have a more complex spatial distribution and large +radiation differences within the same class. Furthermore, +since HSR images can be captured in different angles and +heights, the objects always have multiple scales and show +rotation invariance. These characteristics make anomaly +segmentation for HSR images a challenging task. + The mainstream anomaly segmentation models detect +anomalies in the image space, where the anomaly score is +computed based on the image pixel values. Typical exam- +ples are the autoencoder (AE)-based models (Zavrtanik, +Kristan, and Skoˇcaj 2021; Gong et al. 2019) and the gener- +ative adversarial network (GAN)-based models (Ngoetal. +2019; Zenatietal. 2018b). AE-based models assume that +normal samples can be reconstructed more easily than the +anomalous ones and the reconstruction error indicates the +anomaly segmentation score (Pang et al. 2021). However, +the low-level reconstruction error has been shown to focus +on the pixel-wise error, resulting in abnormal samples also +being reconstructed, especially when the normal distribution +is complex. (Fei et al. 2020; Gong et al. 2019; Zong et al. +Code is available at https://github.com/Jingtao-Li-CVer/ASD. + + + +Figure 1: Anomaly segmentation example for HSR re- +mote sensing images using proposed model. In the forest +scene, the common forest pattern is considered as normal, +and the abnormal objects such as diseased trees (the first +row) and the house in the forest (the second row) are iden- +tified as anomalies. +Anomaly image +Normal images +Anomaly label +Anomaly map +… + +2018). GAN-based models detect anomalies from the gen- +eration performance (Akcay, Atapour-Abarghouei, and +Breckon 2018; Ngo et al. 2019; Xia et al. 2022), where the +superior capability in generating image data also empowers +the detection of abnormal samples (Pang et al. 2021). In +spite of this, the complex distribution of HSR images can +make the generator generate data instances that are out of +the manifold of normal instances (Pang et al. 2021). + Differing from the AE-based and GAN-based models, +deep one-class classification (OCC)-based methods detect +the anomalies in the feature space (Shi, Yang, and Qi 2021; +Lei et al. 2021; Zhao et al. 2022; Li et al. 2022), where the +anomaly score is computed based on the extracted image de- +scriptors. These methods aim to learn discriminative de- +scriptors in the training stage and compute the anomaly +score in the feature space using a measurement such as the +Mahalanobis or Euclidean distance (Reiss et al. 2021; Ruff +et al. 2018; Shi, Yang, and Qi 2021). Because deep OCC- +based methods focus on semantic features rather than low- +level pixel errors, it is more suitable to deal with the anom- +aly segmentation task in HSR imagery which has complex +distribution. However, two barriers exist when applying ex- +isting methods directly. (i) Due to the lack of abnormal sam- +ples, the model training only uses normal samples and is op- +timized to be compact (Ruff et al. 2018; Chalapathy, Menon, +and Chawla 2018), which can easily result in the model col- +lapse problem (Reiss et al. 2021). (ii) The anomalies in HSR +imagery have rich low-level (e.g., texture) and high-level +(e.g., semantic) features, which are both important for the +anomaly segmentation task and real application. Although +the current deep OCC models can capture useful semantic +features, they perform suboptimally than models detecting +in the image space for samples with regular structures (Li et +al. 2021), because low-level features are mostly forgotten in +feature space. + In this paper, we tackle the two problems for the anomaly +segmentation task using HSR images. A novel anomaly seg- +mentation model based on pixel descriptors (ASD) is pro- +posed. (i) In addition to considering the compact property of +the obtained descriptors, the ASD model encourages de- +scriptors to be diverse by increasing the descriptor distance +between the original image and the transformed image with +the use of data augmentation techniques. The transformed +descriptors act as anomalies, to some extent, which en- +hances the anomaly detection ability and prevents simulta- +neous model collapse. (ii) To make the descriptor feature- +rich, a descriptor at different scales is fused for each pixel, +and an auxiliary reconstruction head is designed to force the +descriptor to remember the low-level features. Compact, di- +verse, and feature-rich property optimizes the model to- +gether from the perspective of the feature distance and fea- +ture quantity. ASD sets the first baseline for the anomaly +segmentation task in HSR imagery. + The ASD model was validated on four HSR datasets: the +DeepGlobe land-cover segmentation dataset, the Agricul- +ture-Vision agriculture pattern segmentation dataset, the +Landslide4Sense landslide detection dataset, and the forest +anomaly detection dataset (FAS, made by ourselves). The +ASD model showed an obvious superiority over the recent +state-of-the-art anomaly segmentation models (with an area +under the curve (AUC) improvement of 5–10 points in most +cases). The results obtained on the Landslide4Sense and +FAS datasets confirmed the great application potential of the +ASD model in disaster detection and forest monitoring. +2. Related Work +AE-based models are always composed of an encoding and +decoding network, with the aim being to reconstruct the +original input data (Pimentel et al. 2014). Hawkins et al. +(2022) first introduced the AE into the anomaly detection +field, where the features learned in the latent space can be +used to distinguish normal and anomalous data. The recon- +struction error is considered as the anomaly degree and the +mean square error (MSE) is adopted as the loss function in +most studies (Pang et al. 2021). To promote the performance, +Pathak et al. (2016) blanked the input image randomly and +forced the model to reconstruct the damaged area. Similarly, +the ARNet model was proposed, which erases some input +attributes and reformulates the problem as a restoration task +(Fei et al. 2020). Recently, Zavrtanik et al. (2021) cast the +reconstruction problem as an inpainting problem and recon- +structed the image from partial inpaintings. However, the +extracted low-level features can be shared by both normal +and anomalous data (Fei et al. 2020) when dealing with +complex HSR images. + +GAN-based models aim to generate the image rather than +reconstruct it. As one of the early GAN-based models, the +AnoGAN model assumes that the learned latent space can +represent normal samples well, but not the anomalous sam- +ples (Schlegl et al. 2017). Given a test image, the difference +between the regenerated image obtained using the searched +latent feature and the test image is considered as the anom- +aly degree. The famous GANomaly model improved the +generator architecture from a decoder to an encoder-decoder +encoder design and used high-level features to assist com- +puting the anomaly score (Akcay, Atapour-Abarghouei, and +Breckon 2018). GAN-based models have demonstrated su- +perior capabilities in generating image data, which also em- +powers the detection of abnormal samples (Pang et al. 2021). +In spite of this, the complex distribution of HSR images can +make the generator generate data instances that are out of +the manifold of normal instances (Pang et al. 2021). + + +One-class classification models are also used in some +anomaly segmentation works (Pang et al. 2021). One of their +greatest advantages over the AE-based and GAN-based +models is that the OCC models detect anomalies in the fea- +ture space with high-level semantic information. They first +divide an image into many patches and then learn the corre- +sponding representations. The anomaly score is computed in +the feature space using a measurement such as the Ma- +halanobis or Euclidean distance (Reiss et al. 2021; Ruff et +al. 2018; Shi, Yang, and Qi 2021). Most OCC models are +based on the principle of one-class support vector machine +(OCSVM) (Sch¨olkopf et al. 1999; Andrews, Morton, and +Griffin 2016) or support vector data description (SVDD) +(Tax and Duin 1999; Chalapathy, Menon, and Chawla 2018; +Ruff et al.2018). However, these models mainly consider +the compact property of the obtained one class features, re- +sulting in the model collapse problem (Reiss et al. 2021), +and they lack consideration of the low-level structural fea- +tures. +3. Methodology +Overview. This section describes the core principles of the +proposed ASD model. The overall workflow of the ASD +model is shown in Section 3.1, which includes two steps: +descriptor extracting and anomaly score computation. To +extract the ideal descriptors, descriptor learning is the most +important part and is described detailed in Section 3.2. The +computation method of the anomaly score is given in Sec- +tion 3.3. +3.1. Overall Workflow of The ASD Model +Given an HSR image 𝑿 with size 𝐻 × 𝑊 × 𝐵, where 𝐻, 𝑊, +and 𝐵 are the height, width and bands of the image, the +anomaly segmentation task can be viewed as a mapping +function 𝑓 from the 𝑿 to the anomaly map 𝑨 with size +𝐻 × 𝑊. Each pixel in the anomaly map is in the range [0,1]. +Generally speaking, the higher the value in the anomaly map, +the higher the anomaly degree. + The ASD model separates the function 𝑓 into two steps +and the overall workflow is shown in Figure 2. The first step +𝑓1 extracts the dense descriptor cube 𝑫 for each image pixel, +which is the core part and also the training focus in the ASD +model. The descriptors are expected to contain important +visual characteristics for the anomaly segmentation task. To +incorporate the pixel context and obtain fine pixel corre- +spondence, the patch-based paradigm is chosen to compute +the descriptor 𝐹 for the center pixel 𝑥. In this step, the +𝐻 × 𝑊 patches form the input samples and a descriptor +cube 𝑫 with size 𝐻 × 𝑊 × 𝐿 is output, where 𝐿 is the de- +scriptor length. + The second step 𝑓2 outputs the anomaly map based on the +trained descriptor encoders in the first step. Specifically, the +trained descriptors of the training samples are modeled as a +multivariate Gaussian Distribution (MGD) (Guimaraes et al. +2018) by the Gaussian Density Estimate (GDE). For the test +descriptor, its Mahalanobis distance from the MGD is used +to measure the anomaly score. The formal mapping of 𝑓, 𝑓1, +and 𝑓2 is shown in Eqs. (1-3). + +𝑓: 𝑿 → 𝑨 +(1) +𝑓1 ∶ 𝑿 → 𝑫 +(2) +𝑓2 ∶ 𝑫 → 𝑨 +(3) +3.2. Ideal Descriptors Learning +The descriptors obtained in the first step (as mentioned in +Section 3.1) are expected to contain important visual char- +acteristics for the anomaly segmentation task. To achieve +this aim, ideal descriptors are optimized using three condi- +tions from the characteristics of the anomaly segmentation +task and HSR images. +Compact. One of the characteristics of anomaly segmen- +tation is that only normal samples are used in the training +stage. In other words, all the training samples are of the same +class, which naturally results in compact visual descriptors +in the feature space. This compactness is also a useful su- +pervised signal for the anomaly segmentation task. +To keep 𝑫 compact, an enclosing hypersphere around all +the pixel descriptors is constructed, which is motivated by +the deep SVDD method (Ruff et al. 2018). We let 𝑅 be the +hypersphere radius and 𝐶 be the center. The 𝐿1 loss (Eq. (4)) +aims to minimize the hypersphere radius and the distance +from the obtained pixel descriptors to the center 𝐶, where +the parameter 𝜆 controls the trade-off between the size of +the hypersphere and the number of surrounded descriptors. +The maximum distance between 𝐶 and 𝐹 in 𝑫 is chosen to + + +Figure 2: The overall workflow of the ASD model, which +includes two steps. In the first step, the ASD model ex- +tracts a descriptor for each pixel with the descriptor ex- +tractor. In the second step, the descriptors for normal +scenes are modeled as the Gaussian distribution, and the +Mahalanobis distance between the test descriptor and the +modeled distribution is considered to measure the anom- +aly score. +Normal descriptors +GDE +Test image +Test descriptors +Gaussian +distribution +Anomaly map +Descriptor +extractor +Mahalanobis +distance + +compute the radius 𝑅. Compared to using the mean value, +this setting helps the model focus on special normal samples, +rather than just considering them as noise. + +𝐿1(𝑫) = 𝑅2 + 𝜆 mean{0, max{‖𝐹 − 𝐶‖2 − 𝑅2 | 𝐹 ∈ 𝑫}} (4) + +Diverse. Compactness is the first basic condition. How- +ever, the model can easily collapse if only a compactness +constraint used. In other words, the model would map all the +input samples into the same point. This “cheating” makes +the model lose the anomaly detection ability. To deal with +this problem, the diverse condition is necessary, which +stresses that a different pixel 𝑥 obtains different values of 𝐹. +The key consideration to keeping the descriptors diverse +is to keep the model sensitive to the input sample change. +Considering the fact that training images are always anom- +aly free and real negative samples are difficult to obtain, data +augmentation techniques, such as the channel shuffle oper- +ation, are used to generate negative samples. Formally, the +augmentation operation set 𝑆𝑎 = {𝐴1, 𝐴2, … , 𝐴𝑛} contains 𝑛 +kinds of different augmentation operations. For the original +image 𝑿, the obtained image descriptor cube 𝑫 can be seen +as a positive one. Then, after applying the operations from +𝑆𝑎 on 𝑿 in turn, 𝑿𝑇 can be obtained and the corresponding +cube 𝑫𝑇 is considered to be a negative sample. Eqs. (5-6) +formally express the above process. + +𝑿𝑇 = 𝐴𝑛(… (𝐴2(𝐴1(𝑿𝑇))) +(5) +𝑫 = 𝑓1(𝑿), 𝑫𝑇= 𝑓1(𝑿𝑇) +(6) + + +Both 𝑫 and 𝑫𝑇 have the same shape 𝐻 × 𝑊 × 𝐿. The diver- +sity loss is defined as the average pixel descriptor difference +between 𝑫 and 𝑫𝑇, as shown in Eq. (7). With the 𝐿2 loss, +the model is encouraged to increase the sensitivity to the in- +put difference. + +𝐿2(𝑫, 𝑫𝑇) = 1/{ +1 +𝐻 × 𝑊 ∑ ∑‖𝑫𝑖𝑗 − 𝑫𝑖𝑗 +𝑇 ‖ +2 +𝑊 +𝑗=1 +𝐻 +𝑖=1 +} +(7) +Some technologies have the potential to deal with the +model collapse, such as reducing the model bias (Ruff et al. +2018) or designing early-stopping strategies (Reiss et al. +2021). However, the proposed 𝐿2 loss does not need early- +stopping or change of the model architecture. +Feature-rich. The compact and diverse conditions meas- +ure the descriptors from the perspective of distance. The fea- +ture-rich condition measures the descriptors from the per- +spective of the amount of representative information. In the +ASD model, multi-scale and multi-level features are consid- +ered in particular. +The multi-scale characteristic is an import difference for +HSR images, compared to natural images. For example, +large-scale information is important for rivers and small- +scale information is important for urban buildings. Even for +the same scene, the images are always taken at different +heights, which poses a challenge for the model ability to +catch the multi-scale information. +To enhance the model ability to deal with multi-scale in- +formation, the ASD model uses a resize operation set 𝑆𝑠 = +{𝑈1, 𝑈2, … , 𝑈𝑚} for the input patches. Given an image 𝑿, it +is resized using each operation 𝑈𝑖 in 𝑆𝑠, and obtains 𝑚 dif- +ferent-scale versions 𝑿1, 𝑿2, … , 𝑿𝑚 of the same image. +Then, for each center pixel 𝑥, 𝑚 patches are cropped with +size 𝑃 × 𝑃 from the 𝑚 scaled images. Next, the obtained +pyramid patches are fed into with 𝑚 individual encoders +𝐸1, 𝐸2, … , 𝐸𝑚, and 𝑚 pixel descriptors are obtained, where + + +Figure 3: The descriptor optimization process of the ASD model. (a) For each normal image, its transformed image is +generated using data argumentation techniques for generating the artificially negative samples. (b) The ASD model is de- +signed as a two-head architecture. One head outputs the dense descriptor and the other reconstruction head is designed to +force the obtained descriptors to contain both high-level and low-level features. Pyramid patches are extracted at different +scales for the multi-scale features. (c) To obtain the ideal descriptors, as defined in Section 3.2, the optimization tries to +find a compact hypersphere surrounding all the descriptors of the original image by pulling them to the center, keeping the +descriptors diverse by increasing the distance between the original descriptors and the transformed descriptors. +Pyramid +Patches +Scale 1 +Scale 2 +Scale 3 +Linked +descriptor +Input image +Transformed image +Data +transformation +(a) Data argument +(b) Descriptor extractor +(c) Optimization objective: ++ ++ +Pixel descriptors +Reconstructed image +Pixel descriptors +Reconstructed image +Pull +Push +Push and pull descriptors +pact +Diverse += +Hypersphere radius: +Hypersphere center: +Feature-rich (Multi-level) += + +each descriptor has the same length 𝐿. The 𝑚 pixel de- +scriptors are then concatenated further to form a descriptor +vector with length 𝑚 × 𝐿. The descriptor cube 𝑫𝑐 with size +𝐻 × 𝑊 × (𝑚 × 𝐿) is naturally obtained when all the pixels +in 𝑿 are processed. Finally, a 1 × 1 convolution operation is +used to map the concatenated descriptors into size 𝐿. This is +the process for extracting the final pixel descriptors. Eqs. (8- +10) formally express the above process, which is also the +detailed process of 𝑓1. Figure 3 shows the process when 𝑚 += 3. + +𝑿1, 𝑿2, … , 𝑿𝑚 = 𝑈1(𝑿), 𝑈2(𝑿), … , 𝑈𝑚(𝑿) +(8) +𝑫𝑐 = concat([𝑀(𝑿1, 𝑿2, … , 𝑿𝑚)]) +(9) +𝑫 = Conv1×1(𝑫𝑐) +(10) + + Multi-level features are necessary when dealing with the +various objects in the anomaly segmentation task. Although +the deep architecture extracts high-level semantic infor- +mation through the descriptors, the low-level information +such as texture is gradually forgotten as the network goes +deeper. This is beneficial for objects such as buildings, but +is not expected for some objects such as water and river be- +cause the texture feature is useful for them. +To ensure that both high-level and low-level features are +contained in the descriptors, the ASD model is designed as +a two head architecture. Both heads grow from the concate- +nated descriptor cube 𝑫𝑐. One head uses the 1 × 1 convolu- +tion operation to obtain the final pixel descriptors. The other +head also uses the 1 × 1 convolution but aims to reconstruct +the original pixel. To reconstruct the pixel value, the concat- +enated descriptors are forced to contain the low-level fea- +tures. Note that the reconstruction head is only used in the +training and is abandoned in the test stage. 𝑿′ denotes the +reconstructed image, and the MSE is used to compute the +loss (Eqs. (11-12)). + +𝑿′ = Conv1×1(𝑫𝑐) +(11) +𝐿3(𝑿, 𝑿′) = +1 +𝐻 × 𝑊 ∑ ∑‖𝑿𝑖𝑗 − 𝑿𝑖𝑗 +′ ‖ +2 +𝑊 +𝑗=1 +𝐻 +𝑖=1 + +(12) + + In total, the three properties: compact, diverse and fea- +ture-rich work together to design the model architecture and +optimize the descriptor learning. The optimization objective +of the ASD model is the sum of the above losses, as shown +in Eq. (13). Figure 3 shows the overall descriptor learning +process. + +𝐿𝑜𝑠𝑠 = 𝐿1(𝑫) + 𝐿2(𝑫, 𝑫𝑇) + 𝐿3(𝑿, 𝑿′) +(13) + +3.3. Anomaly Score Computation +When the descriptor optimization process of step 𝑓1 is fin- +ished, the second step 𝑓2 outputs the anomaly map based on +the optimized descriptors. There exist various methods to +complete step 𝑓2. Although non-parametric statistical meth- +ods do not rely on any distribution assumption, it requires a +lot of samples to achieve accurate estimation and can be +computationally expensive e (Pang et al. 2021). Conversely, +parametric density estimation needs fewer samples, and the +Gaussian assumption holds in most cases (Pimentel et al. +2014). + In the ASD model, the Gaussian assumption is adopted to +model the normal descriptors. Using the normal samples in +the training stage, the mean 𝜇 and the covariance matrix 𝚺 +can be estimated. Given a test descriptor 𝑥𝑡, its Mahalanobis +distance from the modeled distribution (as shown in Eq. (14)) +is considered the anomaly degree, which can be converted +to the anomaly score after the normalization. + +𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝑑𝑒𝑔𝑟𝑒𝑒 = √(𝑥𝑡 − 𝜇 )𝑇𝚺−1(𝑥𝑡 − 𝜇) +(14) + +4. Experiments +4.1. Experimental Settings +Datasets +The proposed ASD model was evaluated on four HSR im- +age datasets: DeepGlobe (Demir et al. 2018), Agriculture- +Vision (Chiu et al. 2020), FAS, and Landslide4Sense (Ghor- +banzadeh et al. 2022). The DeepGlobe and Agriculture-Vi- +sion datasets were originally made for the land-cover seg- +mentation and agriculture pattern segmentation tasks, re- +spectively. To adapt these datasets for the anomaly segmen- +tation task, the pixels of the remaining classes were masked +for a fixed normal class in the training process to keep the +anomaly-free characteristic. + To show the application value of the ASD model, the FAS +and Landslide4Sense datasets were used. The FAS dataset +was made by ourselves for the forest monitoring application, +where the common forest pattern (i.e., Figure 1) is treated as +the normal class, and some abnormal objects, such as house, +lake, car, and diseased tree, are considered as anomalies. +The RGB imagery in the FAS dataset was made from UAV- +borne hyperspectral images in forest scene. The pixel reso- +lution is 11 cm and the image size is 120×120. In the Land- +slide4Sense dataset, the anomaly segmentation model was +used to segment the landslide area by learning from the nor- +mal mountain pattern. +Comparative Models and Evaluation Metrics +The ASD model was compared with four state-of-the-art +methods covering both image space and feature space types. + +These methods include GANomaly (Akcay, Atapour-Abar- +ghouei, and Breckon 2018), ARNet (Fei et al. 2020), RIAD +(Zavrtanik, Kristan, and Skocaj 2021) and deep SVDD +(DSVDD) (Ruff et al. 2018). For the GANomaly, RIAD, +and DSVDD, the model hyper-parameters were kept same +as the authors’ open source code. ARNet was implemented +using the same architecture as RIAD. The model perfor- +mance was evaluated using the area under the curve (AUC) +metric and the mean Intersection over Union (mIOU). The +segmentation threshold for the mIOU corresponds to the +left-upper point of the Receiver operating characteristic +(ROC) curve. +Implementation Details +The fast version (Bailer et al. 2018) of the point descriptor +extraction network in the work of Simo-Serra et al. (2015) +acted as the pixel descriptor encoder in the proposed model. +In all the experiments, the models were trained for 100 +epochs, and the batch size was 1. The Adam optimizer with +learning rate 0.0001 was used. 𝜆 was set to 10. 𝑆𝑠 was set to +{0.5,1.0,2.0} and 𝑃 is 15 for all the descriptor encoders. The +first 10 epochs were trained using only the 𝐿3 loss to com- +pute the initial 𝐶. 𝑅 was initialized to 3.0. 𝐶 and 𝑅 were up- +dated after each epoch using all the training descriptors. The +dimension 𝐿 was set to 5. The data augmentation operations +used in the ASD and ARNet model were the GaussNoise, +ChannelShuffle, RandomBrightness, RandomContrast, and +Solarize operations (implemented with the Albumentations +tool (Buslaev et al. 2020)). Due to the AUC computation +burden, 2000 test images in Agriculture-Vision dataset were +chosen to be evaluated. The CPU was an Intel(R) Xeon(R) +CPU E5-2690 v4 @ 2.60 GHz with 62.6 GB memory, and +the GPU was a Tesla P100-PCIE with 16 GB of memory. +4.2. Results on the DeepGlobe Dataset +The quantitative and qualitative results are reported in Table +1 and Figure 4, respectively. In Table 1, the ASD model +achieves the highest AUC values for the four normal classes. +For the Urban land class, the ASD model surpasses the sec- +ond-best model by over 6 points, showing its superiority +when dealing with a complex distribution. In Figure 4, the +anomaly maps obtained by the ASD model are the closest to +the ground truth. +4.3. Results on the Agriculture-Vison Dataset +Table 2 and Figure 4 respectively show the quantitative and +qualitative results for the Agriculture-Vision dataset. In Ta- +ble 2, the ASD model achieves the best AUC results for all +six normal classes. ASD surpasses the second-best model by +5 points for the Drydown class. Except Weed cluster, the +mIOU values of ASD are all close to the optimal value. In +Figure 5, it can be seen that accurate results and fine bound- +aries are obtained by the ASD model for most classes. For + + +Figure 4: The anomaly segmentation results obtained on +the DeepGlobe dataset for each normal class, White pix- +els cover the anomalous region. +Urban +land +Water +Range +land +Barren +land +Agriculture +Forest +land +Image +GT +DSVDD +RIAD +ARNet +GANomaly +ASD +Normal class + + +Figure 5: The anomaly segmentation results obtained on +the Agriculture-Vision dataset for the six normal classes. +Image (RGB) +GT +DSVDD +RIAD +ARNet +GANomaly +ASD +Normal class +Dry down +Nutrient +deficiency +Endrow +Water +Double +plant +Weed +cluster +Image (NIR) +Method +Urban +land +Agricul- +ture +Range +land +Forest +land +Water +Barren +land +AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU +DSVDD 57.0 31.5 60.3 41.4 53.6 16.3 58.7 24.0 37.6 2.2 50.6 17.7 +RIAD +52.3 12.4 65.9 46.3 47.6 7.0 69.4 33.1 57.7 43.0 53.3 13.9 +ARNet +50.2 39.0 60.1 40.9 48.2 6.9 67.6 34.5 54.7 12.5 61.4 33.3 +GANomaly 42.8 44.3 51.7 35.8 55.1 30.6 75.4 41.3 58.8 44.5 36.8 39.7 +ASD +63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 + +Table 1: The anomaly segmentation results obtained on +the DeepGlobe dataset. +Method +Drydown Double +plant +Endrow +Weed +cluster +ND +Water +AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU +DSVDD 60.9 30.8 53.0 14.7 54.6 10.1 48.0 3.67 59.6 24.4 72.1 26.0 +RIAD +62.2 31.0 60.9 25.6 59.3 27.7 55.2 38.2 63.8 25.7 86.1 42.7 +ARNet +61.1 30.6 51.5 15.3 57.1 25.5 53.0 15.9 59.9 26.8 45.3 9.6 +GANomaly 59.5 26.3 49.6 4.2 56.5 26.4 51.1 41.9 62.9 33.1 64.2 20.7 +ASD +67.4 36.4 61.3 24.8 61.1 25.7 58.0 19.7 65.9 31.7 90.0 40.4 + +Table 2: The comparative quantitative anomaly seg- +mentation results on the Agriculture-Vision dataset. +(ND is nutrient deficiency) + + 0.8 + 0.6 +0.4 +0.2 0.8 + 0.6 +0.4 +0.2the normal class of water, only the ASD model outputs a +correct anomaly map, and some models completely reverse +the anomaly regions. +4.4. Results on the FAS and Landslide4Sense Da- +tasets +The FAS and Landslide4Sense datasets were used to show +the application value of the proposed anomaly segmentation +model in forest monitoring and landslide detection. Table 3, +Figure 6, and Figure 7 report the related results. In both da- +tasets, the ASD model achieves the best AUC and mIOU +scores. Satisfactory anomaly maps are obtained, demon- +strating great application value. +4.5. Sensitivity of the Descriptor Scale +Table 4 reports the effect of the multi-scale descriptor on the +anomaly segmentation performance. The multi-scale setting +with 𝑆𝑠 = 0.5, 1.0, 2.0 obtains the optimal AUC values for +four classes. In a real application, although the optimal value +of 𝑆𝑠 may be difficult to establish, Table 4 shows that the +multi-scale setting would ensure satisfactory results. +4.6. Ablation Studies +The core idea of the ASD model is to find ideal descriptors, +so three loss constraints corresponding to the conditions de- +scribed in Section 3.2 were designed. Table 5 illustrates the +effectiveness of three losses for different types of earth vi- +sion scenes 𝐿1 (compact loss) can better handle the scene +with simple spatial distribution, i.e., Agriculture, Forestland, +and Water; (from the first 3 rows). 𝐿3 (feature-rich loss) +works on the complex scenes, i.e., Urban land and Barren +land; (from the 3,5 and 6 rows). 𝐿2 (diversity loss) aims at +further improving segmentation performance by artificial +anomaly samples. (Comparing rows 1 and 3 with 4 and 5, +respectively). +5. Conclusion +In this paper, we have proposed a pixel descriptor based +model for the anomaly segmentation task in HSR imagery. +The core innovations are: 1) The three conditions that the +ideal descriptor should meet are given from the characteris- +tics of the anomaly segmentation task and HSR images. 2) +The corresponding constraints and architecture were de- +signed on this basis. Obvious improvement was achieved on +four datasets (including real anomalies in forest and moun- +tain scenes). Overall, proposed model sets the first baseline +for the anomaly segmentation task of complex HSR imagery. + + +Figure 6: The anomaly segmentation results obtained on +the FAS dataset. The common forest pattern (see Figure +1) is considered as normal, and four anomalies are con- +sidered. +Image +GT +DSVDD +RIAD +ARNet +GANomaly +ASD +PWD +House +Car +Lake +Anomalies + + +Figure 7: The anomaly segmentation results obtained on +the Landslide4Sense dataset. The common mountain pat- +tern is considered as normal, and the landslides are the +anomalies. +GT +DSVDD +RIAD +ARNet +GANomaly +ASD +Image (DEM) +Image (Slope) +Image (RGB) +Dataset +DSVDD +RIAD +ARNet +GANomaly +ASD +AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU +FAS +74.1 46.2 44.3 36.5 82.7 52.9 50.7 24.9 91.0 69.3 +Lanslide4Sense 61.6 20.7 83.7 41.0 78.8 48.7 82.2 39.1 89.8 49.3 + +Table 3: The anomaly segmentation results obtained on +the FAS and Landslide4Sense datasets. + +Constraints +Urban +land +Agricul- +ture +Range +land +Forest +land +Water +Barren +land +AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU +𝐿1 +51.3 38.9 62.5 42.3 52.8 18.1 76.4 42.0 70.2 35.1 56.6 26.9 +𝐿2 +40.9 45.3 59.4 37.9 53.7 40.2 76.7 47.6 71.5 36.1 49.0 34.1 +𝐿3 +62.5 35.9 60.4 38.6 52.9 19.6 75.6 41.5 68.8 36.0 61.2 32.9 +𝐿1+𝐿2 +56.5 32.4 61.0 39.7 54.7 31.8 78.8 48.6 72.5 40.6 54.6 22.8 +𝐿2+𝐿3 +64.5 45.1 62.0 42.5 54.6 22.1 77.4 43.1 73.0 41.7 54.3 23.8 +𝐿1+𝐿3 +64.1 45.1 62.7 41.9 52.9 20.0 77.3 43.1 74.5 41.5 61.7 31.5 +𝐿1+𝐿2+𝐿3 63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 + +Table 5: The ASD model ablation analysis for the three +loss constraints on the anomaly segmentation results ob- +tained using the DeepGlobe dataset. + +Scale +Urban +land +Agricul- +ture +Range +land +Forest +land +Water +Barren +land +AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU +(0.5,0.5,0.5) 61.7 39.9 63.7 42.7 57.4 31.5 77.8 45.5 68.8 38.3 59.6 30.5 +(1.0,1.0,1.0) 63.2 40.0 60.7 40.0 56.1 27.9 78.5 44.1 72.4 35.7 57.5 22.5 +(2.0,2.0,2.0) 58.4 42.5 62.0 40.5 55.5 25.7 75.6 43.5 78.0 41.1 59.0 35.0 +(0.5,1.0,2.0) 63.4 38.5 64.1 42.7 54.3 23.2 79.5 43.1 73.3 39.5 62.5 34.9 + +Table 4: The ASD model sensitivity analysis for the +multi-scale property on the anomaly segmentation re- +sults obtained using the DeepGlobe dataset. + + 0.8 + 0.6 +0.4 +0.2 0.8 + 0.6 +0.4 +0.2Acknowledgments +This work was supported by National Natural Science Foun- +dation +of +China +under +Grant +No.42071350 +and +No.42101327, in part by the Fundamental Research Funds +for the Central Universities under Grant 2042021kf0070, +and LIESMARS Special Research Funding. +References +Akcay, S.; Atapour-Abarghouei, A.; and Breckon, T. 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In In- +ternational conference on learning representations. + diff --git a/2NFQT4oBgHgl3EQf2DZm/content/tmp_files/load_file.txt b/2NFQT4oBgHgl3EQf2DZm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd1ac71c035b37a8d928a32f287867f435d5a2db --- /dev/null +++ b/2NFQT4oBgHgl3EQf2DZm/content/tmp_files/load_file.txt @@ -0,0 +1,1145 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf,len=1144 +page_content='Anomaly Segmentation for High-Resolution Remote Sensing Images Based on Pixel Descriptors Jingtao Li1, Xinyu Wang2*, Hengwei Zhao1, Shaoyu Wang1, Yanfei Zhong1 1 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' China 2 School of Remote Sensing and Information Engineering, Wuhan University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' China {JingtaoLi, wangxinyu, whu_zhaohw, wangshaoyu, zhongyanfei}@whu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='cn Abstract Anomaly segmentation in high spatial resolution (HSR) re- mote sensing imagery is aimed at segmenting anomaly pat- terns of the earth deviating from normal patterns, which plays an important role in various Earth vision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' How- ever, it is a challenging task due to the complex distribution and the irregular shapes of objects, and the lack of abnormal samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To tackle these problems, an anomaly segmentation model based on pixel descriptors (ASD) is proposed for anomaly segmentation in HSR imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Specifically, deep one-class classification is introduced for anomaly segmenta- tion in the feature space with discriminative pixel descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The ASD model incorporates the data argument for generat- ing virtual abnormal samples, which can force the pixel de- scriptors to be compact for normal data and meanwhile to be diverse to avoid the model collapse problems when only pos- itive samples participated in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In addition, the ASD introduced a multi-level and multi-scale feature extrac- tion strategy for learning the low-level and semantic infor- mation to make the pixel descriptors feature-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The pro- posed ASD model was validated using four HSR datasets and compared with the recent state-of-the-art models, showing its potential value in Earth vision applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Introduction Anomaly segmentation is aimed at segmenting the anomaly patterns which deviate from the normal patterns (Pimentel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Due to the lack of abnormal samples, anomaly segmentation is a challenging task, but plays an important role in many computer vision applica- tions, including medical analysis (Fernando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021), in- dustrial defect detection (Bergmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019), video sur- veillance (Liu, Li, and Poczos 2018), and environmental monitoring (Miau and Hung 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Anomaly segmentation in high spatial resolution (HSR) remote sensing images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', Figure 1) is a powerful tool for environmental monitoring (Miau and Hung 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Despite this, few related works have focused on anomaly segmentation in HSR imagery because of the Corresponding author unique characteristics when compared to the industrial and medical images used in most anomaly segmentation tasks, which have a regular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The objects in HSR images typically have a more complex spatial distribution and large radiation differences within the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Furthermore, since HSR images can be captured in different angles and heights, the objects always have multiple scales and show rotation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' These characteristics make anomaly segmentation for HSR images a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The mainstream anomaly segmentation models detect anomalies in the image space, where the anomaly score is computed based on the image pixel values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Typical exam- ples are the autoencoder (AE)-based models (Zavrtanik, Kristan, and Skoˇcaj 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019) and the gener- ative adversarial network (GAN)-based models (Ngoetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Zenatietal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' AE-based models assume that normal samples can be reconstructed more easily than the anomalous ones and the reconstruction error indicates the anomaly segmentation score (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' However, the low-level reconstruction error has been shown to focus on the pixel-wise error, resulting in abnormal samples also being reconstructed, especially when the normal distribution is complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Zong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='com/Jingtao-Li-CVer/ASD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Figure 1: Anomaly segmentation example for HSR re- mote sensing images using proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the forest scene, the common forest pattern is considered as normal, and the abnormal objects such as diseased trees (the first row) and the house in the forest (the second row) are iden- tified as anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Anomaly image Normal images Anomaly label Anomaly map … 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' GAN-based models detect anomalies from the gen- eration performance (Akcay, Atapour-Abarghouei, and Breckon 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ngo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2022), where the superior capability in generating image data also empowers the detection of abnormal samples (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In spite of this, the complex distribution of HSR images can make the generator generate data instances that are out of the manifold of normal instances (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Differing from the AE-based and GAN-based models, deep one-class classification (OCC)-based methods detect the anomalies in the feature space (Shi, Yang, and Qi 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2022), where the anomaly score is computed based on the extracted image de- scriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' These methods aim to learn discriminative de- scriptors in the training stage and compute the anomaly score in the feature space using a measurement such as the Mahalanobis or Euclidean distance (Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Shi, Yang, and Qi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Because deep OCC- based methods focus on semantic features rather than low- level pixel errors, it is more suitable to deal with the anom- aly segmentation task in HSR imagery which has complex distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' However, two barriers exist when applying ex- isting methods directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (i) Due to the lack of abnormal sam- ples, the model training only uses normal samples and is op- timized to be compact (Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Chalapathy, Menon, and Chawla 2018), which can easily result in the model col- lapse problem (Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (ii) The anomalies in HSR imagery have rich low-level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', texture) and high-level (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', semantic) features, which are both important for the anomaly segmentation task and real application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Although the current deep OCC models can capture useful semantic features, they perform suboptimally than models detecting in the image space for samples with regular structures (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021), because low-level features are mostly forgotten in feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In this paper, we tackle the two problems for the anomaly segmentation task using HSR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' A novel anomaly seg- mentation model based on pixel descriptors (ASD) is pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (i) In addition to considering the compact property of the obtained descriptors, the ASD model encourages de- scriptors to be diverse by increasing the descriptor distance between the original image and the transformed image with the use of data augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The transformed descriptors act as anomalies, to some extent, which en- hances the anomaly detection ability and prevents simulta- neous model collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (ii) To make the descriptor feature- rich, a descriptor at different scales is fused for each pixel, and an auxiliary reconstruction head is designed to force the descriptor to remember the low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Compact, di- verse, and feature-rich property optimizes the model to- gether from the perspective of the feature distance and fea- ture quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' ASD sets the first baseline for the anomaly segmentation task in HSR imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The ASD model was validated on four HSR datasets: the DeepGlobe land-cover segmentation dataset, the Agricul- ture-Vision agriculture pattern segmentation dataset, the Landslide4Sense landslide detection dataset, and the forest anomaly detection dataset (FAS, made by ourselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The ASD model showed an obvious superiority over the recent state-of-the-art anomaly segmentation models (with an area under the curve (AUC) improvement of 5–10 points in most cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The results obtained on the Landslide4Sense and FAS datasets confirmed the great application potential of the ASD model in disaster detection and forest monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Related Work AE-based models are always composed of an encoding and decoding network, with the aim being to reconstruct the original input data (Pimentel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (2022) first introduced the AE into the anomaly detection field, where the features learned in the latent space can be used to distinguish normal and anomalous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The recon- struction error is considered as the anomaly degree and the mean square error (MSE) is adopted as the loss function in most studies (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To promote the performance, Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (2016) blanked the input image randomly and forced the model to reconstruct the damaged area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Similarly, the ARNet model was proposed, which erases some input attributes and reformulates the problem as a restoration task (Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Recently, Zavrtanik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (2021) cast the reconstruction problem as an inpainting problem and recon- structed the image from partial inpaintings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' However, the extracted low-level features can be shared by both normal and anomalous data (Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020) when dealing with complex HSR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' GAN-based models aim to generate the image rather than reconstruct it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' As one of the early GAN-based models, the AnoGAN model assumes that the learned latent space can represent normal samples well, but not the anomalous sam- ples (Schlegl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Given a test image, the difference between the regenerated image obtained using the searched latent feature and the test image is considered as the anom- aly degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The famous GANomaly model improved the generator architecture from a decoder to an encoder-decoder encoder design and used high-level features to assist com- puting the anomaly score (Akcay, Atapour-Abarghouei, and Breckon 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' GAN-based models have demonstrated su- perior capabilities in generating image data, which also em- powers the detection of abnormal samples (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In spite of this, the complex distribution of HSR images can make the generator generate data instances that are out of the manifold of normal instances (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' One-class classification models are also used in some anomaly segmentation works (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' One of their greatest advantages over the AE-based and GAN-based models is that the OCC models detect anomalies in the fea- ture space with high-level semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' They first divide an image into many patches and then learn the corre- sponding representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The anomaly score is computed in the feature space using a measurement such as the Ma- halanobis or Euclidean distance (Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Shi, Yang, and Qi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Most OCC models are based on the principle of one-class support vector machine (OCSVM) (Sch¨olkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Andrews, Morton, and Griffin 2016) or support vector data description (SVDD) (Tax and Duin 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Chalapathy, Menon, and Chawla 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' However, these models mainly consider the compact property of the obtained one class features, re- sulting in the model collapse problem (Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021), and they lack consideration of the low-level structural fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Methodology Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' This section describes the core principles of the proposed ASD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The overall workflow of the ASD model is shown in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1, which includes two steps: descriptor extracting and anomaly score computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To extract the ideal descriptors, descriptor learning is the most important part and is described detailed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The computation method of the anomaly score is given in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Overall Workflow of The ASD Model Given an HSR image 𝑿 with size 𝐻 × 𝑊 × 𝐵, where 𝐻, 𝑊, and 𝐵 are the height, width and bands of the image, the anomaly segmentation task can be viewed as a mapping function 𝑓 from the 𝑿 to the anomaly map 𝑨 with size 𝐻 × 𝑊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Each pixel in the anomaly map is in the range [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Generally speaking, the higher the value in the anomaly map, the higher the anomaly degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The ASD model separates the function 𝑓 into two steps and the overall workflow is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The first step 𝑓1 extracts the dense descriptor cube 𝑫 for each image pixel, which is the core part and also the training focus in the ASD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The descriptors are expected to contain important visual characteristics for the anomaly segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To incorporate the pixel context and obtain fine pixel corre- spondence, the patch-based paradigm is chosen to compute the descriptor 𝐹 for the center pixel 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In this step, the 𝐻 × 𝑊 patches form the input samples and a descriptor cube 𝑫 with size 𝐻 × 𝑊 × 𝐿 is output, where 𝐿 is the de- scriptor length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The second step 𝑓2 outputs the anomaly map based on the trained descriptor encoders in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Specifically, the trained descriptors of the training samples are modeled as a multivariate Gaussian Distribution (MGD) (Guimaraes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018) by the Gaussian Density Estimate (GDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For the test descriptor, its Mahalanobis distance from the MGD is used to measure the anomaly score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The formal mapping of 𝑓, 𝑓1, and 𝑓2 is shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (1-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑓: 𝑿 → 𝑨 (1) 𝑓1 ∶ 𝑿 → 𝑫 (2) 𝑓2 ∶ 𝑫 → 𝑨 (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ideal Descriptors Learning The descriptors obtained in the first step (as mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1) are expected to contain important visual char- acteristics for the anomaly segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To achieve this aim, ideal descriptors are optimized using three condi- tions from the characteristics of the anomaly segmentation task and HSR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' One of the characteristics of anomaly segmen- tation is that only normal samples are used in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In other words, all the training samples are of the same class, which naturally results in compact visual descriptors in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' This compactness is also a useful su- pervised signal for the anomaly segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To keep 𝑫 compact, an enclosing hypersphere around all the pixel descriptors is constructed, which is motivated by the deep SVDD method (Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' We let 𝑅 be the hypersphere radius and 𝐶 be the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The 𝐿1 loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (4)) aims to minimize the hypersphere radius and the distance from the obtained pixel descriptors to the center 𝐶, where the parameter 𝜆 controls the trade-off between the size of the hypersphere and the number of surrounded descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The maximum distance between 𝐶 and 𝐹 in 𝑫 is chosen to Figure 2: The overall workflow of the ASD model, which includes two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the first step, the ASD model ex- tracts a descriptor for each pixel with the descriptor ex- tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the second step, the descriptors for normal scenes are modeled as the Gaussian distribution, and the Mahalanobis distance between the test descriptor and the modeled distribution is considered to measure the anom- aly score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Normal descriptors GDE Test image Test descriptors Gaussian distribution Anomaly map Descriptor extractor Mahalanobis distance compute the radius 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Compared to using the mean value, this setting helps the model focus on special normal samples, rather than just considering them as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐿1(𝑫) = 𝑅2 + 𝜆 mean{0, max{‖𝐹 − 𝐶‖2 − 𝑅2 | 𝐹 ∈ 𝑫}} (4) Diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Compactness is the first basic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' How- ever, the model can easily collapse if only a compactness constraint used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In other words, the model would map all the input samples into the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' This “cheating” makes the model lose the anomaly detection ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To deal with this problem, the diverse condition is necessary, which stresses that a different pixel 𝑥 obtains different values of 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The key consideration to keeping the descriptors diverse is to keep the model sensitive to the input sample change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Considering the fact that training images are always anom- aly free and real negative samples are difficult to obtain, data augmentation techniques, such as the channel shuffle oper- ation, are used to generate negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Formally, the augmentation operation set 𝑆𝑎 = {𝐴1, 𝐴2, … , 𝐴𝑛} contains 𝑛 kinds of different augmentation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For the original image 𝑿, the obtained image descriptor cube 𝑫 can be seen as a positive one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Then, after applying the operations from 𝑆𝑎 on 𝑿 in turn, 𝑿𝑇 can be obtained and the corresponding cube 𝑫𝑇 is considered to be a negative sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (5-6) formally express the above process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑿𝑇 = 𝐴𝑛(… (𝐴2(𝐴1(𝑿𝑇))) (5) 𝑫 = 𝑓1(𝑿), 𝑫𝑇= 𝑓1(𝑿𝑇) (6) Both 𝑫 and 𝑫𝑇 have the same shape 𝐻 × 𝑊 × 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The diver- sity loss is defined as the average pixel descriptor difference between 𝑫 and 𝑫𝑇, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' With the 𝐿2 loss, the model is encouraged to increase the sensitivity to the in- put difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐿2(𝑫, 𝑫𝑇) = 1/{ 1 𝐻 × 𝑊 ∑ ∑‖𝑫𝑖𝑗 − 𝑫𝑖𝑗 𝑇 ‖ 2 𝑊 𝑗=1 𝐻 𝑖=1 } (7) Some technologies have the potential to deal with the model collapse, such as reducing the model bias (Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018) or designing early-stopping strategies (Reiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' However, the proposed 𝐿2 loss does not need early- stopping or change of the model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Feature-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The compact and diverse conditions meas- ure the descriptors from the perspective of distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The fea- ture-rich condition measures the descriptors from the per- spective of the amount of representative information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the ASD model, multi-scale and multi-level features are consid- ered in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The multi-scale characteristic is an import difference for HSR images, compared to natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For example, large-scale information is important for rivers and small- scale information is important for urban buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Even for the same scene, the images are always taken at different heights, which poses a challenge for the model ability to catch the multi-scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To enhance the model ability to deal with multi-scale in- formation, the ASD model uses a resize operation set 𝑆𝑠 = {𝑈1, 𝑈2, … , 𝑈𝑚} for the input patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Given an image 𝑿, it is resized using each operation 𝑈𝑖 in 𝑆𝑠, and obtains 𝑚 dif- ferent-scale versions 𝑿1, 𝑿2, … , 𝑿𝑚 of the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Then, for each center pixel 𝑥, 𝑚 patches are cropped with size 𝑃 × 𝑃 from the 𝑚 scaled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Next, the obtained pyramid patches are fed into with 𝑚 individual encoders 𝐸1, 𝐸2, … , 𝐸𝑚, and 𝑚 pixel descriptors are obtained, where Figure 3: The descriptor optimization process of the ASD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (a) For each normal image, its transformed image is generated using data argumentation techniques for generating the artificially negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (b) The ASD model is de- signed as a two-head architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' One head outputs the dense descriptor and the other reconstruction head is designed to force the obtained descriptors to contain both high-level and low-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Pyramid patches are extracted at different scales for the multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (c) To obtain the ideal descriptors, as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2, the optimization tries to find a compact hypersphere surrounding all the descriptors of the original image by pulling them to the center, keeping the descriptors diverse by increasing the distance between the original descriptors and the transformed descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Pyramid Patches Scale 1 Scale 2 Scale 3 Linked descriptor Input image Transformed image Data transformation (a) Data argument (b) Descriptor extractor (c) Optimization objective: + + Pixel descriptors Reconstructed image Pixel descriptors Reconstructed image Pull Push Push and pull descriptors pact Diverse = Hypersphere radius: Hypersphere center: Feature-rich (Multi-level) = each descriptor has the same length 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The 𝑚 pixel de- scriptors are then concatenated further to form a descriptor vector with length 𝑚 × 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The descriptor cube 𝑫𝑐 with size 𝐻 × 𝑊 × (𝑚 × 𝐿) is naturally obtained when all the pixels in 𝑿 are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Finally, a 1 × 1 convolution operation is used to map the concatenated descriptors into size 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' This is the process for extracting the final pixel descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (8- 10) formally express the above process, which is also the detailed process of 𝑓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Figure 3 shows the process when 𝑚 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑿1, 𝑿2, … , 𝑿𝑚 = 𝑈1(𝑿), 𝑈2(𝑿), … , 𝑈𝑚(𝑿) (8) 𝑫𝑐 = concat([𝑀(𝑿1, 𝑿2, … , 𝑿𝑚)]) (9) 𝑫 = Conv1×1(𝑫𝑐) (10) Multi-level features are necessary when dealing with the various objects in the anomaly segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Although the deep architecture extracts high-level semantic infor- mation through the descriptors, the low-level information such as texture is gradually forgotten as the network goes deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' This is beneficial for objects such as buildings, but is not expected for some objects such as water and river be- cause the texture feature is useful for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To ensure that both high-level and low-level features are contained in the descriptors, the ASD model is designed as a two head architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Both heads grow from the concate- nated descriptor cube 𝑫𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' One head uses the 1 × 1 convolu- tion operation to obtain the final pixel descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The other head also uses the 1 × 1 convolution but aims to reconstruct the original pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To reconstruct the pixel value, the concat- enated descriptors are forced to contain the low-level fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Note that the reconstruction head is only used in the training and is abandoned in the test stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑿′ denotes the reconstructed image, and the MSE is used to compute the loss (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (11-12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑿′ = Conv1×1(𝑫𝑐) (11) 𝐿3(𝑿, 𝑿′) = 1 𝐻 × 𝑊 ∑ ∑‖𝑿𝑖𝑗 − 𝑿𝑖𝑗 ′ ‖ 2 𝑊 𝑗=1 𝐻 𝑖=1 (12) In total, the three properties: compact, diverse and fea- ture-rich work together to design the model architecture and optimize the descriptor learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The optimization objective of the ASD model is the sum of the above losses, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Figure 3 shows the overall descriptor learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐿𝑜𝑠𝑠 = 𝐿1(𝑫) + 𝐿2(𝑫, 𝑫𝑇) + 𝐿3(𝑿, 𝑿′) (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Anomaly Score Computation When the descriptor optimization process of step 𝑓1 is fin- ished, the second step 𝑓2 outputs the anomaly map based on the optimized descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' There exist various methods to complete step 𝑓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Although non-parametric statistical meth- ods do not rely on any distribution assumption, it requires a lot of samples to achieve accurate estimation and can be computationally expensive e (Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Conversely, parametric density estimation needs fewer samples, and the Gaussian assumption holds in most cases (Pimentel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the ASD model, the Gaussian assumption is adopted to model the normal descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Using the normal samples in the training stage, the mean 𝜇 and the covariance matrix 𝚺 can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Given a test descriptor 𝑥𝑡, its Mahalanobis distance from the modeled distribution (as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (14)) is considered the anomaly degree, which can be converted to the anomaly score after the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐴𝑛𝑜𝑚𝑎𝑙𝑦 𝑑𝑒𝑔𝑟𝑒𝑒 = √(𝑥𝑡 − 𝜇 )𝑇𝚺−1(𝑥𝑡 − 𝜇) (14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Experimental Settings Datasets The proposed ASD model was evaluated on four HSR im- age datasets: DeepGlobe (Demir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018), Agriculture- Vision (Chiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020), FAS, and Landslide4Sense (Ghor- banzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The DeepGlobe and Agriculture-Vi- sion datasets were originally made for the land-cover seg- mentation and agriculture pattern segmentation tasks, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To adapt these datasets for the anomaly segmen- tation task, the pixels of the remaining classes were masked for a fixed normal class in the training process to keep the anomaly-free characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' To show the application value of the ASD model, the FAS and Landslide4Sense datasets were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The FAS dataset was made by ourselves for the forest monitoring application, where the common forest pattern (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', Figure 1) is treated as the normal class, and some abnormal objects, such as house, lake, car, and diseased tree, are considered as anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The RGB imagery in the FAS dataset was made from UAV- borne hyperspectral images in forest scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The pixel reso- lution is 11 cm and the image size is 120×120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In the Land- slide4Sense dataset, the anomaly segmentation model was used to segment the landslide area by learning from the nor- mal mountain pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Comparative Models and Evaluation Metrics The ASD model was compared with four state-of-the-art methods covering both image space and feature space types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' These methods include GANomaly (Akcay, Atapour-Abar- ghouei, and Breckon 2018), ARNet (Fei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020), RIAD (Zavrtanik, Kristan, and Skocaj 2021) and deep SVDD (DSVDD) (Ruff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For the GANomaly, RIAD, and DSVDD, the model hyper-parameters were kept same as the authors’ open source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' ARNet was implemented using the same architecture as RIAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The model perfor- mance was evaluated using the area under the curve (AUC) metric and the mean Intersection over Union (mIOU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The segmentation threshold for the mIOU corresponds to the left-upper point of the Receiver operating characteristic (ROC) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Implementation Details The fast version (Bailer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018) of the point descriptor extraction network in the work of Simo-Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (2015) acted as the pixel descriptor encoder in the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In all the experiments, the models were trained for 100 epochs, and the batch size was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The Adam optimizer with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0001 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝜆 was set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑆𝑠 was set to {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0} and 𝑃 is 15 for all the descriptor encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The first 10 epochs were trained using only the 𝐿3 loss to com- pute the initial 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝑅 was initialized to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐶 and 𝑅 were up- dated after each epoch using all the training descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The dimension 𝐿 was set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The data augmentation operations used in the ASD and ARNet model were the GaussNoise, ChannelShuffle, RandomBrightness, RandomContrast, and Solarize operations (implemented with the Albumentations tool (Buslaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Due to the AUC computation burden, 2000 test images in Agriculture-Vision dataset were chosen to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The CPU was an Intel(R) Xeon(R) CPU E5-2690 v4 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='60 GHz with 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 GB memory, and the GPU was a Tesla P100-PCIE with 16 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Results on the DeepGlobe Dataset The quantitative and qualitative results are reported in Table 1 and Figure 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In Table 1, the ASD model achieves the highest AUC values for the four normal classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For the Urban land class, the ASD model surpasses the sec- ond-best model by over 6 points, showing its superiority when dealing with a complex distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In Figure 4, the anomaly maps obtained by the ASD model are the closest to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Results on the Agriculture-Vison Dataset Table 2 and Figure 4 respectively show the quantitative and qualitative results for the Agriculture-Vision dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In Ta- ble 2, the ASD model achieves the best AUC results for all six normal classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' ASD surpasses the second-best model by 5 points for the Drydown class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Except Weed cluster, the mIOU values of ASD are all close to the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In Figure 5, it can be seen that accurate results and fine bound- aries are obtained by the ASD model for most classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' For Figure 4: The anomaly segmentation results obtained on the DeepGlobe dataset for each normal class, White pix- els cover the anomalous region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Urban land Water Range land Barren land Agriculture Forest land Image GT DSVDD RIAD ARNet GANomaly ASD Normal class Figure 5: The anomaly segmentation results obtained on the Agriculture-Vision dataset for the six normal classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Image (RGB) GT DSVDD RIAD ARNet GANomaly ASD Normal class Dry down Nutrient deficiency Endrow Water Double plant Weed cluster Image (NIR) Method Urban land Agricul- ture Range land Forest land Water Barren land AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU DSVDD 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 31.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 ASD 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 Table 1: The anomaly segmentation results obtained on the DeepGlobe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Method Drydown Double plant Endrow Weed cluster ND Water AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU DSVDD 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='67 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 24.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 ARNet 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 53.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 ASD 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 Table 2: The comparative quantitative anomaly seg- mentation results on the Agriculture-Vision dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (ND is nutrient deficiency) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2the normal class of water, only the ASD model outputs a correct anomaly map, and some models completely reverse the anomaly regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Results on the FAS and Landslide4Sense Da- tasets The FAS and Landslide4Sense datasets were used to show the application value of the proposed anomaly segmentation model in forest monitoring and landslide detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Table 3, Figure 6, and Figure 7 report the related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In both da- tasets, the ASD model achieves the best AUC and mIOU scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Satisfactory anomaly maps are obtained, demon- strating great application value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Sensitivity of the Descriptor Scale Table 4 reports the effect of the multi-scale descriptor on the anomaly segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The multi-scale setting with 𝑆𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 obtains the optimal AUC values for four classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In a real application, although the optimal value of 𝑆𝑠 may be difficult to establish, Table 4 shows that the multi-scale setting would ensure satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Ablation Studies The core idea of the ASD model is to find ideal descriptors, so three loss constraints corresponding to the conditions de- scribed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Table 5 illustrates the effectiveness of three losses for different types of earth vi- sion scenes 𝐿1 (compact loss) can better handle the scene with simple spatial distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', Agriculture, Forestland, and Water;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (from the first 3 rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐿3 (feature-rich loss) works on the complex scenes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=', Urban land and Barren land;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (from the 3,5 and 6 rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 𝐿2 (diversity loss) aims at further improving segmentation performance by artificial anomaly samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' (Comparing rows 1 and 3 with 4 and 5, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Conclusion In this paper, we have proposed a pixel descriptor based model for the anomaly segmentation task in HSR imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The core innovations are: 1) The three conditions that the ideal descriptor should meet are given from the characteris- tics of the anomaly segmentation task and HSR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2) The corresponding constraints and architecture were de- signed on this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Obvious improvement was achieved on four datasets (including real anomalies in forest and moun- tain scenes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Overall, proposed model sets the first baseline for the anomaly segmentation task of complex HSR imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Figure 6: The anomaly segmentation results obtained on the FAS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The common forest pattern (see Figure 1) is considered as normal, and four anomalies are con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Image GT DSVDD RIAD ARNet GANomaly ASD PWD House Car Lake Anomalies Figure 7: The anomaly segmentation results obtained on the Landslide4Sense dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' The common mountain pat- tern is considered as normal, and the landslides are the anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' GT DSVDD RIAD ARNet GANomaly ASD Image (DEM) Image (Slope) Image (RGB) Dataset DSVDD RIAD ARNet GANomaly ASD AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU FAS 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 Lanslide4Sense 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 Table 3: The anomaly segmentation results obtained on the FAS and Landslide4Sense datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Constraints Urban land Agricul- ture Range land Forest land Water Barren land AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU 𝐿1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 18.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 𝐿2+𝐿3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='1 77.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 Table 5: The ASD model ablation analysis for the three loss constraints on the anomaly segmentation results ob- tained using the DeepGlobe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Scale Urban land Agricul- ture Range land Forest land Water Barren land AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU AUC mIOU (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5,0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='5 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='9 Table 4: The ASD model sensitivity analysis for the multi-scale property on the anomaly segmentation re- sults obtained using the DeepGlobe dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content='42101327, in part by the Fundamental Research Funds for the Central Universities under Grant 2042021kf0070, and LIESMARS Special Research Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' References Akcay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Atapour-Abarghouei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' and Breckon, T.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Cheng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Lumezanu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Cho, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' and Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' Deep autoencoding gaussian- mixture model for unsupervised anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} +page_content=' In In- ternational conference on learning representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFQT4oBgHgl3EQf2DZm/content/2301.13422v1.pdf'} diff --git a/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf b/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e4f11003d21f77d441b3fd4ea5e8ce4035aa658a --- /dev/null +++ b/2dA0T4oBgHgl3EQfM_90/content/2301.02140v1.pdf @@ -0,0 +1,3 @@ +version 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b/3tAyT4oBgHgl3EQfo_h3/content/tmp_files/2301.00517v1.pdf.txt @@ -0,0 +1,2053 @@ +arXiv:2301.00517v1 [math.AG] 2 Jan 2023 +Correspondences in log Hodge cohomology +Charles Godfrey +Pacific Northwest National Laboratory +charles.godfrey@pnnl.gov +January 3, 2023 +Abstract +We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary char- +acteristic. These are represented by classes in the cohomology of sheaves of differential forms with +log poles and, notably, log zeroes on cartesian products of varieties. From one perspective this gen- +eralizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspon- +dences over perfect fields of arbitrary characteristic;from another we provide partial generalizations +of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing +finiteness and strictness conditions on the correspondences considered. +1 +Introduction +Generally speaking, a correspondence between two algebraic varieties푋 and 푌 over a field 푘 is a cycle +or cohomology class on the product 푋×푌. The study of such objects dates back (at least) to Lefschetz +[Lef53], and features prominently in famous conjectures on algebraic cycles (see e.g. [Voi14]) and +Voevodsky’s theory of motives (see e.g. [MVW06]). +In a number of algebro-geometric research areas it has become commonplace to work with pairs +(푋, ∆푋) consisting of a variety 푋 together with a divisor ∆푋 on 푋. Such areas include moduli of +varieties (where pairs generalize the curves with marked points of [DM69]), birational geometry +(where pairs appear naturally, for example as the output of strong resolution of singularities [KM98]) +and logarithmic geometry (in this case vast generalizations of divisors ∆푋 are allowed [Ogu18]). It is +natural to wonder about analogues of correspondencesin this category of pairs, and there have been +efforts in this direction, for example development of categories of logarithmic motives [BPØ20]. +In this paper, we focus on correspondences for logarithmic Hodge cohomology of pairs (푋, ∆푋), +where 푋 is a smooth (but not necessarily proper) variety over a perfect field 푘 and ∆푋 is a simple +normal crossing divisor on 푋. These cohomology groups can be described as +퐻∗(푋, ∆푋) = +⨁ +퐻푞(푋, Ω푝 +푋(log ∆푋)), +(1.1) +where Ω푋(log ∆푋) is the sheaf of differential 1-forms on 푋 with log poles along ∆푋 and Ω푝 +푋(log ∆푋) +the 푝-th exterior power thereof. In addition we consider a generalization where 푋 comes with a +family of supports Φ푋, and the ordinary cohomology groups on the right hand side of eq. (1.1) are +replaced with cohomology with supports in Φ푋, namely 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)). Allowing for supports +greatly expands the applicability of our results: for example, it permits us to construct a correspon- +dence associated to a cycle 푍 ⊂ 푋 × 푌 in a situation where neither 푋 nor 푌 is proper over 푘, but 푍 +is proper over both 푋 and 푌.1 +There are multiple motivations for investigating correspondencesfor this particular cohomology +of pairs: +1One way that such a cycle 푍 might naturally arise is as the closure of the graph of a birational equivalence 푋 ⤏ 푌 of +non-proper varieties. +This work was completed while the author was a PhD student in the University of Washington Department of Mathematics. +The author was partially supported by the University of Washington Department of Mathematics Graduate Research +Fellowship, and by the NSF grant DMS-1440140, administered by the Mathematical Sciences Research Institute, while in +residence at MSRI during the program Birational Geometry and Moduli Spaces. + +• By analogy with the case of varieties (that is, without auxiliary divisors/log structures), we sus- +pect that correspondences at the level of Chow cycles are more fundamental, and that (many) +correspondencesin logarithmic Hodge cohomology are obtained from Chow correspondences +via a cycle morphism. However, as of this writing there is no full-fledged theory of Chow co- +homology of pairs or log schemes (though there has been considerable progress, for instance +in [Bar18; BBG22]). Logarithmic Hodge cohomology is in contrast quite mature, appearing as +early as [Del71]. +• Correspondences in (non-logarithmic) Hodge cohomology have found remarkable applica- +tions. For example, [CR11] used them to prove birational invariance of the cohomology groups +of the structure sheaf 퐻푖(푋, 풪푋) for smooth varieties 푋 over perfect fields of positive character- +istic. In fact, attempting to implement a similar strategy with logarithmic Hodge cohomology +to obtain results on invariance of the cohomology groups 퐻푖(푋, 풪푋(−∆푋)) with respect to (a +restricted class of) birational equivalences was the initial inspiration for this work. Ultimately +that attempt was unsuccessful, as we describe in Appendix A. +• There has been recent interest in logarithmic Hodge cohomology as a representable functor +on a category of motives of log schemes over a perfect field [BPØ20, §9]. While that work does +also construct some correspondences, they are restricted to those associated with logarithmic +Hodge cohomology classes of cycles 푍 ⊂ 푋 × 푌 which are finite over 푋 and obey additional +strictness (in the sense of logarithmic geometry) conditions; we remove these restrictions. +The correspondences we construct are obtained from certain Hodge classes with both log poles +and log zeroes. Our main result is: +Theorem 1.2 (= Theorem 4.1). A class 훾 ∈ 퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) defines +homomorphisms +cor(훾) ∶ 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞+푗−푑푋 +Φ푌 +(푌, Ω푝+푖−푑푋 +푌 +(log ∆푌)) +by the formula cor(훾)(훼) ∶= pr푌∗(pr∗ +푋(훼) ⌣ 훾). Moreover if (푍, ∆푍, Φ푍) is another snc pair with +supports and 훿 ∈ 퐻푗′ +푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ +푌×푍(log ∆푌×푍)(−pr∗ +푌∆푌)), then +pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿)) ∈ 퐻푗+푗′−푑푌 +푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌 +푋×푍 +(log ∆푋×푍)(−pr∗ +푋∆푋)) and +cor(pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿))) = cor(훿)◦ cor(훾) +as homomorphisms 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌 +Φ푍 +(푍, Ω푝+푖+푖′−푑푋−푑푌 +푍 +(log ∆푍)). +In the above, ∆푋푌 ∶= pr∗ +푋∆푋 + pr∗ +푌∆푌, a simple normal crossing divisor on 푋 × 푌. There is a +simple heuristic explanation for the appearance of differential forms in Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋): +working over the complex numbers, in the case where 푋and 푌 are both proper the class cor(훾)(훼) ∶= +pr푌∗(pr∗ +푋(훼) ⌣ 훾) can be computed explicitly as an integral of the form +∫ +푋 +훼(푥) ∧ 훾(푥, 푦), +(1.3) +and this integral will only be finite when the log poles of 훼 along ∆푋 are cancelled by complementary +zeroes of the form 훾(푥, 푦) along the preimage pr∗ +푋∆푋. +Our proof of Theorem 1.2 relies heavily on prior work on both Hodge cohomology with supports +[CR11, §2] and its logarithmic variant [BPØ20, §9]. Section 2 is a rapid summary of those results. +The key new technical ingredient is a base change formula on the interaction of pushforward and +pullback operations in cartesian squares, proved in Section 3. Section 4 includes the proof of our +main theorem. +2 + +1.1 +Acknowledgements +Thanks to Daniel Bragg, Yun Hao, Sarah Scherotzke, Nicolò Sibilla and Mattia Talpo for helpful +conversations, to Lawrence Jack Barrott for illuminating email correspondence regarding logarith- +mic aspects of Chow and Hodge, and to my advisor Sándor Kovács for many insightful discussions. +Thanks also to the participants of the Spring 2019 MSRI graduate student seminar, in particular +Giovanni Inchiostro and organizer Fatemeh Rezaee, for feedback on early work on this paper. +2 +Functoriality properties of log Hodge cohomology with sup- +ports +2.1 +Supports +In order to obtain results that apply to correspondences between varieties 푋 and 푌 where neither 푋 +nor 푌 is proper, it is necessary to work with cohomology with supports, also known as local coho- +mology. A primary source for the material of this subsection is [R&D, §IV]. Let 푋 be a noetherian +scheme. +Definition 2.1 ([R&D, §IV], [CR11, §1.1]). A family of supports Φ on 푋 is a non-empty collection +Φ of closed subsets of 푋 such that +• If 퐶 ∈ Φ and 퐷 ⊂ 퐶 is a closed subset, then 퐷 ∈ Φ. +• If 퐶, 퐷 ∈ Φ then 퐶 ∪ 퐷 ∈ Φ. +Example 2.2. Φ = { all closed subsets of 푋 } is a family of supports. More generally if 풞 is any col- +lection of closed subsets 퐶 ⊂ 푋, there is a smallest family of supports Φ(풞) containing 풞 (explicitly, +Φ(풞) consists of finite unions ⋃ +푖 푍푖 of closed subsets 푍푖 ⊂ 퐶푖 of elements 퐶푖 ∈ 풞). Taking Φ = Φ({푋}) +recovers the previous example. A more interesting example is the case where for some fixed 푝 ∈ ℕ, +Φ = {closed sets 푍 ⊆ 푋 | dim 푍 ≤ 푝}. +There is a close relationship between families of supports on X and certaincollections of specialization- +closed subsets of points on 푋, and we can also consider sheaves of families of supports — for further +details we refer to [R&D, §IV.1]. +If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and Ψ is a family of supports on 푌, then +{푓−1(푍) | 푍 ∈ Ψ} is a family of closed subsets of 푋, and is closed under unions, but is not in general +closed under taking closed subsets. +Definition 2.3. 푓−1(Ψ) is the smallest family of supports on 푋 containing {푓−1(푍) | 푍 ∈ Ψ}. +Let Φ be a family of supports on 푋. The notation/terminology 푓|Φ is proper will mean 푓|퐶 is +proper for every 퐶 ∈ Φ. If 푓|Φ is proper then 푓(퐶) ⊂ 푌 is closed for every 퐶 ∈ Φ and in fact +푓(Φ) = {푓(퐶) ⊂ 푌 | 퐶 ∈ Φ} +(2.4) +is a family of supports on 푌. The key point here is that if 퐷 ⊂ 푓(퐶) is closed, then 푓−1(퐷) ∩ 퐶 ∈ Φ +and 퐷 = 푓(푓−1(퐷) ∩ 퐶). +Definition 2.5. A scheme with supports (푋, Φ푋) is a scheme 푋 together with a family of supports +Φ푋 on 푋. +Definition 2.6. A pushing morphism 푓 ∶ (푋, Φ푋) → (푌, Φ푌) of schemes with supports is a +morphism 푓 ∶ 푋 → 푌 of underlying schemes such that 푓|Φ푋 is proper and 푓(Φ푋) ⊂ Φ푌. A pulling +morphism 푓 ∶ 푋 → 푌 is a morphism 푓 ∶ 푋 → 푌 such that 푓−1(Φ푌) ⊂ Φ푋. +These morphisms provide two different categories with underlying set of objects schemes with +supports (푋, Φ푋), and pushing/pulling morphisms respectively (the verification is elementary; for +instance a composition of pushing morphisms is again a pushing morphism since compositions +of proper morphisms are proper). Schemes with supports provide a natural setting for describing +3 + +functoriality properties of local cohomology. Let ℱ be a sheaf of abelian groups on a scheme with +supports (푋, Φ푋).2 +Definition 2.7. The sheaf of sections with supports of ℱ, denoted ΓΦ(ℱ), is obtained by setting +ΓΦ(ℱ)(푈) = {휎 ∈ ℱ(푈) | supp 휎 ∈ Φ푋|푈 } +(2.8) +for each open 푈 ⊂ 푋 (here Φ푋|푈 is short for 휄−1Φ푋 where 휄 ∶ 푈 → 푋 is the inclusion). More +explicitly: for a local section 휎 ∈ ℱ(푈), 휎 ∈ ΓΦ(ℱ)(푈) means supp 휎 = 퐶 ∩ 푈 for a closed set +퐶 ⊂ Φ푋. +The functor ΓΦ is right adjoint to an exact functor, for instance the inclusion of the subcategory +퐀퐛Φ(푋) ⊂ 퐀퐛(푋) of abelian sheaves on 푋 with supports in Φ; so, ΓΦ is left exact and preserves +injectives. In the case Φ = Φ(푍) for some closed 푍 ⊂ 푋, this is proved in [Stacks, Tag 0A39, Tag 0G6Y, +Tag 0G7F] — the general case can then be obtained by writing ΓΦ as a filtered colimit: +ΓΦ = colim푍∈Φ Γ푍. +The right derived functor of ΓΦ will be denoted 푅ΓΦ. Taking global sections on 푋 gives the sections +with supports of ℱ: ΓΦ(ℱ) ∶= Γ푋(ΓΦ(ℱ)) This is also left exact, and (the cohomologies of) its +derived functor give the cohomology with supports in Φ: 퐻푖 +Φ(푋, ℱ) ∶= 푅푖ΓΦ(ℱ). +Proposition 2.9. Cohomology with supports enjoys the following functoriality properties: +(푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pulling morphism of schemes with supports, ℱ, 풢 are sheaves of +abelian groups on 푋, 푌 respectively, and if +휑 ∶ 풢 → 푓∗ℱ is a morphism of sheaves, +(2.10) +then there is a natural morphism 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ. Similarly if ℱ and 풢 are quasicoherent +then there are natural morphisms 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ. +(푖푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pushing morphism, ℱ, 풢 are sheaves of abelian groups on 푋, 푌 +respectively, and +휓 ∶ 푅푓∗ℱ → 풢 is a morphism in the derived category of 푋, +(2.11) +then there is a natural morphism 푅푓∗푅ΓΦ(ℱ) → 푅ΓΦ풢. +Both parts of the proposition follow from [Stacks, Tag 0G78]; (i) is discussed in detail in [CR11, +§2.1] and (ii) can be extracted from [CR11, §2.2] (although it doesn’t appear to be stated explicitly). +See also [BPØ20, Constructions 9.4.2, 9.5.3] +2.2 +Differential forms with log poles +Let 푘 be a perfect field. +Definition 2.12. A snc pair with supports (푋, ∆푋, Φ푋) over 푘 is a smooth scheme 푋 separated +and of finite type over 푘 with a family of supports Φ푋 together with a reduced, effective divisor ∆푋 +on 푋 such that supp ∆푋 has simple normal crossings, in the sense that for any point 푥 ∈ 푋 there are +regular parameters 푧1, … , 푧푐 ∈ 풪푋,푥 such that supp ∆푋 = 푉(푧1 ⋅ 푧2 ⋯ 푧푟) on a Zariski neighborhood +of 푥.3 The interior 푈푋 of a snc pair with supports (푋, ∆푋, Φ푋) is +푈푋 ∶= 푋 ⧵ supp ∆푋 +(2.13) +The inclusion of 푈푋 in 푋 is denoted by 휄푋 ∶ 푈푋 → 푋. +2Simply put ℱ is a sheaf of abelian groups on 푋. +3This is equivalent to the more general definition [BPØ20, Def. 7.2.1] in the case where the base scheme is Spec 푘, which +is all we need. +4 + +Here supp ∆푋 denotes the support of ∆푋 (if ∆푋 = ∑ +푖 푎푖퐷푖 where the 퐷푖 are prime divisors, then +supp ∆푋 = ∪푖퐷푖). Similarly let 푗푋 ∶ supp ∆푋 → 푋 denote the evident inclusion. +Definition 2.14 (compare with [CR11, Def. 1.1.4]). A pulling morphism 푓 ∶ (푋, ∆푋, Φ푋) → +(푌, ∆푌, Φ푌) of snc pairs with supports is a pulling morphism 푓 ∶ 푋 → 푌 of underlying schemes +with support such that 푓−1(supp ∆푌) ⊂ supp ∆푋; equivalently, 푓 restricts to a morphism 푓|푈푋 ∶ +푈푋 → 푈푌. A pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) of snc pairs with supports is a +pushing morphism of underlying schemes with support such that 푓∗∆푌 = ∆푋. +Note that if 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is a pushing morphism then 푈푋 = 푓−1(푈푌), so for +example if 푓 ∶ 푋 → 푌 is proper then so is the induced map 푈푋 → 푈푌. +Convention 2.15 (compare with [CR11, p. 1.1.5]). A morphism of snc pairs with supports 푓 ∶ +(푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is flat, proper, an immersion, etc. if and only if the same is true of the +underlying morphism of schemes 푓 ∶ 푋 → 푌. A diagram of snc pairs with supports +(푋′, ∆푋′, Φ푋′) +(푋, ∆푋, Φ푋) +(푌′, ∆푌′, Φ푌′) +(푌, ∆푌, Φ푌) +푔′ +푓′ +푓 +푔 +(2.16) +is cartesian if and only if the induced diagram of underlying schemes +푋′ +푋 +푌′ +푌 +푔′ +푓′ +□ +푓 +푔 +(2.17) +is cartesian.4 +The terminology is meant to suggest that pushing (resp. pulling) morphismsinduce pushforward +(resp. pullback) maps on log Hodge cohomology, as we now describe. +If (푋, ∆푋) is an snc pair, or more generally a normal separated scheme of finite type 푋 over 푘 +together with a sequence of effective Cartier divisors 퐷1, … , 퐷푁 ⊆ 푋 with sum ∆푋 = ∑ +푖 퐷푖, then +it comes with a sheaf of differential forms with log poles Ω푋(log ∆푋). In the case where (푋, ∆푋, Φ푋) +is snc, this sheaf and its properties are described in [EV92, §2]. For a definition and treatment of +Ω푋(log ∆푋) in the much greater generality of logarithmic schemes we refer to [Ogu18, §IV]. +In some of the calculations below the following concrete local description will be very useful. +Let 푧1, 푧2, … , 푧푛 be local coordinates at a point 푥 ∈ 푋 such that supp ∆푋 = 푉(푧1푧2 ⋯ 푧푟) in a neigh- +borhood of 푥. Recall that as 푋 is smooth the differentials 푑 푧1, 푑 푧2, … , 푑 푧푛 freely generate Ω푋 on a +neighborhood of 푥. +Lemma 2.18 (see e.g. [EV92, §2]). Thesections 푑 푧1 +푧1 , … , 푑 푧푟 +푧푟 , 푑 푧푟+1, … , 푑 푧푛 freelygenerateΩ푋(log ∆푋) +on a neighborhood of 푥. +Given Ω푋(log ∆푋), we can form the exterior powers +Ω푝 +푋(log ∆푋) ∶= +푝 +⋀ +Ω푋(log ∆푋), +(2.19) +and combining Lemma 2.18 with (2.19) gives concrete local descriptions of the Ω푝 +푋(log ∆푋); in par- +ticular, we see that Ωdim 푋 +푋 +(log ∆푋) = 휔푋(∆푋). +4If we take the red pill of logarithmic geometry, it starts to seem almost more reasonable to only require flatness, properness, +cartesianness and so on of the induced maps of interiors 푈푋 → 푈푌. However we do use the stronger restrictions of the given +definition in some of the proofs below. +5 + +Definition 2.20. The log-Hodge cohomology with supports of a log-smooth pair with supports +(푋, ∆푋, Φ푋) is defined by +퐻푑(푋, ∆푋, Φ푋) = +⨁ +푝+푞=푑 +퐻푞 +Φ(푋, Ω푝 +푋(log ∆푋)) +(2.21) +Here 퐻푞 +Φ denotes local cohomology with respect to the family of supports Φ푋. For connected 푋, we +define 퐻푑(푋, ∆푋, Φ푋) ∶= 퐻2 dim 푋−푑(푋, ∆푋, Φ푋), and in general we set 퐻푑(푋, ∆푋, Φ푋) = ⨁ +푖 퐻푑(푋푖, ∆푋푖, Φ푋푖) +where 푋푖 are the connected components of 푋. +Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be pulling morphism of snc pairs with supports. +Lemma 2.22 ([Ogu18, Prop. 2.3.1] + (2.19)). The map 푓 induces a morphism of sheaves +푓∗Ω푝 +푌(log ∆푌) +푑 푓∨ +����→ Ω푝 +푋(log ∆푋) adjoint to a morphism +푓∗Ω푝 +푌(log ∆푌) +푑푓∨ +����→ Ω푝 +푋(log ∆푋) for all p. +(2.23) +The essential content of this lemma is that when we pull back a log differentialform 휎 on (푌, ∆푌), +it doesn’t develop poles of order ≥ 1 along ∆푋. Combining the previous lemma with proposition 2.9 +gives: +Proposition 2.24 ([BPØ20, §9.1-2], see also [CR11, §2.1]). Foreverypullingmorphism푓 ∶ (푋, ∆푋, Φ푋) → +(푌, ∆푌, Φ푌) there are functorial morphisms +푅ΓΦΩ푝 +푌(log ∆푌) → 푅푓∗푅ΓΦΩ푝 +푌(log ∆푌) for all p +(2.25) +In particular, for each 푝, 푞 there are functorial homomorphisms +푓∗ ∶ 퐻푞 +Φ(푌, Ω푝 +푌(log ∆푌)) → 퐻푞 +Φ(푋, Ω푝 +푋(log ∆푋)) +(2.26) +and hence (summing over 푝 + 푞 = 푑) functorial homomorphisms +푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌) +(2.27) +The maps 푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌) induced by a pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) → +(푌, ∆푌, Φ푌) can be obtained from a combination of Nagata compactification and Grothendieck du- +ality. +Lemma 2.28 ([BPØ20, §9.5], see also [CR11, §2.3]). Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be a pushing +morphism of equidimensional log-smooth pairs with support such that. Then letting 푐 = dim 푌−dim푋, +for each 푝 there are functorial morphisms of complexes of coherent sheaves +푅푓∗푅ΓΦ푋(Ω푝 +푋(log ∆푋)) → 푅ΓΦ푌Ω푝+푐 +푌 +(log ∆푌)[푐] +(2.29) +inducing maps on cohomology +푓∗ ∶ 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞+푐 +Φ푌 (푌, Ω푝+푐 +푌 +(log ∆푌)) +(2.30) +for all 푞. +Since they enter into the calculations below, we give a description of these pushforward mor- +phisms. Before beginning, a word on duality in our current setup: since we are working exclu- +sively over Spec 푘, we can make use of compatible normalized dualizing complexes — namely, if +휋 ∶ 푍 → Spec푘 is a separated finite type 푘-scheme then 휋!풪Spec 푘 is a dualizing complex [Stacks, +Tag 0E2S, Tag 0FVU]. We will make repeated use of the behavior of dualizing with respect to differ- +entials: as a consequence of Lemma 2.18, wedge product gives a perfect pairing +Ω푝 +푋(log ∆푋)(−∆푋) ⊗ Ωdim 푋−푝 +푋 +(log ∆푋) → 휔푋 +(2.31) +6 + +(see also [Har77, Cor. III.7.13]) and so Ωdim 푋−푝 +푋 +(log ∆푋) ≃ 푅ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋), 휔푋). Here +the derived sheaf Hom 푅ℋ표푚푋 agrees with the regular sheaf Hom as Ω푝 +푋(log ∆푋)(−∆푋) is locally +free. On the other hand, the dualizing functor of 푋 is 푅ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋), 휔푋[dim 푋]) where +휔푋 = Ωdim 푋 +푋 +. An upshot is that Grothendieck duality calculations involving the sheaves of differen- +tial forms become more symmetric and predictable if we work with the shifted versions Ω푝 +푋(log ∆푋)(−∆푋)[푝]; +for example then we have the identity +Ωdim 푋−푝 +푋 +(log ∆푋)[dim푋 − 푝] ≃ 푅ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋]) +Now, we need to compactify 푓 ∶ 푋 → 푌. +Theorem 2.32 ([Nag63, §4 Thm. 2], [Con07, Thm. 4.1]). Let 푆 be a quasi-compact quasi-separated +scheme and let 푋 → 푆 be a separated morphism of finite type. Then there is a dense open immersion of +푆-schemes 푋 → 푋 such that 푋 is proper. +Using Theorem 2.32 we obtain morphisms of schemes +푋 +̄푋 +푌 +휄 +푓 +̄푓 +(2.33) +where 휄 ∶ 푋 → ̄푋 is a dense open immersion and ̄푓 ∶ ̄푋 → 푌 is proper. Note that ̄푋 need not be +smooth over 푘, and in the absence of resolutions of singularities5 there is not even a way to make ̄푋 +smooth. This means we cannot hope to upgrade ̄푋to a simple normal crossing pair ( ̄푋, ∆ ̄푋). However, +we do still have a divisor ∆ ̄푋 ∶= ̄푓∗∆푦 on ̄푋. One way to overcome these difficulties is to equip the +possibly singular ̄푋 with a logarithmic structure, in some sense associated to ∆ ̄푋, whose restriction +to 푋 coincides with a logarithmic structure naturally defined by the simple normal crossing divisor +∆푋. +Formally, we use the log structure on ̄푋 pulled back from the log structure on (푌, ∆푌) [Ogu18, +§III.1.6-7] along the morphism ̄푓 ∶ ̄푋 → 푌. Since (푌, ∆푌 = ∑푁 +푖=1 퐷푌 +푖 ) is a simple normal crossing +pair, its associated log structure is Deligne-Faltings [Ogu18, §III.1.7] and can be encoded in the se- +quence of inclusions of ideal sheaves 풪푌(−퐷푌 +푖 ) → 풪푌. The pullback log structure on ̄푋 can then be +encoded in the sequence of inclusions of ideal sheaves +̄푓−1풪푌(−퐷푌 +푖 ) ⋅ 풪 ̄푋 = 풪 ̄푋(− ̄푓∗퐷푌 +푖 ) → 풪 ̄푋. +The pushforward morphisms of Lemma 2.28 are defined using the sheaves of log differential 푝- +forms on ̄푋 over 푘 as described in [Ogu18, §IV.1, V.2] — these will be denoted6 by Ω푝 +푋(log ∆푋). The +essential properties that we need are: +• Ω푝 +푋(log ∆푋) is a coherent sheaf on 푋 together with a functorial morphism +Ω푝 +푌(log ∆푌) → 푓∗Ω푝 +푋(log ∆푋). +Coherence can be obtained as follows: first, the log structure on (푌, ∆푌) is coherent ([Ogu18, +§III.1.9]), and hence so is its pullback to ̄푋 (see for example [Ogu18, Def. III.1.1.5, Rmk III.1.1.6]). +Then [Ogu18, Cor. IV.1.2.8] implies Ω1 +푋(log ∆푋) is a coherent sheaf, and it follows that its 푝-th +exteriorpowersare coherentsheavesas well. The desiredfunctorial morphismcan be obtained +from [Ogu18, Prop. IV.1.2.15]. +5At the time of this writing, this applies to the cases char 푘 = 푝 > 0 and dim 푋 > 3. +6This is an abuse of notation since the construction of this sheaf is (as far as we know) not the same as the one for simple +normal crossing pairs described above Lemma 2.18, however the notation of [Ogu18] seems unsatisfactory for our purposes +as we wish to stress that these are not the ordinary differential forms Ω푝 +푋, +7 + +• There is a natural isomorphism Ω푝 +푋(log ∆푋)|푋 ≃ Ω푝 +푋(∆푋). This can be seen by observing that +the log structures on (푋, ∆푋) and ̄푋 are obtained as pullbacks of the log structure on (푌, ∆푌) +with respect to 푓 and ̄푓 respectively (in the case of (푋, ∆푋) this follows from Definition 2.14, +and in the latter case it is how we defined the log structure on ̄푋). Hence considering eq. (2.33) +we find that the log structure on ̄푋 restricts to that on (푋, ∆푋). +Hence in particular Ω푝 +푋(log ∆푋) is a functorial coherent extension of Ω푝 +푋(∆푋) to the possibly non-snc +log scheme ̄푋. Starting with the log differential +푑 pr∨ +푌 ∶ Ω푝 +푌(log ∆푌)[푝] → 푅푓∗Ω푝 +푋(log ∆푋)[푝], +twisting by −∆푌 and using the projection formula gives a morphism (note: this is where we use the +assumptions that 푓∗∆푌 = ∆푋 and ̄푓∗∆푌 = ∆ ̄푋) +Ω푝 +푌(log ∆푌)(−∆푌)[푝] → 푅푓∗Ω푝 +푋(log ∆푋)(−∆푋)[푝] +(2.34) +to which we apply Grothendieck duality: +Theorem 2.35 (Grothendieck duality, [R&D, Cor. VII.3.4], [Con00, Thm. 3.4.4]). Let 푓 ∶ 푋 → 푌 be a +proper morphism of finite-dimensional noetherian schemes and assume 푌 admits a dualizing complex +(for example 푋 and 푌 could be schemes of finite type over 푘). Then for any pair of objects ℱ∙ ∈ 퐷− +푞푐(푋) +and 풢∙ ∈ 퐷+ +푐 (푌) there is a natural isomorphism +푅푓∗푅퐻표푚푋(ℱ∙, 푓!풢∙) ≃ 푅퐻표푚푌(푅푓∗ℱ∙, 풢∙) in 퐷푏 +푐 (푌) +Combining Theorem 2.35 with eq. (2.34) gives a morphism +푅푓∗푅ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋)[푝], 휔∙ +푋) = 푅ℋ표푚푌(푅푓∗Ω푝 +푋(log ∆푋)(−∆푋)[푝], 휔푌[dim 푌]) +푅ℋ표푚푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌]) +(2.36) +where the equality is Theorem 2.35 and the vertical map is induced by (2.34). Adding supports gives +a morphism +푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω +푝 +푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋]) = 푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω +푝 +푋(log ∆푋)(−∆푋)[푝], 휔∙ +푋) +푅ΓΦ푌푅ℋ표푚푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌]) +(2.37) +where the equality is obtained from the excision property of local cohomology, compatibility of the +dualizing functor with restriction and the natural isomorphism Ω푝 +푋(log ∆푋)|푋 ≃ Ω푝 +푋(∆푋). Using +(2.31) we obtain +Ωdim 푋−푝 +푋 +(log ∆푋) ≃ ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋), 휔푋) = 푅ℋ표푚푋(Ω푝 +푋(log ∆푋)(−∆푋), 휔푋) +where the last equality uses the fact that Ω푝 +푋(log ∆푋)(−∆푋) is locally free. A similar calculation on +푌 transforms (2.37) into: +푅푓∗푅ΓΦ푋Ωdim 푋−푝 +푋 +(log ∆푋)[dim푋 − 푝] → 푅ΓΦ푌Ωdim 푌−푝 +푌 +(log ∆푌)[dim푌 − 푝] +and reindexing like 푝 ↔ dim 푋 − 푝 recovers Lemma 2.28. +8 + +3 +A base change formula +Lemma 3.1 (compare with [CR11, Prop. 2.3.7]). Let +(푋′, ∆푋′, Φ푋′) +(푋, ∆푋, Φ푋) +(푌′, ∆푌′, Φ푌′) +(푌, ∆푌, Φ푌) +□ +푔′ +푓′ +푓 +푔 +(3.2) +be a cartesian diagram of equidimensional snc pairs with supports, where 푓, 푓′ (resp. 푔, 푔′) are pushing +(resp. pulling) morphisms and 푔 is either flat or a closed immersion transverse to 푓. Then +푔∗푓∗ = 푓′ +∗푔′∗ ∶ 퐻∗(푋, ∆푋, Φ푋) → 퐻∗(푌′, ∆푌′, Φ푌′). +We will prove this following Chatzistamatiou and Rülling’s argument [CR11, Prop. 2.3.7] quite +closely, at various points reducing to statements proved therein. In the proofs we will make use of a +slight variant of Definition 2.3. +Definition 3.3. If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and let Φ푌 is a family of +supports on 푌, then +푓−1 +∗ (Φ푌) ∶= {푍 ⊆ 푋 | 푓|푍 is proper and 푓(푍) ∈ Φ푌} +Lemma 3.4. It suffices to prove Lemma 3.1 in the cases where 푓 is either +(푖) a projection morphism of the form pr푌 ∶ (푋 × 푌, pr∗ +푌∆푌, pr−1 +푌∗(Φ푌)) → (푌, ∆푌, Φ푌), or +(푖푖) a closed immersion. +Remark 3.5. This lemma makes essential use of the functoriality part of Lemma 2.28. +Proof. We can decompose (3.2) as a concatenation of cartisian diagrams +(푋′, ∆푋′, Φ푋′) +(푋, ∆푋, Φ푋) +(푋 × 푌′, pr∗ +푌′∆푌, pr−1 +푌′∗(Φ′ +푌)) +(푋 × 푌, pr∗ +푌∆푌, pr−1 +푌∗(Φ푌)) +(푌′, ∆푌′, Φ푌′) +(푌, ∆푌, Φ푌) +(2) +푔′ +ℎ′ +ℎ +(1) +pr푌′ +id×푔 +pr푌 +푔 +(3.6) +where ℎ = id × 푓 is the graph morphism of 푓 and ℎ′ = 푔′ × 푓′. If 푔 is flat or a closed immersion +transverse to 푓 then id × 푔 is flat or a closed immersion transverse to ℎ (by base change). +Here the only new feature not covered in [CR11, Prop. 2.3.7] is the presence of divisors, and we +simply note that ∆푋 = 푓∗∆푋 = ℎ∗pr∗ +푌∆푌 and similarly for ∆푋′, so that both pr푌 and ℎ are pushing +morphisms in the sense of Definition 2.14, and similarly for the left vertical maps. In other words, the +supports and divisors in the middle row have been chosen precisely so that the vertical morphisms +are all “pushing.” +We proceed to consider case (i), and wish to point out that for this case 푔 can be arbitrary (we will +need the flatness/transversality restrictions in case (ii)). In what follows we set 푑푋 = dim 푋, 푑푌 = +dim 푌 and similarly for 푋′, 푌′. Using Theorem 2.32 we obtain a compactification 휄 ∶ 푋 → 푋 over +푘 of the smooth, separated and finite type 푘-scheme 푋 in the upper right corner of (3.2) and (3.6). +9 + +This results in a compactification of the square (1) in (3.6) which we write as +(푋 × 푌′, pr∗ +푌′∆푌, pr−1 +푌′∗(Φ′ +푌)) +(푋 × 푌, pr∗ +푌∆푌, pr−1 +푌∗(Φ푌)) +(푋 × 푌′, pr +∗ +푌′∆푌, pr +−1 +푌′∗(Φ′ +푌)) +(푋 × 푌, pr +∗ +푌∆푌, pr +−1 +푌∗(Φ푌)) +(푌′, ∆푌′, Φ푌′) +(푌, ∆푌, Φ푌) +휄×id +id×푔 +휄×id +pr푌′ +id×푔 +pr푌 +푔 +(3.7) +By the description following Lemma 2.28, we know that +pr푌∗ ∶ 퐻∗(푋 × 푌, pr∗ +푌∆푌, pr−1 +푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌) +stems from a morphism +푅pr푌∗푅ℋ표푚푋×푌(Ω푝 +푋×푌(log pr∗ +푌∆푌)(−pr∗ +푌∆푌)[푝], 휔∙ +푋×푌) → Ω푑푌−푝 +푌 +(log ∆푌)[푑푌 − 푝] +(3.8) +obtained as the Grothendieck dual of a log differential of pr푌 (here and throughout what follows, a +similar statement holds for pr푌′). By an observation of Chatzistamatiou-Rülling , this map factors +as +푅pr푌∗푅ℋ표푚푋×푌(Ω푝 +푋×푌(log pr +∗ +푌∆푌)(−pr +∗ +푌∆푌)[푝], 휔∙ +푋×푌) +→ 푅pr푌∗푅ℋ표푚푋×푌(퐿pr +∗ +푌Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔∙ +푋×푌) +≃ +���������→ +adjunction 푅ℋ표푚푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 푅pr푌∗휔∙ +푋×푌) +����→ +trace 푅ℋ표푚푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔푌[푑푌]) +≃�→ Ω푑푌−푝 +푌 +(log ∆푌)[푑푌 − 푝] +(3.9) +where the adjunction isomorphism is [R&D, Prop. II.5.10], and the map labeled trace is induced by +the Grothendieck trace 푅pr푌∗휔∙ +푋×푌 → 휔푌[푑푌]. If it were the case that 푋 were smooth, then the +usual “box product” decomposition +휔∙ +푋×푌 ≃ 휔푋[푑푋] ⊠ 휔푌[푑푌] ∶= pr∗ +푋 휔푋[푑푋] ⊗ pr푌∗휔푌[푑푌] +together with the perect pairings (2.31) and the local freeness of Ω푝 +푌(log ∆푌)(−∆푌)[푝] would give an +identification +푅ℋ표푚푋×푌(퐿pr +∗ +푌Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔∙ +푋×푌) ≃ pr∗ +푋 휔푋[푑푋] ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)[푑푌 − 푝] (3.10) +In fact a more careful version of this argument, carrying out the above calculation on the smooth +locus 푋 × 푌 and using excision, shows that 퐻∗(푋 × 푌, pr∗ +푌∆푌, pr−1 +푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌) always +factors through the summand 퐻∗ +Φ푋(푋 × 푌, pr∗ +푋 휔푋 ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)). +Our next lemma implies that even when 푋 is not known to be smooth, (3.8) still factors through +something like 푅pr푌∗(pr∗ +푋 휔푋[푑푋] ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)[푑푌 − 푝]), provided we replace pr∗ +푋 휔푋[푑푋] +with pr +! +푌풪푌. +Lemma 3.11 (compare with [CR11, Lem. 2.2.16]). For each 푝 there is a natural map +훾 ∶ pr +! +푌풪푌 ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] → 푅ℋ표푚푋×푌(pr +∗ +푌Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔∙ +푋×푌) +10 + +such that the restriction of 훾 to 푋 × 푌 agrees with the isomorphism +pr∗ +푋 휔푋[푑푋] ⊗ pr∗ +푌 Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +≃�→ 푅ℋ표푚푋×푌(퐿 pr∗ +푌 Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔∙ +푋×푌) +and such that the composition +푅pr푌∗(pr∗ +푋 휔푋[푑푋] ⊗ pr∗ +푌 Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝]) +푅pr푌∗(훾) +��������→ 푅pr푌∗푅ℋ표푚푋×푌(pr∗ +푌 Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔∙ +푋×푌) +≃ +���������→ +adjunction 푅ℋ표푚푋×푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 푅pr푌∗휔∙ +푋×푌) +trace +����→ 푅ℋ표푚푋×푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝], 휔푌[푑푌]) ≃ Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +(3.12) +coincides with the composition +푅pr푌∗(pr +! +푌풪푌 ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝]) +proj. +����→ +form. 푅pr푌∗(pr +! +푌풪푌) ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +tr ⊗id +�����→ Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +(3.13) +By base change for dualizing complexes ([Stacks, Tag 0BZX, Tag 0E2S]) applied to the cartesian +diagram +푋 × 푌 +푋 +푌 +Spec 푘 +(note that this is a very mild situation: 푋 → Spec 푘 is flat and proper and 푌 → Spec 푘 is smooth) we +see that pr +! +푌풪푌 ≃ pr∗ +푋 휔∙ +푋. This makes the map 훾 look even more like (3.10). +Proof. Following [CR11, Lem. 2.2.16] we begin with the morphism +푒 ∶ pr +! +푌풪푌 ⊗퐿 퐿pr +∗ +푌휔∙ +푌 → pr +! +푌휔∙ +푌 =∶ 휔∙ +푋×푌 +of [Con00, p. 4.3.12], which as explained therein agrees with +pr∗ +푋 휔푋[푑푋] ⊗ pr∗ +푌 휔푌[푑푌] +≃�→ 휔푋×푌[푑푋 + 푑푌] +on locus 푋 × 푌,7 and has the property that +푅푝푟푌∗(pr +! +푌풪푌 ⊗퐿 퐿pr +∗ +푌휔∙ +푌) +푅푝푟푌∗휔∙ +푋×푌 +푅푝푟푌∗pr +! +푌풪푌 ⊗퐿 휔∙ +푌 +휔∙ +푌 +푅푝푟푌∗푒 +proj. form +tr +tr ⊗id +7See Conrad’s comment “It is easy to check that 푒푓 coincides with (3.3.21) in the smooth case and is compatible with +composites in f (using (4.3.6).” +11 + +commutes [Con00, Thm. 4.4.1]. We then define our version of 훾 as the composition +pr +! +푌풪푌 ⊗퐿 퐿pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +id⊗퐿(2.31) +���������→ pr +! +푌풪푌 ⊗퐿 퐿pr +∗ +푌푅ℋ표푚푌(Ω푝 +푌(log ∆푌)[푝], 휔∙ +푌) +functoriality +�����������→ +of 퐿pr +∗ +푌,⊗퐿 +푅ℋ표푚푋×푌(퐿pr +∗ +푌Ω푝 +푌(log ∆푌)[푝], pr +! +푌풪푌 ⊗퐿 휔∙ +푌) +induced by +���������→ +푒 +푅ℋ표푚푋×푌(퐿pr +∗ +푌Ω푝 +푌(log ∆푌)[푝], 휔∙ +푋×푌) +(3.14) +Note that we may drop the “퐿”s as Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌) and Ω푝 +푌(log ∆푌) are locally free. Verification +of the stated compatibilities is as in [CR11, Lem. 2.2.16]. +Remark 3.15. It seems like we could have also used the more general version of [Con00, p. 4.3.12] +푒′ ∶ pr +! +푌풪푌 ⊗퐿 퐿pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] → pr +! +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +together with the description +pr +! +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] = 퐷푋×푌(퐿pr +∗ +푌퐷푌(Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝])) +where 퐷푌(−) = 푅ℋ표푚(−, 휔∙ +푌) and similarly for 퐷푋×푌. +Using this modified 훾, we obtain a modified version of the diagram [CR11, p. 732 during Lem. +2.3.4], namely (3.16) in Figure 1). To make this diagram legible, we use a few abbreviations: all func- +tors are derived, we use the dualizing functors of the form 퐷푌(−) = 푅ℋ표푚푌(−, 휔∙ +푌) and we let +푑 = 푑푋 + 푑푌. Lemma 3.11 shows that triangles involving 훾 commute, and (3.9) gives commutativity +of the rest of the diagram. The usefulness of this diagram is that by definition beginning in the top +left corner and following the path →↓ we obtain the pushforward on Hodge cohomology +pr푌∗ Γpr−1 +푌∗ Φ푌Ω푑−푝 +푋×푌(log pr∗ +푌∆푌)[푑 − 푝] → ΓΦ푌Ω푑푌−푝 +×푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +but following ↓→ gives a composition whose behavior with respect to (3.7) is easier to analyze. +Namely, we have a diagram like (3.16) on 푌′, and in fact a map from (3.16) to 푔∗ of the analogous +diagram on 푌′, and hence from the preceding discussion it will suffice to prove commutativity of +(3.17) of Figure 1. +Applying excision together with Lemma 3.11 we may rewrite the top row of (3.17) as +푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌Ω푑−푝 +푋×푌(log pr∗ +푌∆푌)[푑 − 푝] +project +������→ 푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌(pr∗ +푋 휔푋[푑푋] ⊗ pr∗ +푌 Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝]) +proj. +�����→ +form. 푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌(pr∗ +푋 휔푋[푑푋]) ⊗ Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +tr ⊗id +�����→ 푅ΓΦ푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +(3.18) +where the first map is induced by a projection +Ω푑−푝 +푋×푌(log pr∗ +푌∆푌)[푑 − 푝] → pr∗ +푋 휔푋[푑푋] ⊗ pr∗ +푌 Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +coming from a Künneth-type decomposition of Ω푑−푝 +푋×푌(log pr∗ +푌∆푌), the second is the projection for- +mula, and the last map is induced by a trace map with supports defined as the composition +푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌(pr∗ +푋 휔푋[푑푋]) +excision +�������→ 푅pr푌∗푅Γpr−1 +푌∗Φ푌(pr +! +푌풪푌) +Proposition 2.9 +�������������→ 푅ΓΦ푌푅pr푌∗(pr +! +푌풪푌) +tr�→ 푅ΓΦ푌풪푌 +(3.19) +12 + +pr푌∗ Γpr−1 +푌∗ Φ푌 Ω푑−푝 +푋×푌(log pr∗ +푌∆푌)[푑 − 푝] +pr푌∗Γpr−1 +푌∗Φ푌퐷푋×푌(Ω푝 +푋×푌(log pr∗ +푌∆푌)(−pr∗ +푌∆푌)[푝]) +pr푌∗Γpr−1 +푌∗Φ푌퐷푋×푌(pr +∗ +푌Ω푝 +×푌(log ∆푌)(−∆푌)[푝]) +pr푌∗Γpr−1 +푌∗Φ푌(pr +! +푌풪푌 ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝]) +ΓΦ푌 퐷푌(Ω푝 +푌(log ∆푌)(−∆푌)[푝]) = ΓΦ푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +excision +excision+퐿푒푚푚푎 3.11 +푑pr∨ +푌 +푑pr∨ +푌 +(3.13) +pr푌∗(훾) +(3.16) +pr푌∗ Γpr−1 +푌∗Φ푌Ω푑−푝 +푋×푌(log pr∗ +푌∆푌)[푑 − 푝] +pr푌∗Γpr−1 +푌∗Φ푌(pr +! +푌풪푌 ⊗ pr +∗ +푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝]) +ΓΦ푌Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +푔∗ pr푌′∗ Γpr−1 +푌′∗Φ푌′ Ω푑−푝 +푋×푌′(log pr∗ +푌′∆푌′)[푑 − 푝] +푔∗pr푌′∗Γpr−1 +푌′∗Φ푌′ (pr +! +푌′풪푌′ ⊗ pr +∗ +푌′Ω푑푌−푝 +푌′ +(log ∆푌′)(−∆푌)[푑푌 − 푝]) +푔∗ΓΦ푌′ Ω푑푌−푝 +푌′ +(log ∆푌′)(−∆푌′)[푑푌 − 푝] +(3.17) +Figure 1: Modified versions of diagrams appearing in the proof of [CR11, Lem. 2.3.4] (all functors derived) +13 + +Here the second map comes from the functoriality properties of Proposition 2.9, since there is an +inclusion pr−1 +푌∗ Φ푌 ⊆ pr−1 +푌 Φ푌. The decomposition (3.18) maps to a similar decomposition of the bot- +tom row of (3.17), and the only commutativity not guaranteed by standard functoriality properties +(e.g. functoriality of the projection formula appearing in the second map of (3.18)) is that of +푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌(pr∗ +푋 휔푋[푑푋]) ⊗ Ω +푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +푅ΓΦ푌Ω +푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] +푅푔∗(푅 pr푌′∗ 푅Γpr−1 +푌′∗ Φ푌′ (pr∗ +푋 휔푋[푑푋]) ⊗ Ω푑푌−푝 +푌′ +(log ∆푌′)(−∆푌′)[푑푌 − 푝]) +푅푔∗(푅ΓΦ푌′ Ω푑푌−푝 +푌′ +(log ∆푌′)(−∆푌′)[푑푌 − 푝]) +tr ⊗id +tr′ ⊗id +(3.20) +But applying one more projection formula to the bottom row of (3.20), we see (3.20) is obtained by +tensoring the differential +Ω푑푌−푝 +푌 +(log ∆푌)(−∆푌)[푑푌 − 푝] → 푅푔∗Ω푑푌−푝 +푌′ +(log ∆푌′)(−∆푌′)[푑푌 − 푝] +with +푅 pr푌∗ 푅Γpr−1 +푌∗ Φ푌(pr∗ +푋 휔푋[푑푋]) +푅ΓΦ푌풪푌 +푅푔∗(푅 pr푌′∗ 푅Γpr−1 +푌′∗ Φ푌′ (pr∗ +푋 휔푋[푑푋])) +푅푔∗(푅ΓΦ푌′ 풪푌′) +tr ⊗id +tr′ ⊗id +(3.21) +and the commutativity of (3.21) is proved in [CR11, Lem. 2.3.4]. So far we have proved: +Lemma 3.22. Lemma 3.1 holds in case (i) of Lemma 3.4. +It remains to deal with case (ii) of Lemma 3.4, and for this we use the following lemma. +Lemma 3.23 (compare with [CR11, Cor. 2.2.22]). Consider a diagram of pure-dimensional snc pairs +(푋′, ∆푋′) +(푋, ∆푋) +(푌′, ∆푌′) +(푌, ∆푌) +푔′ +횤′ +횤 +푔 +(3.24) +where 횤, 횤′ are pushing closed immersions and dim푌 − dim 푋 = dim 푌′ − dim 푋′ =∶ 푐. Then, for all +푞 the diagram +횤∗Ω푞 +푋(log ∆푋)[푞] +푅푔∗횤′ +∗Ω푞 +푋′(log ∆푋′) +Ω푞+푐 +푌 +(log ∆푌)[푞 + 푐] +푅푔∗Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐] +푑푔′∨ +푑푔∨ +(3.25) +commutes, where the horizontal maps are induced by log differentials and the left vertical map is the +composition +횤∗Ω푞 +푋(log ∆푋)[푞] +≃�→ 횤∗푅ℋ표푚(Ω푑푋−푞 +푋 +(log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙ +푋) +duality +������→ 푅ℋ표푚(횤∗Ω푑푋−푞 +푋 +(log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙ +푌) +푑횤∨ +���→ 푅ℋ표푚(Ω푑푋−푞 +푌 +(log ∆푌)(−∆푌)[푑푋 − 푞], 휔∙ +푌) +≃�→ Ω푞+푐 +푌 +(log ∆푌)[푞 + 푐] +(3.26) +and the right vertical arrow is 푅푔∗ of a similar composition on 푌′. +14 + +Note that the codimension hypotheses hold if 푔 is flat or a closed immersion transverse to 횤. +Proof. While it seems a proof following [CR11, Cor. 2.2.22] step-by-step is possible, we instead reduce +to the case proved there as follows: first, observe that there is an evident map from the cartesian +diagram +푈푋′ +푈푋 +푈푌′ +푈푌 +(3.27) +of interiors to (3.24). Noting that (3.25) will map to a similar diagram obtained from (3.27), that the +compositions (3.26) are at least compatible with Zariski localization, and that the situation of (3.27) +is covered by [CR11, Cor. 2.2.22], it will suffice to show that the natural map +ℎ0푅ℋ표푚푌(횤∗Ω푞 +푋(log ∆푋)[푞], 푅푔∗Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) → ℎ0푅ℋ표푚푈푌(횤∗Ω푞 +푈푋[푞], 푅푔∗Ω푞+푐 +푈푌′ [푞 + 푐]) +(3.28) +is injective. This can be checked Zariski-locally at a point 푥 ∈ 푋 ⊆ 푌, so we may assume 푋 ⊆ 푌 +is a global complete intersection, say of 푡1, … , 푡푐 ∈ 풪푌. In that case the 푡푖 define a Koszul resolu- +tion 풦∙(푡푖) → 풪푋, and because 푋′ = 푌′ ×푌 푋 = 푉(푡1◦푔, ⋯ 푡푐◦푔) is smooth of codimension 푐 by +hypotheses, it must be that the 푡푖◦푔 are also a regular sequence, hence +퐿푖푔∗풪푋 = ℎ−푖푔∗풦∙(푡푖) = {풪푋′, +푖 = 0 +0 +otherwise +in other words 퐿푔∗풪푋 = 풪푋′. Now using the fact that Ω푞 +푋(log ∆푋) is locally free on 푋′ we conclude +퐿푔∗횤∗Ω푞 +푋(log ∆푋)[푞] = 푔∗횤∗Ω푞 +푋(log ∆푋)[푞] = 횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞] +Next, applying derived adjunction to both sides of (3.28) gives a commutative diagram +푅ℋ표푚푌(횤∗Ω +푞 +푋(log ∆푋)[푞], 푅푔∗Ω +푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) +푅ℋ표푚푈푌(횤∗Ω +푞 +푈푋[푞], 푅푔∗Ω +푞+푐 +푈푌′ [푞 + 푐]) +푅푔∗푅ℋ표푚푌′(퐿푔∗횤∗Ω +푞 +푋(log ∆푋)[푞], Ω +푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) +푅푔∗푅ℋ표푚푈푌′ (퐿푔∗횤∗Ω +푞 +푈푋[푞], Ω +푞+푐 +푈푌′ [푞 + 푐]) +푅푔∗푅ℋ표푚푌′(횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞], Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) +푅푔∗푅ℋ표푚푈푌′ (횤′ +∗푔′∗Ω푞 +푈푋[푞], Ω푞+푐 +푈푌′ [푞 + 푐]) +(3.29) +Getting even more Zariski-local we may assume Ω푞 +푋(log ∆푋) is free, say generated by 푑푥1, … , 푑푥푛 +and in that case +푅ℋ표푚푌′(횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞], Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) += ( +∏ +푖 +푅ℋ표푚푌′(풪푋′푑푥푖[푞], 풪푌′[푞 + 푐])) ⊗ Ω푞+푐 +푌′ (log ∆푌′) +(3.30) +and by Grothendieck’s fundamental local isomorphism [Con00, §2.5] +푅ℋ표푚푌′(풪푋′[푞], 풪푌′[푞 + 푐])) ≃ ℰ푥푡푐 +푌′(풪푋′, 풪푌′) ≃ det(ℐ푋′∕ℐ푋′)∨ +(3.31) +(the last 2 as sheaves supported in degree 0). In particular, this is an invertible sheaf on 푋′, and it +follows that the left hand side of (3.30) is a locally free sheaf (supported in degree 0) on 푋′. Recalling +푋′ is smooth and so in particular reduced, and since 푈푌′ ∩ 푋′ is a dense open (this is part of the +15 + +hypothesis that 푋′ → 푌′ is a pulling map) the natural map +ℎ0푅ℋ표푚푌′(횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞], Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) +→ ℎ0푅ℋ표푚푌′(횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞], Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐])|푈푌′ +≃ ℎ0푅ℋ표푚푈푌′ (횤′ +∗푔′∗Ω푞 +푋(log ∆푋)|푈푌′ [푞], Ω푞+푐 +푌′ (log ∆푌′)|푈푌′ [푞 + 푐]) +(3.32) +is injective, where on the third line we have applied localization for ℰ푥푡. Now left-exactness of 푔∗ +gives an injection +ℎ0푅푔∗푅ℋ표푚푌′(횤′ +∗푔′∗Ω푞 +푋(log ∆푋)[푞], Ω푞+푐 +푌′ (log ∆푌′)[푞 + 푐]) +→ ℎ0푅푔∗푅ℋ표푚푈푌′ (횤′ +∗푔′∗Ω푞 +푋(log ∆푋)|푈푌′ [푞], Ω푞+푐 +푌′ (log ∆푌′)|푈푌′ [푞 + 푐]) +(3.33) +To complete the proof, we use (3.29) to identify the map (3.33) with (3.28). +Corollary 3.34. Lemma 3.1 holds in case (ii) of Lemma 3.4. +Proof. This follows by applying cohomology with supports to (3.25). +This completes our proof of Lemma 3.1. +Corollary 3.35 (projection formula, compare with [CR11, Prop. 1.1.16]). Let 푓 ∶ 푋 → 푌 be a map +of smooth schemes admitting two different enhancements to maps of smooth schemes with supports, +(푋, ∆푋, Φ푋) → (푌, ∆푌, 푓(Φ푋)) pushing and (푋, 푓∗(∆′ +푌), 푓−1(Φ푌)) → (푌, ∆′ +푌, Φ푌) pulling +Assume in addition that ∆푋 + 푓∗(∆′ +푌) and ∆푌 + ∆′ +푌 are (reduced) snc divisors. Then +(푋, ∆푋 + 푓∗(∆′ +푌), Φ푋 ∩ 푓−1(Φ푌)) → (푌, ∆푌 + ∆′ +푌, 푓(Φ푋) ∩ Φ푌) +is also a pushing map, and +푓∗(훽 ⌣ 푓∗훼) = 푓∗훽 ⌣ 훼 ∈ 퐻∗(푌, ∆푌 + ∆′ +푌, 푓(Φ푋) ∩ Φ푌) +for any 훼 ∈ 퐻∗(푌, ∆′ +푌, Φ푌) and 훽 ∈ (푋, ∆푋, Φ푋), where ⌣ is the cup product on log Hodge cohomology +defined along the lines of [CR11, §1.1.4, 2.4] +Proof. This is a formal consequence of Lemma 3.1 and can be derived following the proof of [CR11, +Prop. 1.1.16]. Again we use a factorization through the graph like +(푋, ∆푋 + 푓∗(∆′ +푌), Φ푋 ∩ 푓−1(Φ푌)) +(푌, ∆푌 + ∆′ +푌, 푓(Φ푋) ∩ Φ푌) +(푋 × 푋, pr∗ +1 ∆푋 + pr∗ +2 푓∗(∆′ +푌), Φ푋 × 푓−1(Φ푌)) +(푋 × 푌, pr∗ +1 ∆푋 + pr∗ +2 ∆′ +푌, Φ푋 × Φ푌) +(푌 × 푌, pr∗ +1 ∆푌 + pr∗ +2 ∆′ +푌, 푓(Φ푋) × Φ푌) +id푋×id푋 +푓 +id푌×id푌 +id푋×푓 +푓×id푌 +(3.36) +Here 푓 × id푌 on the bottom is a pushing morphism (since 푓|Φ푋 is proper and 푓∗∆푌 = ∆푋) and the +right vertical map id푌 × id푌 is a closed immersion transverse to 푓 × id푌 since the outer rectangle +is cartesian and 푋 is smooth of the correct codimension. This means we are in a situation to apply +Lemma 3.1, and that lemma plus the definition of cup products in terms of pullbacks along diagonals +gives the desired identity. +16 + +4 +Correspondences +Given snc pairs with familes of supports (푋, ∆푋, Φ푋) and (푌, ∆푌, Φ푌) with dimensions 푑푋 and 푑푌, +as in [CR11, §1.3] we may define a family of supports 푃(Φ푋, Φ푌) on 푋 × 푌 by +푃(Φ푋, Φ푌) ∶= {closed subsets 푍 ⊆ 푋 × 푌 | pr푌|푍 is proper and for all 푊 ∈ Φ푋, +pr푌(pr−1 +푋 (푊) ∩ 푍) ∈ Φ푌} +(the conditions of Definition 2.1 are straightforward to verify). For convenience we will let ∆푋×푌 ∶= +pr∗ +푋∆푋 + pr∗ +푌∆푌. +Theorem 4.1. A class 훾 ∈ 퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) defines homomorphisms +cor(훾) ∶ 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞+푗−푑푋 +Φ푌 +(푌, Ω푝+푖−푑푋 +푌 +(log ∆푌)) +by the formula cor(훾)(훼) ∶= pr푌∗(pr∗ +푋(훼) ⌣ 훾). Moreover if (푍, ∆푍, Φ푍) is another snc pair with +supports and 훿 ∈ 퐻푗′ +푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ +푌×푍(log ∆푌×푍)(−pr∗ +푌∆푌)), then +pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿)) ∈ 퐻푗+푗′−푑푌 +푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌 +푋×푍 +(log ∆푋×푍)(−pr∗ +푋∆푋)) and +cor(pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿))) = cor(훿)◦ cor(훾) +as homomorphisms 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌 +Φ푍 +(푍, Ω푝+푖+푖′−푑푋−푑푌 +푍 +(log ∆푍)). +Remark 4.2. The sheavesΩ푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋) are particular instancesof the sheavesΩ푖 +푋(퐴, 퐵) +appearing in [DI87, §4.2]. +Such correspondences involving both log poles and “log zeroes” appear to have been considered +before at least in crystalline cohomology, for example in work of Mieda [Mie09a; Mie09b]. However, +I was unable to find any published proof of Theorem 4.1 in the literature. +Proof. We make two observations: first, using Lemma 2.18 there are natural wedge product pairings +Ω푝 +푋×푌(log ∆푋×푌) ⊗ Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋) +∧�→ Ω푝+푖 +푋×푌(log ∆푌) +Second, essentially by the definition of 푃(Φ푋, Φ푌) the Künneth morphism on cohomology for the +tensor product Ω푝 +푋×푌(log ∆푋×푌) ⊗ Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋) can be enhanced with supports as +퐻푞 +pr−1 +푋 (Φ푋)(푋 × 푌, Ω푝 +푋×푌(log ∆푋×푌)) ⊗ 퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) +→ 퐻푝+푗 +Ψ +(푋 × 푌, Ω푝 +푋×푌(log ∆푋×푌) ⊗ Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) +where Ψ ∶= pr−1 +푌∗(Φ푍) (see [CR11, §1.3.7, Prop. 1.3.10]). Combining these 2 observations gives a +pairing +퐻푞 +pr−1 +푋 (Φ푋)(푋 × 푌, Ω푝 +푋×푌(log ∆푋×푌)) ⊗ 퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) +⌣ +��→ 퐻푝+푗 +Ψ +(푋 × 푌, Ω푝+푖 +푋×푌(log ∆푌)) +Now note that pr푋 ∶ (푋×푌, ∆푋×푌, pr−1 +푋 (Φ푋)) → (푋, ∆푋, Φ푋) is a pulling morphism, so by Proposition 2.24 +there is an induced map pr∗ +푋 ∶ 퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) → 퐻푞 +pr−1 +푋 (Φ푋)(푋 × 푌, Ω푝 +푋×푌(log ∆푋×푌)). On the +other hand since pr푌 ∶ (푋 × 푌, ∆푌, Ψ) → (푌, ∆푌, Φ푌) is a pushing morphism, Lemma 2.28 provides +17 + +a morphism pr푌∗ ∶ 퐻푝+푗 +Ψ +(푋 × 푌, Ω푝+푖 +푋×푌(log ∆푌)) → 퐻푞+푗−푑푋 +Φ푌 +(푌, Ω푝+푖−푑푋 +푌 +(log ∆푌)). Composing, we +obtain the desired homomorphism +퐻푞 +Φ푋(푋, Ω푝 +푋(log ∆푋)) +pr∗ +푋 +���→ 퐻푞 +pr−1 +푋 (Φ푋)(푋 × 푌, Ω푝 +푋×푌(log ∆푋×푌)) +⌣훾 +���→ 퐻푝+푗 +Ψ +(푋 × 푌, Ω푝+푖 +푋×푌(log ∆푌)) +pr푌∗ +����→ 퐻푞+푗−푑푋 +Φ푌 +(푌, Ω푝+푖−푑푋 +푌 +(log ∆푌)) +For the “moreover” half of the lemma, we again begin with a certain wedge product pairing, this +time on 푋 × 푌 × 푍: +Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌)(−pr∗ +푋∆푋) ⊗ Ω푖′ +푋×푌×푍(log pr∗ +푌×푍∆푌×푍)(−pr∗ +푌∆푌) +∧�→ Ω푖+푖′ +푋×푌×푍(log pr∗ +푋×푍∆푋×푍)(−pr∗ +푋∆푋) +(4.3) +If 푉 ∈ 푃(Φ푋, Φ푌), 푊 ∈ 푃(Φ푌, Φ푍) then unravelling definitions (again we refer to [CR11, §1.3.7, +Prop. 1.3.10] for a similar claim) we find: +• pr푋×푍|pr−1 +푋×푌(푉)∩pr−1 +푌×푍(푊) is proper and +• pr푋×푍(pr−1 +푋×푌(푉) ∩ pr−1 +푌×푍(푊)) ∈ 푃(Φ푋, Φ푍) +so that the Künneth morphism on cohomology associated to the left hand side of (4.3) can be en- +hanced with supports like +퐻푗 +pr−1 +푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌)(−pr∗ +푋∆푋)) +⊗ 퐻푗′ +pr−1 +푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′ +푋×푌×푍(log pr∗ +푌×푍∆푌×푍)(−pr∗ +푌∆푌)) +→ 퐻푗+푗′ +Σ +(푋 × 푌 × 푍, Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌)(−pr∗ +푋∆푋) ⊗ Ω푖′ +푋×푌×푍(log pr∗ +푌×푍∆푌×푍)(−pr∗ +푌∆푌)) +where Σ ∶= pr−1 +푋×푍∗(푃(Φ푋, Φ푍)). +Since pr푋×푌 ∶ (푋 × 푌 × 푍, pr∗ +푋×푌∆푋×푌, pr−1 +푋×푌(푃(Φ푋, Φ푌))) → (푋 × 푌, ∆푋×푌, 푃(Φ푋, Φ푌)) is a +pulling morphism, Proposition 2.24 gives an induced morphism +Ω푖 +푋×푌(log ∆푋×푌) → 푅푓∗Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌); +twisting by −∆푋×푌 and applying the projection formula gives a morphism +Ω푖 +푋×푌(log ∆푋×푌)(−∆푋×푌) → 푅푓∗ +(Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌)(−pr∗ +푋×푌∆푋×푌)) +and then taking cohomology with supports along 푃(Φ푋, Φ푌) and using Proposition 2.9 gives a mod- +ified pullback map +퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−∆푋×푌)) +→ 퐻푗 +pr−1 +푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖 +푋×푌×푍(log pr∗ +푋×푌∆푋×푌)(−pr∗ +푋∆푋)) +(4.4) +and a similar argument gives a modified pullback +퐻푗′ +푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ +푌×푍(log ∆푌×푍)(−∆푌×푍)) +→ 퐻푗′ +pr−1 +푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′ +푋×푌×푍(log pr∗ +푌×푍∆푌×푍)(−pr∗ +푋∆푌)) +(4.5) +On the other hand, pr푋×푍 ∶ (푋 × 푌 × 푍, pr∗ +푋×푍∆푋×푌, Σ) → (푋 × 푍, ∆푋×푍, 푃(Φ푋, Φ푍)) is a pushing +morphism and hence by Lemma 2.28 induces morphisms +푅pr푋×푍∗푅ΓΣ(Ωdim푋×푌×푍−푘 +푋×푌×푍 +(log pr∗ +푋×푍∆푋×푌)) → 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘 +푋×푍 +(log ∆푋×푍)[− dim푍] +18 + +for all 푘; twisting by −pr∗ +푋∆푋 and applying the projection formula this becomes +푅pr푋×푍∗푅ΓΣ(Ωdim 푋×푌×푍−푘 +푋×푌×푍 +(log pr∗ +푋×푍∆푋×푌)(−pr∗ +푋∆푋)) +→ 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘 +푋×푍 +(log ∆푋×푍)(−pr∗ +푋∆푋)[− dim푍] +(4.6) +Now letting 푘 = dim 푋 × 푌 × 푍 − 푖 − 푖′, the induced morphisms of cohomology with supports are +퐻푗+푗′ +Σ +(푋 × 푌 × 푍, Ω푖+푖′ +푋×푌×푍(log pr∗ +푋×푍∆푋×푌)(−pr∗ +푋∆푋)) +→ 퐻푗+푗′−dim푍 +푃(Φ푋,Φ푍) +(푋 × 푍, Ω푖+푖′−dim 푍 +푋×푍 +(log ∆푋×푍)(−pr∗ +푋∆푋)) +(4.7) +Combining the above ingredients, we obtain a bilinear pairing +퐻푗 +푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 +푋×푌(log ∆푋×푌)(−∆푋×푌)) ⊗ 퐻푗′ +푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ +푌×푍(log ∆푌×푍)(−∆푌×푍)) +→ 퐻푗+푗′−dim 푍 +푃(Φ푋,Φ푍) +(푋 × 푍, Ω푖+푖′−dim푍 +푋×푍 +(log ∆푋×푍)(−pr∗ +푋∆푋)) +sending 훾 ⊗ 훿 ↦→ pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿)). It remains to be seen that +cor(pr푋×푍∗(pr∗ +푋×푌(훾) ⌣ pr∗ +푌×푍(훿))) = cor(훿)◦ cor(훾) +and for this we will make repeated use of Lemma 3.1. Consider the diagram of smooth schemes +푋 × 푌 × 푍 +푋 × 푌 +푌 × 푍 +푋 +푌 +푍 +∗ +where all morphisms are projections. There are various ways to enhance this to include supports; +here we add the family of supports Ψ on 푋 × 푌 defined above. Then in the cartesian diagram (∗), +pr푌 ∶ (푋 × 푌, Ψ) → (푌, Φ푌) and pr푌×푍 ∶ (푋 × 푌 × 푍, pr−1 +푋×푌Ψ) → (푌 × 푍, pr−1 +푌 Φ푌) are pushing +morphisms, whereas pr푋×푌 and pr푌 are pulling morphisms. At the same time, we have a pulling +morphism pr푋×푍 ∶ (푋 × 푌 × 푍, pr−1 +푋×푍(푃(Φ푌, Φ푍))) → (푌 × 푍, 푃(Φ푌, Φ푍)). To be precise in what +follows, whenever ambiguity is possible we will use notation like pr푋×푌 +푋 +to denote the projection +푋 × 푌 → 푋, pr푋×푌×푍 +푋 +to denote the projection 푋 × 푌 × 푍 → 푋 and so on. +Applying Corollary 3.35 first to pr푋×푍 we see that +pr푌×푍∗(pr∗ +푋×푌(pr푋×푌∗ +푋 +훼 ⌣ 훾) ⌣ pr∗ +푌×푍훿) = pr푌×푍∗(pr∗ +푋×푌(pr푋×푌∗ +푋 +훼 ⌣ 훾)) ⌣ 훿 +and then applying Lemma 3.1 to (∗) shows +pr푌×푍∗(pr∗ +푋×푌(pr푋×푌∗ +푋 +훼 ⌣ 훾)) = pr푌×푍∗ +푌 +(pr푋×푌 +푌∗ (pr푋×푌∗ +푋 +훼 ⌣ 훾)) = pr푌×푍∗ +푌 +cor(훾)(훼) +so that +pr푌×푍∗(pr∗ +푋×푌(pr푋×푌∗ +푋 +훼 ⌣ 훾) ⌣ pr∗ +푌×푍훿) = pr푌×푍∗ +푌 +cor(훾)(훼) ⌣ 훿 +Applying pr푌×푍 +푍∗ +we conclude that +cor 훿(cor 훾)(훼)) = pr푋×푌×푍 +푍∗ +(pr푋×푌×푍∗ +푋 +훼 ⌣ pr∗ +푋×푌훾 ⌣ pr∗ +푌×푍훿) +(4.8) +Finally, we rewrite the right hand side as +pr푋×푍 +푍∗ pr푋×푍∗(pr∗ +푋×푍pr푋×푍∗ +푋 +훼 ⌣ pr∗ +푋×푌훾 ⌣ pr∗ +푌×푍훿) +19 + +and apply Corollary 3.35 to pr푋×푍 (with the pushing morphism (푋 ×푌 ×푍, Σ) → (푋 ×푍, 푃(Φ푋, Φ푍)) +and pulling morphism (푋 × 푌 × 푍, pr푋×푌×푍−1 +푋 +(Φ푋)) → (푋 × 푍, pr푋×푍−1 +푋 +(Φ푋))) to arrive at +pr푋×푍∗(pr∗ +푋×푍pr푋×푍∗ +푋 +훼 ⌣ pr∗ +푋×푌훾 ⌣ pr∗ +푌×푍훿) = pr푋×푍∗ +푋 +훼 ⌣ pr푋×푍∗(pr∗ +푋×푌훾 ⌣ pr∗ +푌×푍훿) +Applying pr푋×푍 +푍∗ +on both sides shows that the right hand side of (4.8) is cor(pr푋×푍∗(pr∗ +푋×푌훾 ⌣ +pr∗ +푌×푍훿)(훼), as desired. +Remark 4.9. 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Lectures on logarithmic algebraic geometry. Vol. 178. Cambridge +Studies in Advanced Mathematics. Cambridge University Press, Cambridge, +2018, pp. xviii+539. isbn: 978-1-107-18773-3. doi: 10.1017/9781316941614. +url: https://doi.org/10.1017/9781316941614. +21 + +[R&D] +RobinHartshorne.Residues and duality.Lecture notes of a seminar on the work +of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. +Lecture Notes in Mathematics, No. 20. Springer-Verlag, Berlin-New York, 1966, +pp. vii+423. +[Stacks] +The Stacks project authors.TheStacks project.2021.url: https://stacks.math.columbia.edu. +[Voi14] +Claire Voisin. Chow Rings, Decomposition of the Diagonal, and the Topology of +Families.PrincetonUniversity Press, 2014. isbn: 9780691160511. url: http://www.jstor.org/stable/j.ctt5hhp7w +(visited on 12/29/2022). +A +Attempts to construct a fundamental class of a thrifty bira- +tional equivalence +As mentioned in Section 1 inspiration for this work was the following remarkable theoremof Chatzistamatiou- +Rülling: +Theorem A.1 ([CR11, Thm. 3.2.8] (see also [CR15, Thm. 1.1], [Kov20, Thm. 1.6])). Let 푘 be a perfect +field and let 푆 be a scheme. Suppose 푋 and 푌 are two separated, finite type 푘-schemes which are +(푖) smooth over 푘 and +(푖푖) properly birational over 푆 in the sense that there is a commutative diagram +푍 +푋 +푌 +푆 +푟 +푠 +푓 +↺ +푔 +(A.2) +with 푟 and 푠 proper birational morphisms. +Let 푛 = dim 푋 = dim 푌 = dim 푍. Then, there are isomorphisms of sheaves +푅푖푓∗풪푋 +∼�→ 푅푖푔∗풪푌 and 푅푖푓∗휔푋 +∼�→ 푅푖푔∗휔푌 for all 푖, +(A.3) +This result implies, for example, that if 푆 is a variety over a perfect field 푘 with a rational res- +olution, that is, a resolution of singularities 푓 ∶ 푋 → 푆 such that 푅푓∗풪푋 = 풪푆, then every other +resolution 푔 ∶ 푌 → 푆 satisfies 푅푔∗풪푌 = 풪푆 and is hence also rational. In characteristic 0 this was +a corollary of Hironaka’s resolution of singularities [Hir64]; in positive characteristic it remained +open until 2011. +The original proof in [CR11, Thm. 3.2.8] makes use of a cycle morphism cl ∶ 퐶퐻∗(푋) → 퐻∗(푋, Ω∗ +푋) +from Chow cohomology to Hodge cohomology, which is ultimately applied to a cycle 푍 ⊂ 푋 × 푌 ob- +tained from a properbirational equivalence. That cycle morphismsatisfies 2 essential properties:the +first is that it is compatible with correspondences: here Chow correspondences are homomorphisms +퐶퐻∗(푋) → 퐶퐻∗(푌) of the form 훼 ↦→ pr푌∗(pr∗ +푋훼 ⌣ 훾) for some 훾 ∈ 퐶퐻∗(푋 × 푌) +where ⌣ is the cup product induced by intersecting cycles; Hodge correspondences are defined in +a similar way. The second key property is a compatibility with the filtrations +퐶퐻푛(푋 × 푌) = 퐹0퐶퐻푛(푋 × 푌) ⊇ 퐹1퐶퐻푛(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐶퐻푛(푋 × 푌) ⊇ 0 +where 퐹푐퐶퐻푛(푋×푌) is the subgroup generated by cycles 푍 ⊆ 푋×푌 such that codim(pr푌푍 ⊆ 푌) ≥ 푐, +and +퐻푛(푋 × 푌, Ω푚 +푋×푌) = 퐹0퐻푛(푋 × 푌, Ω푚 +푋×푌) ⊇ 퐹1퐶퐻∗(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐻푛(푋 × 푌, Ω푚 +푋×푌) ⊇ 0 +22 + +where 퐹푐퐻푛(푋 ×푌, Ω푚 +푋×푌) is the image of the map 퐻푛(푋 ×푌, ⊕푚 +푗=푐Ω푚−푗 +푋 +⊠Ω푗 +푌) → 퐻푛(푋 ×푌, Ω푚 +푋×푌) +coming from the Künneth decomposition. +It is natural to ask if a similar method can be applied to prove an analogue of Theorem A.1 for +pairs, which might read something like Conjecture A.7 below. In order to state this analogue, we +require a few additional definitions. For the remainder of this appendix we work over a fixed perfect +field 푘. +Definition A.4 (slightly simplified version of [Kol13, Def. 1.5]). A pair (푋, ∆푋) over 푘 will mean +• a reduced, equidimensional and 푆2 scheme 푋 of finite type over 푘 admitting a dualizing com- +plex , together with +• a ℚ-Weil divisor ∆푋 = ∑ +푖 푎푖퐷푖 on 푋 such that no irreducible component 퐷푖 of ∆푋 is contained +in Sing(푋). +Definition A.5. A stratum of a simple normal crossing pair (푋, ∆푋 = ∑ +푖 퐷푖) is a connected (equiv- +alently, irreducible) component of an intersection 퐷퐽 = ∩푗∈퐽퐷푗. +Given any pair (푋, ∆푋), there is a largest open set 푈 ⊆ 푋 such that (푈, ∆푋|푈) is a simple nor- +mal crossing pair, and we will refer to the resulting simple normal crossing pair as snc(푋, ∆푋) ∶= +(푈, ∆푋|푈). +Definition A.6 ( compare with [Kol13, Def. 2.79-2.80], [KX16, §1, discussion before Def. 10] ). Let +(푆, ∆푆 = ∑ +푖 퐷푖) be a pair, and assume ∆푆 is reduced and effective. A separated, finite type birational +morphism 푓 ∶ 푋 → 푆 is thrifty with respect to ∆푆 if and only if +(푖) 푓 is an isomorphism over the generic point of every stratum of snc(푆, ∆푆) and +(푖푖) letting ̃퐷푖 = 푓−1 +∗ 퐷푖 for 푖 = 1, … , 푁 be the strict transforms of the divisors 퐷푖, and setting +∆푋 ∶= ∑ +푖 ̃퐷푖, the map 푓 is an isomorphism at the generic point of every stratum of snc(푋, ∆푋). +Conjecture A.7. Let 푘 be a perfect field, let 푆 be a scheme and let (푋, ∆푋) and (푌, ∆푌) be simple +normal crossing pairs over 푘. Suppose (푋, ∆푋) and (푌, ∆푌) are properly birational over 푆 in the sense +that there is a commutative diagram +(푍, ∆푍) +(푋, ∆푋) +(푌, ∆푌) +푆 +푟 +푠 +푓 +↺ +푔 +(A.8) +where 푟, 푠 are proper and birational morphisms, and assume ∆푍 = 푟−1 +∗ ∆푋 = 푠−1 +∗ ∆푌. If 푟 and 푠 are +thrifty, then there are quasi-isomorphisms +푅푓∗풪푋(−∆푋) ≃ 푅푔∗풪푌(−∆푌) and 푅푓∗휔푋(∆푋) ≃ 푅푔∗휔푌(∆푌). +(A.9) +Following [CR11] closely, one might begin by replacing the ordinary sheaves of differentials Ω푋 +appearing in Hodge cohomology with sheaves of differentials with log poles Ω푋(log ∆푋) and attempt +to implement a similar strategy, i.e. starting a cycle 푍 ⊂ 푋×푌 representinga thrifty proper birational +equivalince, producing a correspondence in logarithmic Hodge cohomology and analyzing its prop- +erties. +Ultimately even the correspondences of Section 4 seem to be insufficient to deal with thrifty +proper birational equivalences, as we illustrate in Appendix A.1 below. The problem we encounter +is elementary: looking at the recipe for the Hodge class cl(푍) of a subvariety 푍 ⊆ 푋, where 푍 and 푋 +are smooth an projective (outlined in [Har77, Ex. III.7.4]), we see that cl(푍) ultimately comes from +the trace linear functional tr ∶ 퐻dim 푍(푍, 휔푍) → 푘, or Serre-dually the element 1 ∈ 퐻0(푍, 풪푍). Due +to the introduction of log poles and zeroes in Section 4, trying to follow that recipe we pass through +23 + +cohomology groups of the form 퐻dim 푍(푍, 휔푍(퐷)), or dually 퐻0(푍, 풪푍(−퐷)) where 퐷 is an (often +non-0 in cases of interest) effective Cartier divisor on 푍, and so there simply is no “1” to be had. +Beyond the difficulties described in the previous paragraph, when attempting to formulate a +logarithmic variant of Chatzistamatiou-Rülling’s cycle morphism argument one is hampered by the +fact that we are still in the early days of logarithmic Chow theory . It is not clear to the author which +logarithmic variant of Fulton’s 퐶퐻∗, if any, could be used to construct a logarithmic cycle morphism +with all of the desired properties. Further investigation of this question could be an interesting topic +of future research. +Despite the aforementioned challenges, it is possible to prove a result almost identical to Conjecture A.7 +by entirely different methods [God22].8 +A.1 +Obstructions to obtaining log Hodge correspondences from thrifty bi- +rational equivalences +Let (푋, ∆푋), (푌, ∆푌) be simple normal crossing pairs, and assume in additionthat 푋, 푌 are connected +and proper. Let 푍 ⊆ 푋 × 푌 be a smooth closed subvariety with codimension 푐. In this situation the +fundamental class of cl(푍) ∈ 퐻푐(푋 × 푌, Ω푐 +푋×푌) (no log poles yet) can be described using only Serre +duality, as follows (we refer to [Har77, Ex. III.7.4]). the composition +퐻dim푍(푋 × 푌, Ωdim푍 +푋×푌 ) → 퐻dim 푍(푍, Ωdim 푍 +푍 +) +tr�→ 푘 +(A.10) +(where tr is the trace map of Serre duality) is an element of +퐻dim푍(푋 × 푌, Ωdim푍 +푋×푌 )∨ ≃ 퐻푐(푋 × 푌, Ω푐 +푋×푌) +(A.11) +which we may define to be cl(푍).9 In light of Theorem 4.1 we might hope to modify eqs. (A.10) +and (A.11) to obtain a class in 퐻푐(푋 ×푌, Ω푐 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)). Let us focus on the case where +• pr푋|푍 ∶ 푍 → 푋, pr푌|푍 ∶ 푍 → 푌 are both thrifty and birational, so in particular 푐 = dim 푋 = +dim 푌 =∶ 푑 and +• (pr푋|푍)−1 +∗ ∆푋 = (pr푌|푍)−1 +∗ ∆푌 =∶ ∆푍 +To keep the notation under control, set 휋푋 ∶= pr푋|푍 and 휋푌 ∶= pr푌|푍. +In this situation letting 휄 ∶ 푍 → 푋 × 푌 be the inclusion there is a natural map +푑휄∨ ∶ Ω푑 +푋×푌(log ∆푋×푌) → 휄∗Ω푑 +푍(log ∆푋×푌|푍) and twisting by −pr∗ +푌∆푌 gives a map +Ω푑 +푋×푌(log ∆푋×푌)(−pr∗ +푌∆푌) → 휄∗Ω푑 +푍(log ∆푋×푌|푍)(−pr∗ +푌∆푌|푍) = 휄∗Ω푑 +푍(log ∆푋×푌|푍)(−휋∗ +푌∆푌) +To identify Ω푑 +푍(log ∆푋×푌|푍)(−pr∗ +푋∆푋|푍), write +(휋푋)∗∆푋 = (휋푋)−1 +∗ ∆푋 + 퐸푋 = ∆푍 + 퐸푋 and +(휋푌)∗∆푌 = (휋푌)−1 +∗ ∆푌 + 퐸푌 = ∆푍 + 퐸푌 +so that ∆푋×푌|푍 = (휋푋)∗∆푋 + (휋푌)∗∆푌 = 2∆푍 + 퐸푋 + 퐸푌. While the hypotheses guarantee ∆푍 is +reduced it may be that 퐸푋, 퐸푌 are non-reduced — however something can be said about their multi- +plicities. If 퐸푋 = ∑ +푖 푎푖 +푋퐸푖 +푋, 퐸푌 = ∑ +푖 푎푖 +푌퐸푖 +푌 where the 퐸푖 +푋, 퐸푖 +푌 are irreducible, then by a generalization +of [Har77, Prop. 3.6] (see also [Kol13, §2.10]), +푎푖 +푋 = mlt(휋푋(퐸푖 +푋) ⊆ ∆푋) +and since ∆푋 is a reduced effective simple normal crossing divisor, if in addition we write ∆푋 = +∑ +푖 퐷푖 +푋, then mlt(휋푋(퐸푖 +푋) ⊆ ∆푋) = |{푖 | 휋푋(퐸푖 +푋) ⊆ 퐷푖 +푋}|. The thriftiness hypothesis that 휋푋(퐸푖 +푋) is not +8The reason the result is only “almost identical” is that in [God22] we require ostensibly stronger hypotheses on the +base scheme 푆 (namely that it is excellent and noetherian), but it is possible that even in the situation of Theorem A.1 +and Conjecture A.7 one can reduce to this case, for example using noetherian approximation. +9It may then be non-trivial to verify this agrees with other definitions, especially if we worry about signs, but we will not +need that level of detail for what follows. +24 + +a stratum then implies 푎푖 +푋 = mlt(휋푋(퐸푖 +푋) ⊆ ∆푋) < codim(휋푋(퐸푖 +푋) ⊂ 푋). Since differentials with log +poles are insensitive to multiplicities, we have +Ω푑 +푍(log ∆푋×푌|푍) = 휔푍(∆푍 + 퐸red +푋 ++ 퐸red +푌 ) +where −red denotes the associated reduced effective divisor. Then +Ω푑 +푍(log ∆푋×푌|푍)(−휋∗ +푌∆푌) = 휔푍(∆푍 + 퐸red +푋 ++ 퐸red +푌 +− ∆푍 − 퐸푌) +휔푍(퐸red +푋 ++ (퐸red +푌 +− 퐸푌)) = 휔푍( +∑ +푖 +퐸푖 +푋 + +∑ +푖 +(1 − 푎푖 +푌)퐸푖 +푌) +The upshot is that we have an induced map +퐻푑(푋 × 푌, Ω푑 +푋×푌(log ∆푋×푌)(−pr∗ +푌∆푌)) → 퐻푑(푍, 휔푍(퐸red +푋 ++ (퐸red +푌 +− 퐸푌))) +(A.12) +Here the left hand side is Serre dual to 퐻푑(푋 × 푌, Ω푑 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)), so the 푘-linear dual +of (A.12) is a morphism +퐻푑(푍, 휔푍(퐸red +푋 ++ (퐸red +푌 +− 퐸푌)))∨ → 퐻푑(푋 × 푌, Ω푑 +푋×푌(log ∆푋×푌)(−pr∗ +푋∆푋)) +Unfortunately10 퐻푑(푍, 휔푍(퐸red +푋 + (퐸red +푌 − 퐸푌))) is often 0. If 퐸푋 and 퐸푌 are both reduced (an explicit +example where this holds will be given below), then 퐻푑(푍, 휔푍(퐸red +푋 +(퐸red +푌 −퐸푌))) = 퐻푑(푍, 휔푍(퐸푋)). +If in addition 퐸푋 ≠ 0, we obtain 퐻푑(푍, 휔푍(퐸푋)) = 0 by an extremely weak (but characteristic inde- +pendent) sort of Kodaira vanishing: +Lemma A.13. Let 푍 be a proper variety over a field 푘 with dimension 푑, and assume 푍 is normal and +Cohen-Macaulay. If 퐷 ⊂ 푍 is a non-0 effective Cartier divisor on 푍 then 퐻푑(푍, 휔푍(퐷)) = 0. +Proof. By Serre duality 퐻푑(푍, 휔푍(퐷)) = 퐻0(푍, 풪푍(−퐷)), which vanishes by the classic fact that “a +nontrivial line bundle and its inverse can’t both have non-0 global sections.” Since I am not aware +of a specific reference, here is a proof: +Suppose towards contraditction that there is a non-0 global section 휎 ∈ 퐻0(푍, 풪푍(−퐷)) — then +the composition +풪푍 +풪푍(−퐷) +풪푍 +휎 +휏 +is non-0. By [Stacks, Tag 0358] 퐻0(푍, 풪푍) is a (normal) domain, and since it’s also a finite dimen- +sional 푘-vector space it must be an extension field of 푘. But then 휏 ∈ 퐻0(푍, 풪푍) is invertible hence +surjective, so 풪푍(−퐷) → 풪푍 is surjective, which is a contradiction since by hypothesis the cokernel +풪퐷 ≠ 0. +Example A.14. Let 푋 = ℙ2 and let ∆푋 ⊂ 푋 be a line. Let 푝 ∈ 퐿 be a 푘-point, let 푌 = Bl푝 푋 and +let ∆푌 = ̃퐿 = the strict transform of 퐿. Finally let 푓 ∶ 푌 → 푋 be the blowup map and let 푍 = +(푓 ×id)(푌) ⊂ 푋 ×푌. In this case (with all notation as above) 휋푋◦(푓 ×id) = 푓 and 휋푌◦(푓 ×id) = id푌, +so under the isomorphism 푓 × id ∶ 푌 ≃ 푍, 퐸푋 is the exceptional divisor of 푓 (with multiplicity 1). +On the other hand 퐸푌 = 0. In particular 퐸푋 and 퐸푌 are reduced and 퐸푋 ≠ 0 so from the above +discussion 퐻2(푍, 휔푍(퐸푋)) = 0. +10at least for the purposes of constructing log Hodge cohomology classes of subvarieties ... +25 + diff --git a/3tAyT4oBgHgl3EQfo_h3/content/tmp_files/load_file.txt b/3tAyT4oBgHgl3EQfo_h3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a03c7a831e543ca8c99b76585e8565a9ea4bc93 --- /dev/null +++ b/3tAyT4oBgHgl3EQfo_h3/content/tmp_files/load_file.txt @@ -0,0 +1,1025 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf,len=1024 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='00517v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='AG] 2 Jan 2023 Correspondences in log Hodge cohomology Charles Godfrey Pacific Northwest National Laboratory charles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='godfrey@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='gov January 3, 2023 Abstract We construct correspondences in logarithmic Hodge theory over a perfect field of arbitrary char- acteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' These are represented by classes in the cohomology of sheaves of differential forms with log poles and, notably, log zeroes on cartesian products of varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' From one perspective this gen- eralizes work of Chatzistamatiou and Rülling, who developed (non-logarithmic) Hodge correspon- dences over perfect fields of arbitrary characteristic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='from another we provide partial generalizations of more recent work of Binda, Park and Østvær on logarithmic Hodge correspondences by relaxing finiteness and strictness conditions on the correspondences considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1 Introduction Generally speaking, a correspondence between two algebraic varieties푋 and 푌 over a field 푘 is a cycle or cohomology class on the product 푋×푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The study of such objects dates back (at least) to Lefschetz [Lef53], and features prominently in famous conjectures on algebraic cycles (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Voi14]) and Voevodsky’s theory of motives (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [MVW06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In a number of algebro-geometric research areas it has become commonplace to work with pairs (푋, ∆푋) consisting of a variety 푋 together with a divisor ∆푋 on 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Such areas include moduli of varieties (where pairs generalize the curves with marked points of [DM69]), birational geometry (where pairs appear naturally, for example as the output of strong resolution of singularities [KM98]) and logarithmic geometry (in this case vast generalizations of divisors ∆푋 are allowed [Ogu18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It is natural to wonder about analogues of correspondencesin this category of pairs, and there have been efforts in this direction, for example development of categories of logarithmic motives [BPØ20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In this paper, we focus on correspondences for logarithmic Hodge cohomology of pairs (푋, ∆푋), where 푋 is a smooth (but not necessarily proper) variety over a perfect field 푘 and ∆푋 is a simple normal crossing divisor on 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' These cohomology groups can be described as 퐻∗(푋, ∆푋) = ⨁ 퐻푞(푋, Ω푝 푋(log ∆푋)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1) where Ω푋(log ∆푋) is the sheaf of differential 1-forms on 푋 with log poles along ∆푋 and Ω푝 푋(log ∆푋) the 푝-th exterior power thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In addition we consider a generalization where 푋 comes with a family of supports Φ푋, and the ordinary cohomology groups on the right hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1) are replaced with cohomology with supports in Φ푋, namely 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Allowing for supports greatly expands the applicability of our results: for example, it permits us to construct a correspon- dence associated to a cycle 푍 ⊂ 푋 × 푌 in a situation where neither 푋 nor 푌 is proper over 푘, but 푍 is proper over both 푋 and 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 There are multiple motivations for investigating correspondencesfor this particular cohomology of pairs: 1One way that such a cycle 푍 might naturally arise is as the closure of the graph of a birational equivalence 푋 ⤏ 푌 of non-proper varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This work was completed while the author was a PhD student in the University of Washington Department of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The author was partially supported by the University of Washington Department of Mathematics Graduate Research Fellowship, and by the NSF grant DMS-1440140, administered by the Mathematical Sciences Research Institute, while in residence at MSRI during the program Birational Geometry and Moduli Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' By analogy with the case of varieties (that is, without auxiliary divisors/log structures), we sus- pect that correspondences at the level of Chow cycles are more fundamental, and that (many) correspondencesin logarithmic Hodge cohomology are obtained from Chow correspondences via a cycle morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' However, as of this writing there is no full-fledged theory of Chow co- homology of pairs or log schemes (though there has been considerable progress, for instance in [Bar18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' BBG22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Logarithmic Hodge cohomology is in contrast quite mature, appearing as early as [Del71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Correspondences in (non-logarithmic) Hodge cohomology have found remarkable applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For example, [CR11] used them to prove birational invariance of the cohomology groups of the structure sheaf 퐻푖(푋, 풪푋) for smooth varieties 푋 over perfect fields of positive character- istic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In fact, attempting to implement a similar strategy with logarithmic Hodge cohomology to obtain results on invariance of the cohomology groups 퐻푖(푋, 풪푋(−∆푋)) with respect to (a restricted class of) birational equivalences was the initial inspiration for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ultimately that attempt was unsuccessful, as we describe in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' There has been recent interest in logarithmic Hodge cohomology as a representable functor on a category of motives of log schemes over a perfect field [BPØ20, §9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' While that work does also construct some correspondences, they are restricted to those associated with logarithmic Hodge cohomology classes of cycles 푍 ⊂ 푋 × 푌 which are finite over 푋 and obey additional strictness (in the sense of logarithmic geometry) conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' we remove these restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The correspondences we construct are obtained from certain Hodge classes with both log poles and log zeroes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Our main result is: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2 (= Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A class 훾 ∈ 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) defines homomorphisms cor(훾) ∶ 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞+푗−푑푋 Φ푌 (푌, Ω푝+푖−푑푋 푌 (log ∆푌)) by the formula cor(훾)(훼) ∶= pr푌∗(pr∗ 푋(훼) ⌣ 훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Moreover if (푍, ∆푍, Φ푍) is another snc pair with supports and 훿 ∈ 퐻푗′ 푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ 푌×푍(log ∆푌×푍)(−pr∗ 푌∆푌)), then pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿)) ∈ 퐻푗+푗′−푑푌 푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌 푋×푍 (log ∆푋×푍)(−pr∗ 푋∆푋)) and cor(pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿))) = cor(훿)◦ cor(훾) as homomorphisms 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌 Φ푍 (푍, Ω푝+푖+푖′−푑푋−푑푌 푍 (log ∆푍)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In the above, ∆푋푌 ∶= pr∗ 푋∆푋 + pr∗ 푌∆푌, a simple normal crossing divisor on 푋 × 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' There is a simple heuristic explanation for the appearance of differential forms in Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋): working over the complex numbers, in the case where 푋and 푌 are both proper the class cor(훾)(훼) ∶= pr푌∗(pr∗ 푋(훼) ⌣ 훾) can be computed explicitly as an integral of the form ∫ 푋 훼(푥) ∧ 훾(푥, 푦), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3) and this integral will only be finite when the log poles of 훼 along ∆푋 are cancelled by complementary zeroes of the form 훾(푥, 푦) along the preimage pr∗ 푋∆푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2 relies heavily on prior work on both Hodge cohomology with supports [CR11, §2] and its logarithmic variant [BPØ20, §9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Section 2 is a rapid summary of those results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The key new technical ingredient is a base change formula on the interaction of pushforward and pullback operations in cartesian squares, proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Section 4 includes the proof of our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 Acknowledgements Thanks to Daniel Bragg, Yun Hao, Sarah Scherotzke, Nicolò Sibilla and Mattia Talpo for helpful conversations, to Lawrence Jack Barrott for illuminating email correspondence regarding logarith- mic aspects of Chow and Hodge, and to my advisor Sándor Kovács for many insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Thanks also to the participants of the Spring 2019 MSRI graduate student seminar, in particular Giovanni Inchiostro and organizer Fatemeh Rezaee, for feedback on early work on this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2 Functoriality properties of log Hodge cohomology with sup- ports 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 Supports In order to obtain results that apply to correspondences between varieties 푋 and 푌 where neither 푋 nor 푌 is proper, it is necessary to work with cohomology with supports, also known as local coho- mology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A primary source for the material of this subsection is [R&D, §IV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푋 be a noetherian scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 ([R&D, §IV], [CR11, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A family of supports Φ on 푋 is a non-empty collection Φ of closed subsets of 푋 such that If 퐶 ∈ Φ and 퐷 ⊂ 퐶 is a closed subset, then 퐷 ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 퐶, 퐷 ∈ Φ then 퐶 ∪ 퐷 ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Φ = { all closed subsets of 푋 } is a family of supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' More generally if 풞 is any col- lection of closed subsets 퐶 ⊂ 푋, there is a smallest family of supports Φ(풞) containing 풞 (explicitly, Φ(풞) consists of finite unions ⋃ 푖 푍푖 of closed subsets 푍푖 ⊂ 퐶푖 of elements 퐶푖 ∈ 풞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Taking Φ = Φ({푋}) recovers the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A more interesting example is the case where for some fixed 푝 ∈ ℕ, Φ = {closed sets 푍 ⊆ 푋 | dim 푍 ≤ 푝}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' There is a close relationship between families of supports on X and certaincollections of specialization- closed subsets of points on 푋, and we can also consider sheaves of families of supports — for further details we refer to [R&D, §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and Ψ is a family of supports on 푌, then {푓−1(푍) | 푍 ∈ Ψ} is a family of closed subsets of 푋, and is closed under unions, but is not in general closed under taking closed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푓−1(Ψ) is the smallest family of supports on 푋 containing {푓−1(푍) | 푍 ∈ Ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let Φ be a family of supports on 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The notation/terminology 푓|Φ is proper will mean 푓|퐶 is proper for every 퐶 ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 푓|Φ is proper then 푓(퐶) ⊂ 푌 is closed for every 퐶 ∈ Φ and in fact 푓(Φ) = {푓(퐶) ⊂ 푌 | 퐶 ∈ Φ} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4) is a family of supports on 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The key point here is that if 퐷 ⊂ 푓(퐶) is closed, then 푓−1(퐷) ∩ 퐶 ∈ Φ and 퐷 = 푓(푓−1(퐷) ∩ 퐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A scheme with supports (푋, Φ푋) is a scheme 푋 together with a family of supports Φ푋 on 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A pushing morphism 푓 ∶ (푋, Φ푋) → (푌, Φ푌) of schemes with supports is a morphism 푓 ∶ 푋 → 푌 of underlying schemes such that 푓|Φ푋 is proper and 푓(Φ푋) ⊂ Φ푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A pulling morphism 푓 ∶ 푋 → 푌 is a morphism 푓 ∶ 푋 → 푌 such that 푓−1(Φ푌) ⊂ Φ푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' These morphisms provide two different categories with underlying set of objects schemes with supports (푋, Φ푋), and pushing/pulling morphisms respectively (the verification is elementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' for instance a composition of pushing morphisms is again a pushing morphism since compositions of proper morphisms are proper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Schemes with supports provide a natural setting for describing 3 functoriality properties of local cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let ℱ be a sheaf of abelian groups on a scheme with supports (푋, Φ푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The sheaf of sections with supports of ℱ, denoted ΓΦ(ℱ), is obtained by setting ΓΦ(ℱ)(푈) = {휎 ∈ ℱ(푈) | supp 휎 ∈ Φ푋|푈 } (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) for each open 푈 ⊂ 푋 (here Φ푋|푈 is short for 휄−1Φ푋 where 휄 ∶ 푈 → 푋 is the inclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' More explicitly: for a local section 휎 ∈ ℱ(푈), 휎 ∈ ΓΦ(ℱ)(푈) means supp 휎 = 퐶 ∩ 푈 for a closed set 퐶 ⊂ Φ푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The functor ΓΦ is right adjoint to an exact functor, for instance the inclusion of the subcategory 퐀퐛Φ(푋) ⊂ 퐀퐛(푋) of abelian sheaves on 푋 with supports in Φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' so, ΓΦ is left exact and preserves injectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In the case Φ = Φ(푍) for some closed 푍 ⊂ 푋, this is proved in [Stacks, Tag 0A39, Tag 0G6Y, Tag 0G7F] — the general case can then be obtained by writing ΓΦ as a filtered colimit: ΓΦ = colim푍∈Φ Γ푍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The right derived functor of ΓΦ will be denoted 푅ΓΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Taking global sections on 푋 gives the sections with supports of ℱ: ΓΦ(ℱ) ∶= Γ푋(ΓΦ(ℱ)) This is also left exact, and (the cohomologies of) its derived functor give the cohomology with supports in Φ: 퐻푖 Φ(푋, ℱ) ∶= 푅푖ΓΦ(ℱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cohomology with supports enjoys the following functoriality properties: (푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pulling morphism of schemes with supports, ℱ, 풢 are sheaves of abelian groups on 푋, 푌 respectively, and if 휑 ∶ 풢 → 푓∗ℱ is a morphism of sheaves, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10) then there is a natural morphism 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Similarly if ℱ and 풢 are quasicoherent then there are natural morphisms 푅ΓΦ풢 → 푅푓∗푅ΓΦℱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (푖푖) If 푓 ∶ (푋, Φ푋) → (푌, Φ푌) is a pushing morphism, ℱ, 풢 are sheaves of abelian groups on 푋, 푌 respectively, and 휓 ∶ 푅푓∗ℱ → 풢 is a morphism in the derived category of 푋, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11) then there is a natural morphism 푅푓∗푅ΓΦ(ℱ) → 푅ΓΦ풢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Both parts of the proposition follow from [Stacks, Tag 0G78];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (i) is discussed in detail in [CR11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1] and (ii) can be extracted from [CR11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2] (although it doesn’t appear to be stated explicitly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' See also [BPØ20, Constructions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2 Differential forms with log poles Let 푘 be a perfect field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A snc pair with supports (푋, ∆푋, Φ푋) over 푘 is a smooth scheme 푋 separated and of finite type over 푘 with a family of supports Φ푋 together with a reduced, effective divisor ∆푋 on 푋 such that supp ∆푋 has simple normal crossings, in the sense that for any point 푥 ∈ 푋 there are regular parameters 푧1, … , 푧푐 ∈ 풪푋,푥 such that supp ∆푋 = 푉(푧1 ⋅ 푧2 ⋯ 푧푟) on a Zariski neighborhood of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3 The interior 푈푋 of a snc pair with supports (푋, ∆푋, Φ푋) is 푈푋 ∶= 푋 ⧵ supp ∆푋 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='13) The inclusion of 푈푋 in 푋 is denoted by 휄푋 ∶ 푈푋 → 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2Simply put ℱ is a sheaf of abelian groups on 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3This is equivalent to the more general definition [BPØ20, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1] in the case where the base scheme is Spec 푘, which is all we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4 Here supp ∆푋 denotes the support of ∆푋 (if ∆푋 = ∑ 푖 푎푖퐷푖 where the 퐷푖 are prime divisors, then supp ∆푋 = ∪푖퐷푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Similarly let 푗푋 ∶ supp ∆푋 → 푋 denote the evident inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='14 (compare with [CR11, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A pulling morphism 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) of snc pairs with supports is a pulling morphism 푓 ∶ 푋 → 푌 of underlying schemes with support such that 푓−1(supp ∆푌) ⊂ supp ∆푋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' equivalently, 푓 restricts to a morphism 푓|푈푋 ∶ 푈푋 → 푈푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) of snc pairs with supports is a pushing morphism of underlying schemes with support such that 푓∗∆푌 = ∆푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Note that if 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is a pushing morphism then 푈푋 = 푓−1(푈푌), so for example if 푓 ∶ 푋 → 푌 is proper then so is the induced map 푈푋 → 푈푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='15 (compare with [CR11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A morphism of snc pairs with supports 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) is flat, proper, an immersion, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' if and only if the same is true of the underlying morphism of schemes 푓 ∶ 푋 → 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A diagram of snc pairs with supports (푋′, ∆푋′, Φ푋′) (푋, ∆푋, Φ푋) (푌′, ∆푌′, Φ푌′) (푌, ∆푌, Φ푌) 푔′ 푓′ 푓 푔 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16) is cartesian if and only if the induced diagram of underlying schemes 푋′ 푋 푌′ 푌 푔′ 푓′ □ 푓 푔 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='17) is cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4 The terminology is meant to suggest that pushing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' pulling) morphismsinduce pushforward (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' pullback) maps on log Hodge cohomology, as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If (푋, ∆푋) is an snc pair, or more generally a normal separated scheme of finite type 푋 over 푘 together with a sequence of effective Cartier divisors 퐷1, … , 퐷푁 ⊆ 푋 with sum ∆푋 = ∑ 푖 퐷푖, then it comes with a sheaf of differential forms with log poles Ω푋(log ∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In the case where (푋, ∆푋, Φ푋) is snc, this sheaf and its properties are described in [EV92, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For a definition and treatment of Ω푋(log ∆푋) in the much greater generality of logarithmic schemes we refer to [Ogu18, §IV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In some of the calculations below the following concrete local description will be very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푧1, 푧2, … , 푧푛 be local coordinates at a point 푥 ∈ 푋 such that supp ∆푋 = 푉(푧1푧2 ⋯ 푧푟) in a neigh- borhood of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Recall that as 푋 is smooth the differentials 푑 푧1, 푑 푧2, … , 푑 푧푛 freely generate Ω푋 on a neighborhood of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [EV92, §2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Thesections 푑 푧1 푧1 , … , 푑 푧푟 푧푟 , 푑 푧푟+1, … , 푑 푧푛 freelygenerateΩ푋(log ∆푋) on a neighborhood of 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Given Ω푋(log ∆푋), we can form the exterior powers Ω푝 푋(log ∆푋) ∶= 푝 ⋀ Ω푋(log ∆푋), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='19) and combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18 with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='19) gives concrete local descriptions of the Ω푝 푋(log ∆푋);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' in par- ticular, we see that Ωdim 푋 푋 (log ∆푋) = 휔푋(∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4If we take the red pill of logarithmic geometry, it starts to seem almost more reasonable to only require flatness, properness, cartesianness and so on of the induced maps of interiors 푈푋 → 푈푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' However we do use the stronger restrictions of the given definition in some of the proofs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The log-Hodge cohomology with supports of a log-smooth pair with supports (푋, ∆푋, Φ푋) is defined by 퐻푑(푋, ∆푋, Φ푋) = ⨁ 푝+푞=푑 퐻푞 Φ(푋, Ω푝 푋(log ∆푋)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='21) Here 퐻푞 Φ denotes local cohomology with respect to the family of supports Φ푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For connected 푋, we define 퐻푑(푋, ∆푋, Φ푋) ∶= 퐻2 dim 푋−푑(푋, ∆푋, Φ푋), and in general we set 퐻푑(푋, ∆푋, Φ푋) = ⨁ 푖 퐻푑(푋푖, ∆푋푖, Φ푋푖) where 푋푖 are the connected components of 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be pulling morphism of snc pairs with supports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='22 ([Ogu18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1] + (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The map 푓 induces a morphism of sheaves 푓∗Ω푝 푌(log ∆푌) 푑 푓∨ ����→ Ω푝 푋(log ∆푋) adjoint to a morphism 푓∗Ω푝 푌(log ∆푌) 푑푓∨ ����→ Ω푝 푋(log ∆푋) for all p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='23) The essential content of this lemma is that when we pull back a log differentialform 휎 on (푌, ∆푌), it doesn’t develop poles of order ≥ 1 along ∆푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Combining the previous lemma with proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9 gives: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='24 ([BPØ20, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1-2], see also [CR11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Foreverypullingmorphism푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) there are functorial morphisms 푅ΓΦΩ푝 푌(log ∆푌) → 푅푓∗푅ΓΦΩ푝 푌(log ∆푌) for all p (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='25) In particular, for each 푝, 푞 there are functorial homomorphisms 푓∗ ∶ 퐻푞 Φ(푌, Ω푝 푌(log ∆푌)) → 퐻푞 Φ(푋, Ω푝 푋(log ∆푋)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='26) and hence (summing over 푝 + 푞 = 푑) functorial homomorphisms 푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='27) The maps 푓∗ ∶ 퐻푑(푋, ∆푋, Φ푋) → 퐻푑(푌, ∆푌, Φ푌) induced by a pushing morphism 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) can be obtained from a combination of Nagata compactification and Grothendieck du- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28 ([BPØ20, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5], see also [CR11, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푓 ∶ (푋, ∆푋, Φ푋) → (푌, ∆푌, Φ푌) be a pushing morphism of equidimensional log-smooth pairs with support such that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then letting 푐 = dim 푌−dim푋, for each 푝 there are functorial morphisms of complexes of coherent sheaves 푅푓∗푅ΓΦ푋(Ω푝 푋(log ∆푋)) → 푅ΓΦ푌Ω푝+푐 푌 (log ∆푌)[푐] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='29) inducing maps on cohomology 푓∗ ∶ 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞+푐 Φ푌 (푌, Ω푝+푐 푌 (log ∆푌)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='30) for all 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Since they enter into the calculations below, we give a description of these pushforward mor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Before beginning, a word on duality in our current setup: since we are working exclu- sively over Spec 푘, we can make use of compatible normalized dualizing complexes — namely, if 휋 ∶ 푍 → Spec푘 is a separated finite type 푘-scheme then 휋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='풪Spec 푘 is a dualizing complex [Stacks, Tag 0E2S, Tag 0FVU].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' We will make repeated use of the behavior of dualizing with respect to differ- entials: as a consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18, wedge product gives a perfect pairing Ω푝 푋(log ∆푋)(−∆푋) ⊗ Ωdim 푋−푝 푋 (log ∆푋) → 휔푋 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='31) 6 (see also [Har77, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='13]) and so Ωdim 푋−푝 푋 (log ∆푋) ≃ 푅ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋), 휔푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Here the derived sheaf Hom 푅ℋ표푚푋 agrees with the regular sheaf Hom as Ω푝 푋(log ∆푋)(−∆푋) is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' On the other hand, the dualizing functor of 푋 is 푅ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋), 휔푋[dim 푋]) where 휔푋 = Ωdim 푋 푋 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' An upshot is that Grothendieck duality calculations involving the sheaves of differen- tial forms become more symmetric and predictable if we work with the shifted versions Ω푝 푋(log ∆푋)(−∆푋)[푝];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' for example then we have the identity Ωdim 푋−푝 푋 (log ∆푋)[dim푋 − 푝] ≃ 푅ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋]) Now, we need to compactify 푓 ∶ 푋 → 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='32 ([Nag63, §4 Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2], [Con07, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푆 be a quasi-compact quasi-separated scheme and let 푋 → 푆 be a separated morphism of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then there is a dense open immersion of 푆-schemes 푋 \ue0b4→ 푋 such that 푋 is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='32 we obtain morphisms of schemes 푋 ̄푋 푌 휄 푓 ̄푓 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='33) where 휄 ∶ 푋 → ̄푋 is a dense open immersion and ̄푓 ∶ ̄푋 → 푌 is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Note that ̄푋 need not be smooth over 푘, and in the absence of resolutions of singularities5 there is not even a way to make ̄푋 smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This means we cannot hope to upgrade ̄푋to a simple normal crossing pair ( ̄푋, ∆ ̄푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' However, we do still have a divisor ∆ ̄푋 ∶= ̄푓∗∆푦 on ̄푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' One way to overcome these difficulties is to equip the possibly singular ̄푋 with a logarithmic structure, in some sense associated to ∆ ̄푋, whose restriction to 푋 coincides with a logarithmic structure naturally defined by the simple normal crossing divisor ∆푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Formally, we use the log structure on ̄푋 pulled back from the log structure on (푌, ∆푌) [Ogu18, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6-7] along the morphism ̄푓 ∶ ̄푋 → 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Since (푌, ∆푌 = ∑푁 푖=1 퐷푌 푖 ) is a simple normal crossing pair, its associated log structure is Deligne-Faltings [Ogu18, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7] and can be encoded in the se- quence of inclusions of ideal sheaves 풪푌(−퐷푌 푖 ) \ue0b4→ 풪푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The pullback log structure on ̄푋 can then be encoded in the sequence of inclusions of ideal sheaves ̄푓−1풪푌(−퐷푌 푖 ) ⋅ 풪 ̄푋 = 풪 ̄푋(− ̄푓∗퐷푌 푖 ) \ue0b4→ 풪 ̄푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The pushforward morphisms of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28 are defined using the sheaves of log differential 푝- forms on ̄푋 over 푘 as described in [Ogu18, §IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2] — these will be denoted6 by Ω푝 푋(log ∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The essential properties that we need are: Ω푝 푋(log ∆푋) is a coherent sheaf on 푋 together with a functorial morphism Ω푝 푌(log ∆푌) → 푓∗Ω푝 푋(log ∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Coherence can be obtained as follows: first, the log structure on (푌, ∆푌) is coherent ([Ogu18, §III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9]), and hence so is its pullback to ̄푋 (see for example [Ogu18, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5, Rmk III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then [Ogu18, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8] implies Ω1 푋(log ∆푋) is a coherent sheaf, and it follows that its 푝-th exteriorpowersare coherentsheavesas well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The desiredfunctorial morphismcan be obtained from [Ogu18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 5At the time of this writing, this applies to the cases char 푘 = 푝 > 0 and dim 푋 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 6This is an abuse of notation since the construction of this sheaf is (as far as we know) not the same as the one for simple normal crossing pairs described above Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18, however the notation of [Ogu18] seems unsatisfactory for our purposes as we wish to stress that these are not the ordinary differential forms Ω푝 푋, 7 There is a natural isomorphism Ω푝 푋(log ∆푋)|푋 ≃ Ω푝 푋(∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This can be seen by observing that the log structures on (푋, ∆푋) and ̄푋 are obtained as pullbacks of the log structure on (푌, ∆푌) with respect to 푓 and ̄푓 respectively (in the case of (푋, ∆푋) this follows from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='14, and in the latter case it is how we defined the log structure on ̄푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Hence considering eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='33) we find that the log structure on ̄푋 restricts to that on (푋, ∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Hence in particular Ω푝 푋(log ∆푋) is a functorial coherent extension of Ω푝 푋(∆푋) to the possibly non-snc log scheme ̄푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Starting with the log differential 푑 pr∨ 푌 ∶ Ω푝 푌(log ∆푌)[푝] → 푅푓∗Ω푝 푋(log ∆푋)[푝], twisting by −∆푌 and using the projection formula gives a morphism (note: this is where we use the assumptions that 푓∗∆푌 = ∆푋 and ̄푓∗∆푌 = ∆ ̄푋) Ω푝 푌(log ∆푌)(−∆푌)[푝] → 푅푓∗Ω푝 푋(log ∆푋)(−∆푋)[푝] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='34) to which we apply Grothendieck duality: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 (Grothendieck duality, [R&D, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4], [Con00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푓 ∶ 푋 → 푌 be a proper morphism of finite-dimensional noetherian schemes and assume 푌 admits a dualizing complex (for example 푋 and 푌 could be schemes of finite type over 푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then for any pair of objects ℱ∙ ∈ 퐷− 푞푐(푋) and 풢∙ ∈ 퐷+ 푐 (푌) there is a natural isomorphism 푅푓∗푅퐻표푚푋(ℱ∙, 푓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='풢∙) ≃ 푅퐻표푚푌(푅푓∗ℱ∙, 풢∙) in 퐷푏 푐 (푌) Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='34) gives a morphism 푅푓∗푅ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋)[푝], 휔∙ 푋) = 푅ℋ표푚푌(푅푓∗Ω푝 푋(log ∆푋)(−∆푋)[푝], 휔푌[dim 푌]) 푅ℋ표푚푌(Ω푝 푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='36) where the equality is Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 and the vertical map is induced by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Adding supports gives a morphism 푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω 푝 푋(log ∆푋)(−∆푋)[푝], 휔푋[dim 푋]) = 푅푓∗푅ΓΦ푋푅ℋ표푚푋(Ω 푝 푋(log ∆푋)(−∆푋)[푝], 휔∙ 푋) 푅ΓΦ푌푅ℋ표푚푌(Ω푝 푌(log ∆푌)(−∆푌)[푝], 휔푌[dim 푌]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='37) where the equality is obtained from the excision property of local cohomology, compatibility of the dualizing functor with restriction and the natural isomorphism Ω푝 푋(log ∆푋)|푋 ≃ Ω푝 푋(∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='31) we obtain Ωdim 푋−푝 푋 (log ∆푋) ≃ ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋), 휔푋) = 푅ℋ표푚푋(Ω푝 푋(log ∆푋)(−∆푋), 휔푋) where the last equality uses the fact that Ω푝 푋(log ∆푋)(−∆푋) is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A similar calculation on 푌 transforms (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='37) into: 푅푓∗푅ΓΦ푋Ωdim 푋−푝 푋 (log ∆푋)[dim푋 − 푝] → 푅ΓΦ푌Ωdim 푌−푝 푌 (log ∆푌)[dim푌 − 푝] and reindexing like 푝 ↔ dim 푋 − 푝 recovers Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 8 3 A base change formula Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 (compare with [CR11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let (푋′, ∆푋′, Φ푋′) (푋, ∆푋, Φ푋) (푌′, ∆푌′, Φ푌′) (푌, ∆푌, Φ푌) □ 푔′ 푓′ 푓 푔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2) be a cartesian diagram of equidimensional snc pairs with supports, where 푓, 푓′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푔, 푔′) are pushing (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' pulling) morphisms and 푔 is either flat or a closed immersion transverse to 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then 푔∗푓∗ = 푓′ ∗푔′∗ ∶ 퐻∗(푋, ∆푋, Φ푋) → 퐻∗(푌′, ∆푌′, Φ푌′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' We will prove this following Chatzistamatiou and Rülling’s argument [CR11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7] quite closely, at various points reducing to statements proved therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In the proofs we will make use of a slight variant of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 푓 ∶ 푋 → 푌 is a morphism of noetherian schemes and let Φ푌 is a family of supports on 푌, then 푓−1 ∗ (Φ푌) ∶= {푍 ⊆ 푋 | 푓|푍 is proper and 푓(푍) ∈ Φ푌} Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It suffices to prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 in the cases where 푓 is either (푖) a projection morphism of the form pr푌 ∶ (푋 × 푌, pr∗ 푌∆푌, pr−1 푌∗(Φ푌)) → (푌, ∆푌, Φ푌), or (푖푖) a closed immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This lemma makes essential use of the functoriality part of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' We can decompose (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2) as a concatenation of cartisian diagrams (푋′, ∆푋′, Φ푋′) (푋, ∆푋, Φ푋) (푋 × 푌′, pr∗ 푌′∆푌, pr−1 푌′∗(Φ′ 푌)) (푋 × 푌, pr∗ 푌∆푌, pr−1 푌∗(Φ푌)) (푌′, ∆푌′, Φ푌′) (푌, ∆푌, Φ푌) (2) 푔′ ℎ′ ℎ (1) pr푌′ id×푔 pr푌 푔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6) where ℎ = id × 푓 is the graph morphism of 푓 and ℎ′ = 푔′ × 푓′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 푔 is flat or a closed immersion transverse to 푓 then id × 푔 is flat or a closed immersion transverse to ℎ (by base change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Here the only new feature not covered in [CR11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7] is the presence of divisors, and we simply note that ∆푋 = 푓∗∆푋 = ℎ∗pr∗ 푌∆푌 and similarly for ∆푋′, so that both pr푌 and ℎ are pushing morphisms in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='14, and similarly for the left vertical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In other words, the supports and divisors in the middle row have been chosen precisely so that the vertical morphisms are all “pushing.” We proceed to consider case (i), and wish to point out that for this case 푔 can be arbitrary (we will need the flatness/transversality restrictions in case (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In what follows we set 푑푋 = dim 푋, 푑푌 = dim 푌 and similarly for 푋′, 푌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='32 we obtain a compactification 휄 ∶ 푋 \ue0b4→ 푋 over 푘 of the smooth, separated and finite type 푘-scheme 푋 in the upper right corner of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 9 This results in a compactification of the square (1) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6) which we write as (푋 × 푌′, pr∗ 푌′∆푌, pr−1 푌′∗(Φ′ 푌)) (푋 × 푌, pr∗ 푌∆푌, pr−1 푌∗(Φ푌)) (푋 × 푌′, pr ∗ 푌′∆푌, pr −1 푌′∗(Φ′ 푌)) (푋 × 푌, pr ∗ 푌∆푌, pr −1 푌∗(Φ푌)) (푌′, ∆푌′, Φ푌′) (푌, ∆푌, Φ푌) 휄×id id×푔 휄×id pr푌′ id×푔 pr푌 푔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7) By the description following Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28, we know that pr푌∗ ∶ 퐻∗(푋 × 푌, pr∗ 푌∆푌, pr−1 푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌) stems from a morphism 푅pr푌∗푅ℋ표푚푋×푌(Ω푝 푋×푌(log pr∗ 푌∆푌)(−pr∗ 푌∆푌)[푝], 휔∙ 푋×푌) → Ω푑푌−푝 푌 (log ∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) obtained as the Grothendieck dual of a log differential of pr푌 (here and throughout what follows, a similar statement holds for pr푌′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' By an observation of Chatzistamatiou-Rülling , this map factors as 푅pr푌∗푅ℋ표푚푋×푌(Ω푝 푋×푌(log pr ∗ 푌∆푌)(−pr ∗ 푌∆푌)[푝], 휔∙ 푋×푌) → 푅pr푌∗푅ℋ표푚푋×푌(퐿pr ∗ 푌Ω푝 푌(log ∆푌)(−∆푌)[푝], 휔∙ 푋×푌) ≃ ���������→ adjunction 푅ℋ표푚푌(Ω푝 푌(log ∆푌)(−∆푌)[푝], 푅pr푌∗휔∙ 푋×푌) ����→ trace 푅ℋ표푚푌(Ω푝 푌(log ∆푌)(−∆푌)[푝], 휔푌[푑푌]) ≃�→ Ω푑푌−푝 푌 (log ∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9) where the adjunction isomorphism is [R&D, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10], and the map labeled trace is induced by the Grothendieck trace 푅pr푌∗휔∙ 푋×푌 → 휔푌[푑푌].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If it were the case that 푋 were smooth, then the usual “box product” decomposition 휔∙ 푋×푌 ≃ 휔푋[푑푋] ⊠ 휔푌[푑푌] ∶= pr∗ 푋 휔푋[푑푋] ⊗ pr푌∗휔푌[푑푌] together with the perect pairings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='31) and the local freeness of Ω푝 푌(log ∆푌)(−∆푌)[푝] would give an identification 푅ℋ표푚푋×푌(퐿pr ∗ 푌Ω푝 푌(log ∆푌)(−∆푌)[푝], 휔∙ 푋×푌) ≃ pr∗ 푋 휔푋[푑푋] ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10) In fact a more careful version of this argument, carrying out the above calculation on the smooth locus 푋 × 푌 and using excision, shows that 퐻∗(푋 × 푌, pr∗ 푌∆푌, pr−1 푌∗(Φ푌)) → 퐻∗(푌, ∆푌, Φ푌) always factors through the summand 퐻∗ Φ푋(푋 × 푌, pr∗ 푋 휔푋 ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Our next lemma implies that even when 푋 is not known to be smooth, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) still factors through something like 푅pr푌∗(pr∗ 푋 휔푋[푑푋] ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)[푑푌 − 푝]), provided we replace pr∗ 푋 휔푋[푑푋] with pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11 (compare with [CR11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For each 푝 there is a natural map 훾 ∶ pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] → 푅ℋ표푚푋×푌(pr ∗ 푌Ω푝 푌(log ∆푌)(−∆푌)[푝],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 휔∙ 푋×푌) 10 such that the restriction of 훾 to 푋 × 푌 agrees with the isomorphism pr∗ 푋 휔푋[푑푋] ⊗ pr∗ 푌 Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] ≃�→ 푅ℋ표푚푋×푌(퐿 pr∗ 푌 Ω푝 푌(log ∆푌)(−∆푌)[푝],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 휔∙ 푋×푌) and such that the composition 푅pr푌∗(pr∗ 푋 휔푋[푑푋] ⊗ pr∗ 푌 Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝]) 푅pr푌∗(훾) ��������→ 푅pr푌∗푅ℋ표푚푋×푌(pr∗ 푌 Ω푝 푌(log ∆푌)(−∆푌)[푝],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 휔∙ 푋×푌) ≃ ���������→ adjunction 푅ℋ표푚푋×푌(Ω푝 푌(log ∆푌)(−∆푌)[푝],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푅pr푌∗휔∙ 푋×푌) trace ����→ 푅ℋ표푚푋×푌(Ω푝 푌(log ∆푌)(−∆푌)[푝],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 휔푌[푑푌]) ≃ Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12) coincides with the composition 푅pr푌∗(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝]) proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' ����→ form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푅pr푌∗(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌) ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] tr ⊗id �����→ Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='13) By base change for dualizing complexes ([Stacks, Tag 0BZX, Tag 0E2S]) applied to the cartesian diagram 푋 × 푌 푋 푌 Spec 푘 (note that this is a very mild situation: 푋 → Spec 푘 is flat and proper and 푌 → Spec 푘 is smooth) we see that pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ≃ pr∗ 푋 휔∙ 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This makes the map 훾 look even more like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Following [CR11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16] we begin with the morphism 푒 ∶ pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 퐿pr ∗ 푌휔∙ 푌 → pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌휔∙ 푌 =∶ 휔∙ 푋×푌 of [Con00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12], which as explained therein agrees with pr∗ 푋 휔푋[푑푋] ⊗ pr∗ 푌 휔푌[푑푌] ≃�→ 휔푋×푌[푑푋 + 푑푌] on locus 푋 × 푌,7 and has the property that 푅푝푟푌∗(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 퐿pr ∗ 푌휔∙ 푌) 푅푝푟푌∗휔∙ 푋×푌 푅푝푟푌∗pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 휔∙ 푌 휔∙ 푌 푅푝푟푌∗푒 proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' form tr tr ⊗id 7See Conrad’s comment “It is easy to check that 푒푓 coincides with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='21) in the smooth case and is compatible with composites in f (using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6).” 11 commutes [Con00, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' We then define our version of 훾 as the composition pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 퐿pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] id⊗퐿(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='31) ���������→ pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 퐿pr ∗ 푌푅ℋ표푚푌(Ω푝 푌(log ∆푌)[푝], 휔∙ 푌) functoriality �����������→ of 퐿pr ∗ 푌,⊗퐿 푅ℋ표푚푋×푌(퐿pr ∗ 푌Ω푝 푌(log ∆푌)[푝], pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 휔∙ 푌) induced by ���������→ 푒 푅ℋ표푚푋×푌(퐿pr ∗ 푌Ω푝 푌(log ∆푌)[푝], 휔∙ 푋×푌) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='14) Note that we may drop the “퐿”s as Ω푑푌−푝 푌 (log ∆푌)(−∆푌) and Ω푝 푌(log ∆푌) are locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Verification of the stated compatibilities is as in [CR11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It seems like we could have also used the more general version of [Con00, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12] 푒′ ∶ pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗퐿 퐿pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] → pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] together with the description pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] = 퐷푋×푌(퐿pr ∗ 푌퐷푌(Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝])) where 퐷푌(−) = 푅ℋ표푚(−, 휔∙ 푌) and similarly for 퐷푋×푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Using this modified 훾, we obtain a modified version of the diagram [CR11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 732 during Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4], namely (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16) in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' To make this diagram legible, we use a few abbreviations: all func- tors are derived, we use the dualizing functors of the form 퐷푌(−) = 푅ℋ표푚푌(−, 휔∙ 푌) and we let 푑 = 푑푋 + 푑푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11 shows that triangles involving 훾 commute, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9) gives commutativity of the rest of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The usefulness of this diagram is that by definition beginning in the top left corner and following the path →↓ we obtain the pushforward on Hodge cohomology pr푌∗ Γpr−1 푌∗ Φ푌Ω푑−푝 푋×푌(log pr∗ 푌∆푌)[푑 − 푝] → ΓΦ푌Ω푑푌−푝 ×푌 (log ∆푌)(−∆푌)[푑푌 − 푝] but following ↓→ gives a composition whose behavior with respect to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7) is easier to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Namely, we have a diagram like (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16) on 푌′, and in fact a map from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16) to 푔∗ of the analogous diagram on 푌′, and hence from the preceding discussion it will suffice to prove commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='17) of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Applying excision together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11 we may rewrite the top row of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='17) as 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌Ω푑−푝 푋×푌(log pr∗ 푌∆푌)[푑 − 푝] project ������→ 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌(pr∗ 푋 휔푋[푑푋] ⊗ pr∗ 푌 Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝]) proj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' �����→ form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌(pr∗ 푋 휔푋[푑푋]) ⊗ Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] tr ⊗id �����→ 푅ΓΦ푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18) where the first map is induced by a projection Ω푑−푝 푋×푌(log pr∗ 푌∆푌)[푑 − 푝] → pr∗ 푋 휔푋[푑푋] ⊗ pr∗ 푌 Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] coming from a Künneth-type decomposition of Ω푑−푝 푋×푌(log pr∗ 푌∆푌), the second is the projection for- mula, and the last map is induced by a trace map with supports defined as the composition 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌(pr∗ 푋 휔푋[푑푋]) excision �������→ 푅pr푌∗푅Γpr−1 푌∗Φ푌(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9 �������������→ 푅ΓΦ푌푅pr푌∗(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌) tr�→ 푅ΓΦ푌풪푌 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='19) 12 pr푌∗ Γpr−1 푌∗ Φ푌 Ω푑−푝 푋×푌(log pr∗ 푌∆푌)[푑 − 푝] pr푌∗Γpr−1 푌∗Φ푌퐷푋×푌(Ω푝 푋×푌(log pr∗ 푌∆푌)(−pr∗ 푌∆푌)[푝]) pr푌∗Γpr−1 푌∗Φ푌퐷푋×푌(pr ∗ 푌Ω푝 ×푌(log ∆푌)(−∆푌)[푝]) pr푌∗Γpr−1 푌∗Φ푌(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝]) ΓΦ푌 퐷푌(Ω푝 푌(log ∆푌)(−∆푌)[푝]) = ΓΦ푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] excision excision+퐿푒푚푚푎 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11 푑pr∨ 푌 푑pr∨ 푌 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='13) pr푌∗(훾) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16) pr푌∗ Γpr−1 푌∗Φ푌Ω푑−푝 푋×푌(log pr∗ 푌∆푌)[푑 − 푝] pr푌∗Γpr−1 푌∗Φ푌(pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌풪푌 ⊗ pr ∗ 푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝]) ΓΦ푌Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] 푔∗ pr푌′∗ Γpr−1 푌′∗Φ푌′ Ω푑−푝 푋×푌′(log pr∗ 푌′∆푌′)[푑 − 푝] 푔∗pr푌′∗Γpr−1 푌′∗Φ푌′ (pr !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 푌′풪푌′ ⊗ pr ∗ 푌′Ω푑푌−푝 푌′ (log ∆푌′)(−∆푌)[푑푌 − 푝]) 푔∗ΓΦ푌′ Ω푑푌−푝 푌′ (log ∆푌′)(−∆푌′)[푑푌 − 푝] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='17) Figure 1: Modified versions of diagrams appearing in the proof of [CR11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4] (all functors derived) 13 Here the second map comes from the functoriality properties of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9, since there is an inclusion pr−1 푌∗ Φ푌 ⊆ pr−1 푌 Φ푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18) maps to a similar decomposition of the bot- tom row of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='17), and the only commutativity not guaranteed by standard functoriality properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' functoriality of the projection formula appearing in the second map of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18)) is that of 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌(pr∗ 푋 휔푋[푑푋]) ⊗ Ω 푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] 푅ΓΦ푌Ω 푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] 푅푔∗(푅 pr푌′∗ 푅Γpr−1 푌′∗ Φ푌′ (pr∗ 푋 휔푋[푑푋]) ⊗ Ω푑푌−푝 푌′ (log ∆푌′)(−∆푌′)[푑푌 − 푝]) 푅푔∗(푅ΓΦ푌′ Ω푑푌−푝 푌′ (log ∆푌′)(−∆푌′)[푑푌 − 푝]) tr ⊗id tr′ ⊗id (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='20) But applying one more projection formula to the bottom row of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='20), we see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='20) is obtained by tensoring the differential Ω푑푌−푝 푌 (log ∆푌)(−∆푌)[푑푌 − 푝] → 푅푔∗Ω푑푌−푝 푌′ (log ∆푌′)(−∆푌′)[푑푌 − 푝] with 푅 pr푌∗ 푅Γpr−1 푌∗ Φ푌(pr∗ 푋 휔푋[푑푋]) 푅ΓΦ푌풪푌 푅푔∗(푅 pr푌′∗ 푅Γpr−1 푌′∗ Φ푌′ (pr∗ 푋 휔푋[푑푋])) 푅푔∗(푅ΓΦ푌′ 풪푌′) tr ⊗id tr′ ⊗id (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='21) and the commutativity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='21) is proved in [CR11, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' So far we have proved: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 holds in case (i) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It remains to deal with case (ii) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4, and for this we use the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='23 (compare with [CR11, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Consider a diagram of pure-dimensional snc pairs (푋′, ∆푋′) (푋, ∆푋) (푌′, ∆푌′) (푌, ∆푌) 푔′ 횤′ 횤 푔 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='24) where 횤, 횤′ are pushing closed immersions and dim푌 − dim 푋 = dim 푌′ − dim 푋′ =∶ 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then, for all 푞 the diagram 횤∗Ω푞 푋(log ∆푋)[푞] 푅푔∗횤′ ∗Ω푞 푋′(log ∆푋′) Ω푞+푐 푌 (log ∆푌)[푞 + 푐] 푅푔∗Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐] 푑푔′∨ 푑푔∨ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='25) commutes, where the horizontal maps are induced by log differentials and the left vertical map is the composition 횤∗Ω푞 푋(log ∆푋)[푞] ≃�→ 횤∗푅ℋ표푚(Ω푑푋−푞 푋 (log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙ 푋) duality ������→ 푅ℋ표푚(횤∗Ω푑푋−푞 푋 (log ∆푋)(−∆푋)[푑푋 − 푞], 휔∙ 푌) 푑횤∨ ���→ 푅ℋ표푚(Ω푑푋−푞 푌 (log ∆푌)(−∆푌)[푑푋 − 푞], 휔∙ 푌) ≃�→ Ω푞+푐 푌 (log ∆푌)[푞 + 푐] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='26) and the right vertical arrow is 푅푔∗ of a similar composition on 푌′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 14 Note that the codimension hypotheses hold if 푔 is flat or a closed immersion transverse to 횤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' While it seems a proof following [CR11, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='22] step-by-step is possible, we instead reduce to the case proved there as follows: first, observe that there is an evident map from the cartesian diagram 푈푋′ 푈푋 푈푌′ 푈푌 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='27) of interiors to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Noting that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='25) will map to a similar diagram obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='27), that the compositions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='26) are at least compatible with Zariski localization, and that the situation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='27) is covered by [CR11, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='22], it will suffice to show that the natural map ℎ0푅ℋ표푚푌(횤∗Ω푞 푋(log ∆푋)[푞], 푅푔∗Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐]) → ℎ0푅ℋ표푚푈푌(횤∗Ω푞 푈푋[푞], 푅푔∗Ω푞+푐 푈푌′ [푞 + 푐]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This can be checked Zariski-locally at a point 푥 ∈ 푋 ⊆ 푌, so we may assume 푋 ⊆ 푌 is a global complete intersection, say of 푡1, … , 푡푐 ∈ 풪푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In that case the 푡푖 define a Koszul resolu- tion 풦∙(푡푖) → 풪푋, and because 푋′ = 푌′ ×푌 푋 = 푉(푡1◦푔, ⋯ 푡푐◦푔) is smooth of codimension 푐 by hypotheses, it must be that the 푡푖◦푔 are also a regular sequence, hence 퐿푖푔∗풪푋 = ℎ−푖푔∗풦∙(푡푖) = {풪푋′, 푖 = 0 0 otherwise in other words 퐿푔∗풪푋 = 풪푋′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Now using the fact that Ω푞 푋(log ∆푋) is locally free on 푋′ we conclude 퐿푔∗횤∗Ω푞 푋(log ∆푋)[푞] = 푔∗횤∗Ω푞 푋(log ∆푋)[푞] = 횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞] Next, applying derived adjunction to both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28) gives a commutative diagram 푅ℋ표푚푌(횤∗Ω 푞 푋(log ∆푋)[푞], 푅푔∗Ω 푞+푐 푌′ (log ∆푌′)[푞 + 푐]) 푅ℋ표푚푈푌(횤∗Ω 푞 푈푋[푞], 푅푔∗Ω 푞+푐 푈푌′ [푞 + 푐]) 푅푔∗푅ℋ표푚푌′(퐿푔∗횤∗Ω 푞 푋(log ∆푋)[푞], Ω 푞+푐 푌′ (log ∆푌′)[푞 + 푐]) 푅푔∗푅ℋ표푚푈푌′ (퐿푔∗횤∗Ω 푞 푈푋[푞], Ω 푞+푐 푈푌′ [푞 + 푐]) 푅푔∗푅ℋ표푚푌′(횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞], Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐]) 푅푔∗푅ℋ표푚푈푌′ (횤′ ∗푔′∗Ω푞 푈푋[푞], Ω푞+푐 푈푌′ [푞 + 푐]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='29) Getting even more Zariski-local we may assume Ω푞 푋(log ∆푋) is free, say generated by 푑푥1, … , 푑푥푛 and in that case 푅ℋ표푚푌′(횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞], Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐]) = ( ∏ 푖 푅ℋ표푚푌′(풪푋′푑푥푖[푞], 풪푌′[푞 + 푐])) ⊗ Ω푞+푐 푌′ (log ∆푌′) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='30) and by Grothendieck’s fundamental local isomorphism [Con00, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5] 푅ℋ표푚푌′(풪푋′[푞], 풪푌′[푞 + 푐])) ≃ ℰ푥푡푐 푌′(풪푋′, 풪푌′) ≃ det(ℐ푋′∕ℐ푋′)∨ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='31) (the last 2 as sheaves supported in degree 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In particular, this is an invertible sheaf on 푋′, and it follows that the left hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='30) is a locally free sheaf (supported in degree 0) on 푋′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Recalling 푋′ is smooth and so in particular reduced, and since 푈푌′ ∩ 푋′ is a dense open (this is part of the 15 hypothesis that 푋′ → 푌′ is a pulling map) the natural map ℎ0푅ℋ표푚푌′(횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞], Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐]) → ℎ0푅ℋ표푚푌′(횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞], Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐])|푈푌′ ≃ ℎ0푅ℋ표푚푈푌′ (횤′ ∗푔′∗Ω푞 푋(log ∆푋)|푈푌′ [푞], Ω푞+푐 푌′ (log ∆푌′)|푈푌′ [푞 + 푐]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='32) is injective, where on the third line we have applied localization for ℰ푥푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Now left-exactness of 푔∗ gives an injection ℎ0푅푔∗푅ℋ표푚푌′(횤′ ∗푔′∗Ω푞 푋(log ∆푋)[푞], Ω푞+푐 푌′ (log ∆푌′)[푞 + 푐]) → ℎ0푅푔∗푅ℋ표푚푈푌′ (횤′ ∗푔′∗Ω푞 푋(log ∆푋)|푈푌′ [푞], Ω푞+푐 푌′ (log ∆푌′)|푈푌′ [푞 + 푐]) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='33) To complete the proof, we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='29) to identify the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='33) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 holds in case (ii) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This follows by applying cohomology with supports to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This completes our proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 (projection formula, compare with [CR11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푓 ∶ 푋 → 푌 be a map of smooth schemes admitting two different enhancements to maps of smooth schemes with supports, (푋, ∆푋, Φ푋) → (푌, ∆푌, 푓(Φ푋)) pushing and (푋, 푓∗(∆′ 푌), 푓−1(Φ푌)) → (푌, ∆′ 푌, Φ푌) pulling Assume in addition that ∆푋 + 푓∗(∆′ 푌) and ∆푌 + ∆′ 푌 are (reduced) snc divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then (푋, ∆푋 + 푓∗(∆′ 푌), Φ푋 ∩ 푓−1(Φ푌)) → (푌, ∆푌 + ∆′ 푌, 푓(Φ푋) ∩ Φ푌) is also a pushing map, and 푓∗(훽 ⌣ 푓∗훼) = 푓∗훽 ⌣ 훼 ∈ 퐻∗(푌, ∆푌 + ∆′ 푌, 푓(Φ푋) ∩ Φ푌) for any 훼 ∈ 퐻∗(푌, ∆′ 푌, Φ푌) and 훽 ∈ (푋, ∆푋, Φ푋), where ⌣ is the cup product on log Hodge cohomology defined along the lines of [CR11, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This is a formal consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 and can be derived following the proof of [CR11, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Again we use a factorization through the graph like (푋, ∆푋 + 푓∗(∆′ 푌), Φ푋 ∩ 푓−1(Φ푌)) (푌, ∆푌 + ∆′ 푌, 푓(Φ푋) ∩ Φ푌) (푋 × 푋, pr∗ 1 ∆푋 + pr∗ 2 푓∗(∆′ 푌), Φ푋 × 푓−1(Φ푌)) (푋 × 푌, pr∗ 1 ∆푋 + pr∗ 2 ∆′ 푌, Φ푋 × Φ푌) (푌 × 푌, pr∗ 1 ∆푌 + pr∗ 2 ∆′ 푌, 푓(Φ푋) × Φ푌) id푋×id푋 푓 id푌×id푌 id푋×푓 푓×id푌 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='36) Here 푓 × id푌 on the bottom is a pushing morphism (since 푓|Φ푋 is proper and 푓∗∆푌 = ∆푋) and the right vertical map id푌 × id푌 is a closed immersion transverse to 푓 × id푌 since the outer rectangle is cartesian and 푋 is smooth of the correct codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' This means we are in a situation to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1, and that lemma plus the definition of cup products in terms of pullbacks along diagonals gives the desired identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 16 4 Correspondences Given snc pairs with familes of supports (푋, ∆푋, Φ푋) and (푌, ∆푌, Φ푌) with dimensions 푑푋 and 푑푌, as in [CR11, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3] we may define a family of supports 푃(Φ푋, Φ푌) on 푋 × 푌 by 푃(Φ푋, Φ푌) ∶= {closed subsets 푍 ⊆ 푋 × 푌 | pr푌|푍 is proper and for all 푊 ∈ Φ푋, pr푌(pr−1 푋 (푊) ∩ 푍) ∈ Φ푌} (the conditions of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 are straightforward to verify).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For convenience we will let ∆푋×푌 ∶= pr∗ 푋∆푋 + pr∗ 푌∆푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A class 훾 ∈ 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) defines homomorphisms cor(훾) ∶ 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞+푗−푑푋 Φ푌 (푌, Ω푝+푖−푑푋 푌 (log ∆푌)) by the formula cor(훾)(훼) ∶= pr푌∗(pr∗ 푋(훼) ⌣ 훾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Moreover if (푍, ∆푍, Φ푍) is another snc pair with supports and 훿 ∈ 퐻푗′ 푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ 푌×푍(log ∆푌×푍)(−pr∗ 푌∆푌)), then pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿)) ∈ 퐻푗+푗′−푑푌 푃(Φ푋,Φ푍)(푋 × 푍, Ω푖+푖′−푑푌 푋×푍 (log ∆푋×푍)(−pr∗ 푋∆푋)) and cor(pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿))) = cor(훿)◦ cor(훾) as homomorphisms 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞+푗+푗′−푑푋−푑푌 Φ푍 (푍, Ω푝+푖+푖′−푑푋−푑푌 푍 (log ∆푍)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The sheavesΩ푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋) are particular instancesof the sheavesΩ푖 푋(퐴, 퐵) appearing in [DI87, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Such correspondences involving both log poles and “log zeroes” appear to have been considered before at least in crystalline cohomology, for example in work of Mieda [Mie09a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Mie09b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' However, I was unable to find any published proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' We make two observations: first, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='18 there are natural wedge product pairings Ω푝 푋×푌(log ∆푋×푌) ⊗ Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋) ∧�→ Ω푝+푖 푋×푌(log ∆푌) Second,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' essentially by the definition of 푃(Φ푋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Φ푌) the Künneth morphism on cohomology for the tensor product Ω푝 푋×푌(log ∆푋×푌) ⊗ Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋) can be enhanced with supports as 퐻푞 pr−1 푋 (Φ푋)(푋 × 푌,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ω푝 푋×푌(log ∆푋×푌)) ⊗ 퐻푗 푃(Φ푋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Φ푌)(푋 × 푌,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) → 퐻푝+푗 Ψ (푋 × 푌,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ω푝 푋×푌(log ∆푋×푌) ⊗ Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) where Ψ ∶= pr−1 푌∗(Φ푍) (see [CR11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Combining these 2 observations gives a pairing 퐻푞 pr−1 푋 (Φ푋)(푋 × 푌, Ω푝 푋×푌(log ∆푋×푌)) ⊗ 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) ⌣ ��→ 퐻푝+푗 Ψ (푋 × 푌, Ω푝+푖 푋×푌(log ∆푌)) Now note that pr푋 ∶ (푋×푌, ∆푋×푌, pr−1 푋 (Φ푋)) → (푋, ∆푋, Φ푋) is a pulling morphism, so by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='24 there is an induced map pr∗ 푋 ∶ 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) → 퐻푞 pr−1 푋 (Φ푋)(푋 × 푌, Ω푝 푋×푌(log ∆푋×푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' On the other hand since pr푌 ∶ (푋 × 푌, ∆푌, Ψ) → (푌, ∆푌, Φ푌) is a pushing morphism, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28 provides 17 a morphism pr푌∗ ∶ 퐻푝+푗 Ψ (푋 × 푌, Ω푝+푖 푋×푌(log ∆푌)) → 퐻푞+푗−푑푋 Φ푌 (푌, Ω푝+푖−푑푋 푌 (log ∆푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Composing, we obtain the desired homomorphism 퐻푞 Φ푋(푋, Ω푝 푋(log ∆푋)) pr∗ 푋 ���→ 퐻푞 pr−1 푋 (Φ푋)(푋 × 푌, Ω푝 푋×푌(log ∆푋×푌)) ⌣훾 ���→ 퐻푝+푗 Ψ (푋 × 푌, Ω푝+푖 푋×푌(log ∆푌)) pr푌∗ ����→ 퐻푞+푗−푑푋 Φ푌 (푌, Ω푝+푖−푑푋 푌 (log ∆푌)) For the “moreover” half of the lemma, we again begin with a certain wedge product pairing, this time on 푋 × 푌 × 푍: Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌)(−pr∗ 푋∆푋) ⊗ Ω푖′ 푋×푌×푍(log pr∗ 푌×푍∆푌×푍)(−pr∗ 푌∆푌) ∧�→ Ω푖+푖′ 푋×푌×푍(log pr∗ 푋×푍∆푋×푍)(−pr∗ 푋∆푋) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3) If 푉 ∈ 푃(Φ푋, Φ푌), 푊 ∈ 푃(Φ푌, Φ푍) then unravelling definitions (again we refer to [CR11, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10] for a similar claim) we find: pr푋×푍|pr−1 푋×푌(푉)∩pr−1 푌×푍(푊) is proper and pr푋×푍(pr−1 푋×푌(푉) ∩ pr−1 푌×푍(푊)) ∈ 푃(Φ푋, Φ푍) so that the Künneth morphism on cohomology associated to the left hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3) can be en- hanced with supports like 퐻푗 pr−1 푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌)(−pr∗ 푋∆푋)) ⊗ 퐻푗′ pr−1 푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′ 푋×푌×푍(log pr∗ 푌×푍∆푌×푍)(−pr∗ 푌∆푌)) → 퐻푗+푗′ Σ (푋 × 푌 × 푍, Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌)(−pr∗ 푋∆푋) ⊗ Ω푖′ 푋×푌×푍(log pr∗ 푌×푍∆푌×푍)(−pr∗ 푌∆푌)) where Σ ∶= pr−1 푋×푍∗(푃(Φ푋, Φ푍)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Since pr푋×푌 ∶ (푋 × 푌 × 푍, pr∗ 푋×푌∆푋×푌, pr−1 푋×푌(푃(Φ푋, Φ푌))) → (푋 × 푌, ∆푋×푌, 푃(Φ푋, Φ푌)) is a pulling morphism, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='24 gives an induced morphism Ω푖 푋×푌(log ∆푋×푌) → 푅푓∗Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' twisting by −∆푋×푌 and applying the projection formula gives a morphism Ω푖 푋×푌(log ∆푋×푌)(−∆푋×푌) → 푅푓∗ (Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌)(−pr∗ 푋×푌∆푋×푌)) and then taking cohomology with supports along 푃(Φ푋, Φ푌) and using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9 gives a mod- ified pullback map 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−∆푋×푌)) → 퐻푗 pr−1 푋×푌(푃(Φ푋,Φ푌))(푋 × 푌 × 푍, Ω푖 푋×푌×푍(log pr∗ 푋×푌∆푋×푌)(−pr∗ 푋∆푋)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4) and a similar argument gives a modified pullback 퐻푗′ 푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ 푌×푍(log ∆푌×푍)(−∆푌×푍)) → 퐻푗′ pr−1 푌×푍(푃(Φ푌,Φ푍))(푋 × 푌 × 푍, Ω푖′ 푋×푌×푍(log pr∗ 푌×푍∆푌×푍)(−pr∗ 푋∆푌)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5) On the other hand, pr푋×푍 ∶ (푋 × 푌 × 푍, pr∗ 푋×푍∆푋×푌, Σ) → (푋 × 푍, ∆푋×푍, 푃(Φ푋, Φ푍)) is a pushing morphism and hence by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='28 induces morphisms 푅pr푋×푍∗푅ΓΣ(Ωdim푋×푌×푍−푘 푋×푌×푍 (log pr∗ 푋×푍∆푋×푌)) → 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘 푋×푍 (log ∆푋×푍)[− dim푍] 18 for all 푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' twisting by −pr∗ 푋∆푋 and applying the projection formula this becomes 푅pr푋×푍∗푅ΓΣ(Ωdim 푋×푌×푍−푘 푋×푌×푍 (log pr∗ 푋×푍∆푋×푌)(−pr∗ 푋∆푋)) → 푅Γ푃(Φ푋,Φ푍)Ωdim 푋×푍−푘 푋×푍 (log ∆푋×푍)(−pr∗ 푋∆푋)[− dim푍] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6) Now letting 푘 = dim 푋 × 푌 × 푍 − 푖 − 푖′, the induced morphisms of cohomology with supports are 퐻푗+푗′ Σ (푋 × 푌 × 푍, Ω푖+푖′ 푋×푌×푍(log pr∗ 푋×푍∆푋×푌)(−pr∗ 푋∆푋)) → 퐻푗+푗′−dim푍 푃(Φ푋,Φ푍) (푋 × 푍, Ω푖+푖′−dim 푍 푋×푍 (log ∆푋×푍)(−pr∗ 푋∆푋)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7) Combining the above ingredients, we obtain a bilinear pairing 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−∆푋×푌)) ⊗ 퐻푗′ 푃(Φ푌,Φ푍)(푌 × 푍, Ω푖′ 푌×푍(log ∆푌×푍)(−∆푌×푍)) → 퐻푗+푗′−dim 푍 푃(Φ푋,Φ푍) (푋 × 푍, Ω푖+푖′−dim푍 푋×푍 (log ∆푋×푍)(−pr∗ 푋∆푋)) sending 훾 ⊗ 훿 ↦→ pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It remains to be seen that cor(pr푋×푍∗(pr∗ 푋×푌(훾) ⌣ pr∗ 푌×푍(훿))) = cor(훿)◦ cor(훾) and for this we will make repeated use of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Consider the diagram of smooth schemes 푋 × 푌 × 푍 푋 × 푌 푌 × 푍 푋 푌 푍 ∗ where all morphisms are projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' There are various ways to enhance this to include supports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' here we add the family of supports Ψ on 푋 × 푌 defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then in the cartesian diagram (∗), pr푌 ∶ (푋 × 푌, Ψ) → (푌, Φ푌) and pr푌×푍 ∶ (푋 × 푌 × 푍, pr−1 푋×푌Ψ) → (푌 × 푍, pr−1 푌 Φ푌) are pushing morphisms, whereas pr푋×푌 and pr푌 are pulling morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' At the same time, we have a pulling morphism pr푋×푍 ∶ (푋 × 푌 × 푍, pr−1 푋×푍(푃(Φ푌, Φ푍))) → (푌 × 푍, 푃(Φ푌, Φ푍)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' To be precise in what follows, whenever ambiguity is possible we will use notation like pr푋×푌 푋 to denote the projection 푋 × 푌 → 푋, pr푋×푌×푍 푋 to denote the projection 푋 × 푌 × 푍 → 푋 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Applying Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 first to pr푋×푍 we see that pr푌×푍∗(pr∗ 푋×푌(pr푋×푌∗ 푋 훼 ⌣ 훾) ⌣ pr∗ 푌×푍훿) = pr푌×푍∗(pr∗ 푋×푌(pr푋×푌∗ 푋 훼 ⌣ 훾)) ⌣ 훿 and then applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 to (∗) shows pr푌×푍∗(pr∗ 푋×푌(pr푋×푌∗ 푋 훼 ⌣ 훾)) = pr푌×푍∗ 푌 (pr푋×푌 푌∗ (pr푋×푌∗ 푋 훼 ⌣ 훾)) = pr푌×푍∗ 푌 cor(훾)(훼) so that pr푌×푍∗(pr∗ 푋×푌(pr푋×푌∗ 푋 훼 ⌣ 훾) ⌣ pr∗ 푌×푍훿) = pr푌×푍∗ 푌 cor(훾)(훼) ⌣ 훿 Applying pr푌×푍 푍∗ we conclude that cor 훿(cor 훾)(훼)) = pr푋×푌×푍 푍∗ (pr푋×푌×푍∗ 푋 훼 ⌣ pr∗ 푋×푌훾 ⌣ pr∗ 푌×푍훿) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) Finally, we rewrite the right hand side as pr푋×푍 푍∗ pr푋×푍∗(pr∗ 푋×푍pr푋×푍∗ 푋 훼 ⌣ pr∗ 푋×푌훾 ⌣ pr∗ 푌×푍훿) 19 and apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='35 to pr푋×푍 (with the pushing morphism (푋 ×푌 ×푍, Σ) → (푋 ×푍, 푃(Φ푋, Φ푍)) and pulling morphism (푋 × 푌 × 푍, pr푋×푌×푍−1 푋 (Φ푋)) → (푋 × 푍, pr푋×푍−1 푋 (Φ푋))) to arrive at pr푋×푍∗(pr∗ 푋×푍pr푋×푍∗ 푋 훼 ⌣ pr∗ 푋×푌훾 ⌣ pr∗ 푌×푍훿) = pr푋×푍∗ 푋 훼 ⌣ pr푋×푍∗(pr∗ 푋×푌훾 ⌣ pr∗ 푌×푍훿) Applying pr푋×푍 푍∗ on both sides shows that the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) is cor(pr푋×푍∗(pr∗ 푋×푌훾 ⌣ pr∗ 푌×푍훿)(훼), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' There is a Grothendieck-Serre dual approach to such correspondences, where classes 훾 ∈ 퐻푗 푃(Φ푋,Φ푌)(푋 × 푌, Ω푖 푋×푌(log ∆푋×푌)(−pr∗ 푌∆푌)) define homomorphisms 퐻푞(푋, Ω푝 푋(log ∆푋)(−∆푋)) → 퐻푞+푗−푑푋(푌, Ω푝+푖−푑푋 푌 (log ∆푌)(−∆푌)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The construction is formally similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' References [Bar18] Lawrence Jack Barrott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='“Logarithmic Chow theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='In: arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='03746 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [BBG22] Christian Böhning, Hans-Christian Graf von Bothmer, and Michel van Garrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Prelog Chow rings and degenerations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Rendiconti del Circolo Matematico di Palermo Series 2 (2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [BPØ20] Federico Binda, Doosung Park, and Paul Arne Østvær.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='“Triangulated Categories of Logarithmic Motives over a Field”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12298 [math] (Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' arXiv: 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12298 [math].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Con00] Brian Conrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Grothendieck duality and base change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Springer-Verlag, Berlin, 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' vi+296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 3-540-41134-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/b75857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/b75857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Con07] Brian Conrad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Deligne’s notes on Nagata compactifications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ramanu- jan Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3 (2007), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 205–257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0970-1249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [CR11] Andre Chatzistamatiou and Kay Rülling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Higher direct images of the structure sheaf in positive characteristic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='In: Algebra Number Theory 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 693– 775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='issn: 1937-0652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2140/ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2140/ant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [CR15] Andre Chatzistamatiou and Kay Rülling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Vanishing of the higher direct im- ages of the structure sheaf”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2131–2144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0010-437X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1112/S0010437X15007435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1112/S0010437X15007435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Del71] Pierre Deligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Théorie de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 40 (1971), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='issn: 0073-8301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/item?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='id=PMIHES_1971__40__5_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [DI87] Pierre Deligne and Luc Illusie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Relèvements modulo 푝2 et décomposition du complexe de de Rham”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 247–270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0020- 9910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/BF01389078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/BF01389078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [DM69] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Deligne and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Mumford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “The Irreducibility of the Space of Curves of given Genus”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Hautes Études Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 36 (1969), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 75–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0073-8301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='numdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/item?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='id=PMIHES_1969__36__75_0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [EV92] Hélène Esnault and Eckart Viehweg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lectures on vanishing theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' DMV Seminar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Birkhäuser Verlag, Basel, 1992, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' vi+164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 3-7643-2822-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/978-3-0348-8600-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/978-3-0348-8600-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 20 [God22] Charles Godfrey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Higher direct images ofsnc ideal sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='01142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='01142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Har77] Robin Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Graduate Texts in Mathematics, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Springer-Verlag, New York-Heidelberg, 1977, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' xvi+496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 0-387-90244- 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Hir64] Heisuke Hironaka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Resolution of singularities of an algebraic variety over a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' I, II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (2) 79 (1964), 109–203;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (2) 79 (1964), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='205–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='issn: 0003-486X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2307/1970547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2307/1970547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [KM98] János Kollár and Shigefumi Mori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Birational geometry ofalgebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge Tracts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' With the collaboration of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Clemens and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Corti, Translated from the 1998 Japanese original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge Univer- sity Press, Cambridge, 1998, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='viii+254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 0-521-63277-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/CBO9780511662560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/CBO9780511662560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Kol13] János Kollár.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Singularities of the minimal model program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge Tracts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' With a collaborationof Sándor Kovács.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge Uni- versity Press, Cambridge, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='x+370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='isbn: 978-1-107-03534-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/CBO9781139547895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/CBO9781139547895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Kov20] Sándor J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Kovács.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Rational Singularities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='02269 [math] (July 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='02269 [math].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [KX16] János Kollár and Chenyang Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “The dual complex of Calabi-Yau pairs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: In- vent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='527–557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0020-9910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/s00222-015-0640-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1007/s00222-015-0640-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Lef53] Solomon Lefschetz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Princeton University Press, Princeton, NJ, 1953, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' ix+233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Mie09a] Yoichi Mieda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Cycle classes, Lefschetz trace formula and integrality for p-adic cohomology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Algebraic Number Theory and Related Topics 2007, RIMS Kôkyûroku Bessatsu B12 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='kurims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='jp/~kenkyubu/bessatsu/open/B12/pdf/B12_005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Mie09b] Yoichi Mieda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “Integral Log Crystalline Cohomology and Algebraic Correspon- dences”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: Proceedings of Kinosaki Algebraic Geometry Symposium (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='jp/~mieda/pdf/kinosaki2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [MVW06] Carlo Mazza, Vladimir Voevodsky, and Charles Weibel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lecture Notes on Mo- tivic Cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Clay Mathematics Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' American Mathemat- ical Society, Providence, RI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Clay Mathematics Institute, Cambridge, MA, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' xiv+216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 978-0-8218-3847-1 0-8218-3847-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Nag63] Masayoshi Nagata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' “A generalization of the imbedding problem of an abstract variety in a complete variety”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Kyoto Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3 (1963), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 89–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' issn: 0023-608X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1215/kjm/1250524859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1215/kjm/1250524859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Ogu18] Arthur Ogus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lectures on logarithmic algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge Studies in Advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Cambridge University Press, Cambridge, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' xviii+539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 978-1-107-18773-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/9781316941614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1017/9781316941614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 21 [R&D] RobinHartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Residues and duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='Lecture notes of a seminar on the work of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Grothendieck, given at Harvard 1963/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' With an appendix by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Deligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Lecture Notes in Mathematics, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Springer-Verlag, Berlin-New York, 1966, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' vii+423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Stacks] The Stacks project authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='TheStacks project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='url: https://stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' [Voi14] Claire Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Chow Rings, Decomposition of the Diagonal, and the Topology of Families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='PrincetonUniversity Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' isbn: 9780691160511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='org/stable/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='ctt5hhp7w (visited on 12/29/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A Attempts to construct a fundamental class of a thrifty bira- tional equivalence As mentioned in Section 1 inspiration for this work was the following remarkable theoremof Chatzistamatiou- Rülling: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 ([CR11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8] (see also [CR15, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1], [Kov20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푘 be a perfect field and let 푆 be a scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Suppose 푋 and 푌 are two separated, finite type 푘-schemes which are (푖) smooth over 푘 and (푖푖) properly birational over 푆 in the sense that there is a commutative diagram 푍 푋 푌 푆 푟 푠 푓 ↺ 푔 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2) with 푟 and 푠 proper birational morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푛 = dim 푋 = dim 푌 = dim 푍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then, there are isomorphisms of sheaves 푅푖푓∗풪푋 ∼�→ 푅푖푔∗풪푌 and 푅푖푓∗휔푋 ∼�→ 푅푖푔∗휔푌 for all 푖, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='3) This result implies, for example, that if 푆 is a variety over a perfect field 푘 with a rational res- olution, that is, a resolution of singularities 푓 ∶ 푋 → 푆 such that 푅푓∗풪푋 = 풪푆, then every other resolution 푔 ∶ 푌 → 푆 satisfies 푅푔∗풪푌 = 풪푆 and is hence also rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In characteristic 0 this was a corollary of Hironaka’s resolution of singularities [Hir64];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' in positive characteristic it remained open until 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The original proof in [CR11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8] makes use of a cycle morphism cl ∶ 퐶퐻∗(푋) → 퐻∗(푋, Ω∗ 푋) from Chow cohomology to Hodge cohomology, which is ultimately applied to a cycle 푍 ⊂ 푋 × 푌 ob- tained from a properbirational equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' That cycle morphismsatisfies 2 essential properties:the first is that it is compatible with correspondences: here Chow correspondences are homomorphisms 퐶퐻∗(푋) → 퐶퐻∗(푌) of the form 훼 ↦→ pr푌∗(pr∗ 푋훼 ⌣ 훾) for some 훾 ∈ 퐶퐻∗(푋 × 푌) where ⌣ is the cup product induced by intersecting cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Hodge correspondences are defined in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The second key property is a compatibility with the filtrations 퐶퐻푛(푋 × 푌) = 퐹0퐶퐻푛(푋 × 푌) ⊇ 퐹1퐶퐻푛(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐶퐻푛(푋 × 푌) ⊇ 0 where 퐹푐퐶퐻푛(푋×푌) is the subgroup generated by cycles 푍 ⊆ 푋×푌 such that codim(pr푌푍 ⊆ 푌) ≥ 푐, and 퐻푛(푋 × 푌, Ω푚 푋×푌) = 퐹0퐻푛(푋 × 푌, Ω푚 푋×푌) ⊇ 퐹1퐶퐻∗(푋 × 푌) ⊇ ⋯ ⊇ 퐹dim푌퐻푛(푋 × 푌, Ω푚 푋×푌) ⊇ 0 22 where 퐹푐퐻푛(푋 ×푌, Ω푚 푋×푌) is the image of the map 퐻푛(푋 ×푌, ⊕푚 푗=푐Ω푚−푗 푋 ⊠Ω푗 푌) → 퐻푛(푋 ×푌, Ω푚 푋×푌) coming from the Künneth decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It is natural to ask if a similar method can be applied to prove an analogue of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 for pairs, which might read something like Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In order to state this analogue, we require a few additional definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' For the remainder of this appendix we work over a fixed perfect field 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4 (slightly simplified version of [Kol13, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A pair (푋, ∆푋) over 푘 will mean a reduced, equidimensional and 푆2 scheme 푋 of finite type over 푘 admitting a dualizing com- plex , together with a ℚ-Weil divisor ∆푋 = ∑ 푖 푎푖퐷푖 on 푋 such that no irreducible component 퐷푖 of ∆푋 is contained in Sing(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A stratum of a simple normal crossing pair (푋, ∆푋 = ∑ 푖 퐷푖) is a connected (equiv- alently, irreducible) component of an intersection 퐷퐽 = ∩푗∈퐽퐷푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Given any pair (푋, ∆푋), there is a largest open set 푈 ⊆ 푋 such that (푈, ∆푋|푈) is a simple nor- mal crossing pair, and we will refer to the resulting simple normal crossing pair as snc(푋, ∆푋) ∶= (푈, ∆푋|푈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6 ( compare with [Kol13, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='79-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='80], [KX16, §1, discussion before Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 10] ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let (푆, ∆푆 = ∑ 푖 퐷푖) be a pair, and assume ∆푆 is reduced and effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' A separated, finite type birational morphism 푓 ∶ 푋 → 푆 is thrifty with respect to ∆푆 if and only if (푖) 푓 is an isomorphism over the generic point of every stratum of snc(푆, ∆푆) and (푖푖) letting ̃퐷푖 = 푓−1 ∗ 퐷푖 for 푖 = 1, … , 푁 be the strict transforms of the divisors 퐷푖, and setting ∆푋 ∶= ∑ 푖 ̃퐷푖, the map 푓 is an isomorphism at the generic point of every stratum of snc(푋, ∆푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푘 be a perfect field, let 푆 be a scheme and let (푋, ∆푋) and (푌, ∆푌) be simple normal crossing pairs over 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Suppose (푋, ∆푋) and (푌, ∆푌) are properly birational over 푆 in the sense that there is a commutative diagram (푍, ∆푍) (푋, ∆푋) (푌, ∆푌) 푆 푟 푠 푓 ↺ 푔 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8) where 푟, 푠 are proper and birational morphisms, and assume ∆푍 = 푟−1 ∗ ∆푋 = 푠−1 ∗ ∆푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 푟 and 푠 are thrifty, then there are quasi-isomorphisms 푅푓∗풪푋(−∆푋) ≃ 푅푔∗풪푌(−∆푌) and 푅푓∗휔푋(∆푋) ≃ 푅푔∗휔푌(∆푌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9) Following [CR11] closely, one might begin by replacing the ordinary sheaves of differentials Ω푋 appearing in Hodge cohomology with sheaves of differentials with log poles Ω푋(log ∆푋) and attempt to implement a similar strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' starting a cycle 푍 ⊂ 푋×푌 representinga thrifty proper birational equivalince, producing a correspondence in logarithmic Hodge cohomology and analyzing its prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Ultimately even the correspondences of Section 4 seem to be insufficient to deal with thrifty proper birational equivalences, as we illustrate in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The problem we encounter is elementary: looking at the recipe for the Hodge class cl(푍) of a subvariety 푍 ⊆ 푋, where 푍 and 푋 are smooth an projective (outlined in [Har77, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4]), we see that cl(푍) ultimately comes from the trace linear functional tr ∶ 퐻dim 푍(푍, 휔푍) → 푘, or Serre-dually the element 1 ∈ 퐻0(푍, 풪푍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Due to the introduction of log poles and zeroes in Section 4, trying to follow that recipe we pass through 23 cohomology groups of the form 퐻dim 푍(푍, 휔푍(퐷)), or dually 퐻0(푍, 풪푍(−퐷)) where 퐷 is an (often non-0 in cases of interest) effective Cartier divisor on 푍, and so there simply is no “1” to be had.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Beyond the difficulties described in the previous paragraph, when attempting to formulate a logarithmic variant of Chatzistamatiou-Rülling’s cycle morphism argument one is hampered by the fact that we are still in the early days of logarithmic Chow theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' It is not clear to the author which logarithmic variant of Fulton’s 퐶퐻∗, if any, could be used to construct a logarithmic cycle morphism with all of the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Further investigation of this question could be an interesting topic of future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Despite the aforementioned challenges, it is possible to prove a result almost identical to Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7 by entirely different methods [God22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 Obstructions to obtaining log Hodge correspondences from thrifty bi- rational equivalences Let (푋, ∆푋), (푌, ∆푌) be simple normal crossing pairs, and assume in additionthat 푋, 푌 are connected and proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푍 ⊆ 푋 × 푌 be a smooth closed subvariety with codimension 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In this situation the fundamental class of cl(푍) ∈ 퐻푐(푋 × 푌, Ω푐 푋×푌) (no log poles yet) can be described using only Serre duality, as follows (we refer to [Har77, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' the composition 퐻dim푍(푋 × 푌, Ωdim푍 푋×푌 ) → 퐻dim 푍(푍, Ωdim 푍 푍 ) tr�→ 푘 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10) (where tr is the trace map of Serre duality) is an element of 퐻dim푍(푋 × 푌, Ωdim푍 푋×푌 )∨ ≃ 퐻푐(푋 × 푌, Ω푐 푋×푌) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11) which we may define to be cl(푍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='9 In light of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 we might hope to modify eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='11) to obtain a class in 퐻푐(푋 ×푌, Ω푐 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let us focus on the case where pr푋|푍 ∶ 푍 → 푋, pr푌|푍 ∶ 푍 → 푌 are both thrifty and birational, so in particular 푐 = dim 푋 = dim 푌 =∶ 푑 and (pr푋|푍)−1 ∗ ∆푋 = (pr푌|푍)−1 ∗ ∆푌 =∶ ∆푍 To keep the notation under control, set 휋푋 ∶= pr푋|푍 and 휋푌 ∶= pr푌|푍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In this situation letting 휄 ∶ 푍 → 푋 × 푌 be the inclusion there is a natural map 푑휄∨ ∶ Ω푑 푋×푌(log ∆푋×푌) → 휄∗Ω푑 푍(log ∆푋×푌|푍) and twisting by −pr∗ 푌∆푌 gives a map Ω푑 푋×푌(log ∆푋×푌)(−pr∗ 푌∆푌) → 휄∗Ω푑 푍(log ∆푋×푌|푍)(−pr∗ 푌∆푌|푍) = 휄∗Ω푑 푍(log ∆푋×푌|푍)(−휋∗ 푌∆푌) To identify Ω푑 푍(log ∆푋×푌|푍)(−pr∗ 푋∆푋|푍), write (휋푋)∗∆푋 = (휋푋)−1 ∗ ∆푋 + 퐸푋 = ∆푍 + 퐸푋 and (휋푌)∗∆푌 = (휋푌)−1 ∗ ∆푌 + 퐸푌 = ∆푍 + 퐸푌 so that ∆푋×푌|푍 = (휋푋)∗∆푋 + (휋푌)∗∆푌 = 2∆푍 + 퐸푋 + 퐸푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' While the hypotheses guarantee ∆푍 is reduced it may be that 퐸푋, 퐸푌 are non-reduced — however something can be said about their multi- plicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 퐸푋 = ∑ 푖 푎푖 푋퐸푖 푋, 퐸푌 = ∑ 푖 푎푖 푌퐸푖 푌 where the 퐸푖 푋, 퐸푖 푌 are irreducible, then by a generalization of [Har77, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='6] (see also [Kol13, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='10]), 푎푖 푋 = mlt(휋푋(퐸푖 푋) ⊆ ∆푋) and since ∆푋 is a reduced effective simple normal crossing divisor, if in addition we write ∆푋 = ∑ 푖 퐷푖 푋, then mlt(휋푋(퐸푖 푋) ⊆ ∆푋) = |{푖 | 휋푋(퐸푖 푋) ⊆ 퐷푖 푋}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' The thriftiness hypothesis that 휋푋(퐸푖 푋) is not 8The reason the result is only “almost identical” is that in [God22] we require ostensibly stronger hypotheses on the base scheme 푆 (namely that it is excellent and noetherian), but it is possible that even in the situation of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='1 and Conjecture A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='7 one can reduce to this case, for example using noetherian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 9It may then be non-trivial to verify this agrees with other definitions, especially if we worry about signs, but we will not need that level of detail for what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 24 a stratum then implies 푎푖 푋 = mlt(휋푋(퐸푖 푋) ⊆ ∆푋) < codim(휋푋(퐸푖 푋) ⊂ 푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Since differentials with log poles are insensitive to multiplicities, we have Ω푑 푍(log ∆푋×푌|푍) = 휔푍(∆푍 + 퐸red 푋 + 퐸red 푌 ) where −red denotes the associated reduced effective divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Then Ω푑 푍(log ∆푋×푌|푍)(−휋∗ 푌∆푌) = 휔푍(∆푍 + 퐸red 푋 + 퐸red 푌 − ∆푍 − 퐸푌) 휔푍(퐸red 푋 + (퐸red 푌 − 퐸푌)) = 휔푍( ∑ 푖 퐸푖 푋 + ∑ 푖 (1 − 푎푖 푌)퐸푖 푌) The upshot is that we have an induced map 퐻푑(푋 × 푌, Ω푑 푋×푌(log ∆푋×푌)(−pr∗ 푌∆푌)) → 퐻푑(푍, 휔푍(퐸red 푋 + (퐸red 푌 − 퐸푌))) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12) Here the left hand side is Serre dual to 퐻푑(푋 × 푌, Ω푑 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)), so the 푘-linear dual of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='12) is a morphism 퐻푑(푍, 휔푍(퐸red 푋 + (퐸red 푌 − 퐸푌)))∨ → 퐻푑(푋 × 푌, Ω푑 푋×푌(log ∆푋×푌)(−pr∗ 푋∆푋)) Unfortunately10 퐻푑(푍, 휔푍(퐸red 푋 + (퐸red 푌 − 퐸푌))) is often 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 퐸푋 and 퐸푌 are both reduced (an explicit example where this holds will be given below), then 퐻푑(푍, 휔푍(퐸red 푋 +(퐸red 푌 −퐸푌))) = 퐻푑(푍, 휔푍(퐸푋)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If in addition 퐸푋 ≠ 0, we obtain 퐻푑(푍, 휔푍(퐸푋)) = 0 by an extremely weak (but characteristic inde- pendent) sort of Kodaira vanishing: Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푍 be a proper variety over a field 푘 with dimension 푑, and assume 푍 is normal and Cohen-Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' If 퐷 ⊂ 푍 is a non-0 effective Cartier divisor on 푍 then 퐻푑(푍, 휔푍(퐷)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' By Serre duality 퐻푑(푍, 휔푍(퐷)) = 퐻0(푍, 풪푍(−퐷)), which vanishes by the classic fact that “a nontrivial line bundle and its inverse can’t both have non-0 global sections.” Since I am not aware of a specific reference, here is a proof: Suppose towards contraditction that there is a non-0 global section 휎 ∈ 퐻0(푍, 풪푍(−퐷)) — then the composition 풪푍 풪푍(−퐷) 풪푍 휎 휏 is non-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' By [Stacks, Tag 0358] 퐻0(푍, 풪푍) is a (normal) domain, and since it’s also a finite dimen- sional 푘-vector space it must be an extension field of 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' But then 휏 ∈ 퐻0(푍, 풪푍) is invertible hence surjective, so 풪푍(−퐷) \ue0b4→ 풪푍 is surjective, which is a contradiction since by hypothesis the cokernel 풪퐷 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푋 = ℙ2 and let ∆푋 ⊂ 푋 be a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Let 푝 ∈ 퐿 be a 푘-point, let 푌 = Bl푝 푋 and let ∆푌 = ̃퐿 = the strict transform of 퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' Finally let 푓 ∶ 푌 → 푋 be the blowup map and let 푍 = (푓 ×id)(푌) ⊂ 푋 ×푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In this case (with all notation as above) 휋푋◦(푓 ×id) = 푓 and 휋푌◦(푓 ×id) = id푌, so under the isomorphism 푓 × id ∶ 푌 ≃ 푍, 퐸푋 is the exceptional divisor of 푓 (with multiplicity 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' On the other hand 퐸푌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' In particular 퐸푋 and 퐸푌 are reduced and 퐸푋 ≠ 0 so from the above discussion 퐻2(푍, 휔푍(퐸푋)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 10at least for the purposes of constructing log Hodge cohomology classes of subvarieties .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfo_h3/content/2301.00517v1.pdf'} diff --git a/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/2301.12146v1.pdf.txt b/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/2301.12146v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8474f700610a3610eec487bb4d01a67dbb3d4b8e --- /dev/null +++ b/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/2301.12146v1.pdf.txt @@ -0,0 +1,1007 @@ +arXiv:2301.12146v1 [math.NT] 28 Jan 2023 +On Tribonacci Sequences +Luke Pebody +Saturday 28, January 2023 +Abstract +Let a tribonacci sequence be a sequence of integers satisfying ak = +ak−1 + ak−2 + ak−3 for all k ≥ 4. +For any positive integers k and n, +denote by fk(n) the number of tribonacci sequences with a1, a2, a3 > 0 +and with ak = n. +For all n, there is a maximum k such that fk(n) is non-zero. Answering +a question of Spiro [1], we show that there is a finite upper bound (we +specifically prove 561001) on fk(n) for any positive integer n ≥ 3 and this +maximum k. +We do this by showing that fk(n) has transitions in n around constant +multiples of φ3k/2 (where φ is the real root of φ3 = φ2 + φ + 1): there +exists a constant C such that fk(n) > 0 whenever n > Cφ3k/2 and for any +constant T , the values of fk(n) with n < T φ3k/2 have an upper bound +independent of k. +1 +Introduction +A tribonacci sequence of length k is a sequence of integers ⟨ai⟩k +i=1 such that +ai = ai−1 + ai−2 + ai−3 for all 4 ≤ i ≤ k. +We say that such a sequence +terminates at ak and that it is positive if a1, a2, a3 > 0 - note that this easily +implies that ai > 0 for all i. Denote by fk(n) the number of tribonacci sequences +of length k terminating at n. +Clearly f1(n) = 1 for all n > 0, the only tribonacci sequence of length 1 +terminating at n being ⟨n⟩. Further, f2(n) = f3(n) = ∞ as we can choose any +values for the proceeding terms. +For n ≥ 3, there exists a tribonacci sequence of length longer than 3 termi- +nating at n, for example ⟨n − 2, 1, 1, n⟩. However for any tribonacci sequence +⟨ai⟩k +i=1 of length k, and for any 4 ≤ i ≤ k, ai = ai−1 + ai−2 + ai−3 ≥ ai−1 + 2, +so by induction ai ≥ 2i− 5 for all 3 ≤ i ≤ k, and hence if n < 2k − 5, fk(n) = 0. +Let t(n) be the largest number such that ft(n)(n) > 0. +Let p(n) denote the number of positive tribonacci sequences of length t(n) +terminating at n, so p(n) = ft(n)(n). +Clearly, since t(1) = t(2) = 3 it follows that p(1) = p(2) = ∞. Spiro [1] asks +Question 1. Does there exist some absolute constant c such that for all n ≥ 3, +p(n) ≤ c for all n? +1 + +The purpose of this paper is to give a positive answer to this question. Indeed +we will show +Theorem 2. For any integer n ≥ 3, there are at most 561001 positive tribonacci +sequences of length t(n) terminating at n. +It turns out the key question for our proof is the minimum size of the vector + + +a1 +a2 +a3 + + where ⟨ai⟩n +i=1 is a non-zero tribonacci sequence terminating at an = 0. +In Section 2 we will show a lower bound on such a sequence of the order of φn/2, +which will allow us to prove +Theorem 3. For any positive integers n, k with k ≥ 4, the number of positive +sequences of length k terminating at n is at most +⌈1500 +n +φ3k/2 ⌉2. +In Section 3 we turn to trying to put an upper bound on numbers that don’t +have any positive tribonacci sequences of length k terminating at them. This +is an instance of the Coin Problem, also known as calculating the Frobenius +Number. We construct two specific tribonacci sequences terminating at an = 0 +with + + +a1 +a2 +a3 + + being of the order of φn/2 and with the integers a1, a2, a3 having +specified signs, allowing us to prove +Theorem 4. For any integer n above 0.2φ3k/2, there exists a positive tribonacci +sequence of length k terminating at n. +This will be all that is required. +Proof of Theorem 2. There is no sequence of length t(n) + 1 terminating at n. +Hence by Theorem 4, it follows that n < 0.2φ3(t(n)+1)/2 = 0.2φ3/2φ3t(n)/2. +Thus from Theorem 3, it follows that there are at most +⌈1500 +n +φ3t(n)/2 ⌉2 ≤ ⌈15000.2φ3/2φ3t(n)/2 +φ3t(n)/2 +⌉2 +≤ ⌈300φ3/2⌉2 = 7492 = 561001 +positive tribonacci sequences of length t(n) terminating at n. +In Section 4, we will investigate which recurrence relations of the form xn = +axn−1+bxn−2+cxn−3 for non-negative a and b and for positive c the arguments +in this paper can be carried across to. We will extend the result in the earlier +sections to the following case. +2 + +Theorem 5. Suppose a, b, c are non-negative integers with a + b > 0, c = 1 +and such that x3 − ax2 − bx − c = 0 has exactly one real root. +Then there +is an absolute bound T such that if positive integers k ≥ 4 and n are such +that there are no positive sequences ⟨ai⟩k+1 +i=1 satisfying the recurrence relation +ai = aai−1 + bai−2 + cai−3 of length k + 1 terminating at n, then there are at +most T such sequences of length k terminating at n. +We will leave open the question of which linear recurrences satisfy this prop- +erty, but will at least demonstrate an example of a recurrence that does not. +In particular we will show the existence of positive integers k and n such +that there is no positive sequence ⟨ai⟩k+1 +i=1 satisfying the recurrence relation +ai = ai−1 + ai−2 + 2ai−3 of length k + 1 terminating at n, but for which the +number of such sequences of length k terminating at n is unbounded. +2 +Lower Bound +Let us say a sequence ⟨ai⟩∞ +i=1 is a reverse-tribonacci sequence if for all i ≥ 0, +ai = ai+1+ai+2+ai+3. Let us write out the expression for the reverse-tribonacci +sequence starting ⟨0, k, l⟩. Recall that φ is the real solution to φ3 = φ2 + φ + 1. +We write the complex roots as φ1 and φ2 = φ1. +Lemma 6. For all integers k, l, if ⟨ai⟩∞ +i=1 is a reverse-tribonacci sequence with +a1 = 0, a2 = k and a3 = l, then for all i, ai can be expressed as +ai = αφ−i + (kψ1 + lζ1)φ−i +1 + (kψ2 + lζ2)φ−i +2 += (αφ−3i/2 + β cos(γ − δi))φi/2, +where +ψ1 = +φ3 +1 + φ2 +1 +φ2 +1 + 2φ1 + 3 +ψ2 = ψ1 +ζ1 = +φ3 +1 +φ2 +1 + 2φ1 + 3 +ζ2 = ζ1 +α = kφ2 + (k + l)φ3 +φ2 + 2φ + 3 +βeγi = 2(kψ1 + lζ1) and +eδi = φ1 +� +φ. +Proof. Any two-way infinite tribonacci sequence ⟨ai⟩∞ +−∞ can be written as ai = +pφi + qφi +1 + rφi +2 for some p, q and r. +Thus any reverse-tribonacci sequence ⟨ai⟩∞ +−∞ can be written as ai = pφ−i + +qφ−i +1 + rφ−i +2 +for some p, q, r. Solving for the p, q, r that give a1 = 0, a2 = k and +a3 = l leads to the above expression. +3 + +Note that in the above expressions, ψ1, ψ2, ζ1, ζ2 and δ are constants that +do not depend on k and l. +Lemma 7. For any integers k and l, if α and β are defined as in Lemma 6, +then |α| ≤ |k| + |l| and β ≥ |k|+|l| +31 +. +Proof. α is roughly 0.9546k+0.6184l, which is clearly bounded above by |k|+|l|. +ψ1 is roughly 0.02267 − 0.217i and ζ1 is roughly 0.1908 − 0.0187i. As such, if +k and l are non-negative then the real part of 2(kψ1 + lζ1) (and hence β) is at +least 0.04(k + l) > |k|+|l| +31 +. +For k positive and l negative, the minimum value of +β +k−l is approximately +0.03221 > +1 +31, and is achieved around k = −0.3653l. +Finally we need a simple trigonometric property +Lemma 8. For any real numbers p and q with π +2 < q < π, the larger of | cos(p)| +and | cos(p + q)| is at least cos(q/2). +Proof. Note that cos(q) < 0. Thus +2(cos(p)2 + cos(p + q)2) = 2 cos(p)2 + 2 cos(p + q)2 += cos(2p) + cos(2p + 2q) + 2 += 2 cos(2p + q) cos(q) + 2 +≥ 2 + 2 cos(q) += 4 cos(q/2)2. +Thus either cos(p)2 ≥ cos(q/2)2 or cos(p + q)2 ≥ cos(q/2)2. +This allows us to put a lower bound on the size of at least one of each +consecutive pair of a reverse-tribonacci sequence. +Corollary 9. Given a non-zero integer reverse-tribonacci sequence ⟨ai⟩∞ +i=1 with +a1 = 0, for every integer n ≥ 2, either |an| > 0.01φn/2 or |an+1| > 0.01φ(n+1)/2 +(or both). +Proof. For n ≥ 2 if an and an+1 are both 0, then a1 is the same sign as an−1. +Since a1 = 0, it follows that the entire series must be 0. Since the sequence is +non-zero, it follows that either |an| ≥ 1 or |an+1 ≥ 1. Since 1 > 0.01φn/2 for +n ≤ 15, we have proved the statement for n ≤ 14. Thus we may assume n ≥ 15. +By Lemma 6, +ai +φi/2 can be written as αφ−3i/2 + β cos(γ − δi). +Now by Lemma 8, at least one of | cos(γ − δn)| and | cos(γ − δ(n − 1))| is +at least cos(δ/2) (δ = 2.176 is between π +2 and π). Let t be the choice from +{n − 1, n} that maximises | cos(γ − δt)|. +By Lemma 7, if we write α′ = +α +|k|+|l| and β′ = +β +|k|+|l| then |α′| ≤ 1 and +β′ > +1 +31. +4 + +Therefore +| at +φt/2 | = |αφ−3t/2 + β cos(γ − δt)| +≥ |α′φ−3t/2 + β′ cos(γ − δt)| +≥ |β′ cos(γ − δt)| − |α′φ−3t/2| +≥ 1 +31 cos(δ/2) − φ−3t/2 ≥ cos(δ/2) +31 +− φ−22.5 > 0.01. +Then we have a bound on the size of tribonacci sequences terminating at 0. +Corollary 10. For n ≥ 3, if ⟨ai⟩n +i=1 is a non-zero integer tribonacci sequence +terminating at 0 then either |a1| > 0.01φn/2 or |a2| > 0.01φ(n−1)/2 (or both). +Proof. Let k = an−1 and l = an−2. Then if ⟨bi⟩∞ +i=1 is the reverse-tribonacci +sequence with b1 = 0, b2 = k and b3 = l, then ai = bn+1−i for all 1 ≤ i ≤ n. +Then this is just a restatement of Corollary 9. +This is all we need to prove Theorem 3. +Proof of Theorem 3. Partition the tribonacci sequences of length k ≥ 4 termi- +nating at n ⟨ai⟩k +i=1 by the pair (⌊ +a1 +0.01φk/2 ⌋, ⌊ +a2 +0.01φk−1/2 ⌋). +If two sequences ⟨ai⟩k +i=1 and ⟨bi⟩k +i=1 have the same pair, then |a1 − b1| < +0.01φk/2 and |a2 − b2| < 0.01φ(k−1)/2 and hence, by Corollary 10, either ⟨ai − +bi⟩k +i=1 is zero everywhere or does not terminate at 0. +Thus each distinct tribonacci sequence of length k terminating at n has a +distinct pair. +Define tribonacci sequence by x1 = 1, x2 = 0, x3 = 0. Then if a1, a2, . . . , ak +is a positive tribonacci sequence, ai ≥ xia4 for i = 2, 3 and 4 and therefore +ak ≥ xka4. Now xk < φk/11 for all k ≥ 4 and hence a1 + a2 + a3 ≤ 11n +φk for all +tribonacci sequences of length k terminating at n. +Thus ⌊ +a1 +0.01φk/2 ⌋ is at most 1100n +φ3k/2 and ⌊ +a2 +0.01φ(k−1)/2 ⌋ is at most +1100n +φ3k−1/2 < 1500n +φ3k/2 . +It follows that the number of Tribonacci sequences of length k ≥ 4 termi- +nating at n is at most ⌈ 1500n +φ3k/2 ⌉2. +Note we have not worked hard here to get the best bound. In a previous +draft we had a much more complicated proof of an upper bound which showed, +in place of Corollary 10, that if ⟨ai⟩n +i=1 terminated at 0 then +� +a2 +1 + a2 +2 + a2 +3 > +0.28φn/2, which led to an upper bound for the main theorem of 42875. +3 +Upper Bound +In this section, we turn to numbers which are not the terminus for any tribonacci +sequence of length k, working towards a proof of Theorem 4. +5 + +To that end, define three infinite tribonacci sequences ⟨pi⟩∞ +i=1, ⟨qi⟩∞ +i=1 and +⟨ri⟩∞ +i=1 by (p1, p2, p3) = (1, 0, 0), (q1, q2, q3) = (0, 1, 0) and (r1, r2, r3) = (0, 0, 1). +It is clear that for any tribonacci sequence ⟨ai⟩n +i=1, an = a1pn + b1qn + c1rn. +Thus we are simply looking to get an upper bound on the largest number which +cannot be written as a positive integral linear combination of pn, qn and rn. +This is called the Frobenius Number of pn, qn and rn. +First let us see that a finite bound does exist. +Lemma 11. For all k ≥ 1, pk, qk and rk have no non-trivial common divisor. +Proof. If pk, qk and rk had a non-trivial common divisor t > 1 then t would be +a common divisor of the terminus of every tribonacci sequence of length k, from +which it would follow that t would in fact be a common divisor of pk+l for all +l ≥ 0 (since ⟨pi+l⟩k +1 is a tribonacci sequence of length k). +Then, since pi = pi+3 − (pi+1 + pi+2), it would follow that t would be a +common divisor of pk−1, pk−2 and all the way back to p0 = 1 by induction, +causing a contradiction. +We will use the following bound, which might be originally due to Killing- +bergtro. +Theorem 12. Suppose p, q and r are integers with no non-trivial common di- +visor and let us suppose ap = bq + cr and dq = ep + fr where a, c, d, f > 0 and +b, e ≥ 0. Then for every integer N ≥ ap + dq + r, N can be written in the form +xp + yq + zr for some positive integers p, q, r. +Proof. Let x, y, z be positive integers such that px + qy + rz is equivalent to +N (mod r), but for which px + qy + rz is minimal (such a triple x, y, z exist +because, as is well known, if p, q and r have no non-trivial common divisor then +all sufficiently large integers can be written in the form px + qy + rz, and many +of these sufficiently large integers are equivalent to N (mod r).) +Since px+qy+rz is minimal, px+qy+rz−r cannot be written as a positive +linear combination of x, y and z. +Thus in each of the equations +px + qy + rz − r = px ++ qy ++ r(z − 1) +px + qy + rz − r = p(x − a) ++ q(y + b) ++ r(z + c − 1) +px + qy + rz − r = p(x + e) ++ q(y − d) ++ r(z + f − 1), +it must follow that one of the coefficients must not be positive. Two of the +coefficients in each equation are clearly positive, so it follows that x ≤ a, y ≤ d +and z ≤ 1, so px + qy + rz ≤ pa + qd + r ≤ N. Since N and px + qy + rz +are equivalent modulo r, there exists a non-negative integer t such that N = +px + qy + rz + rt. Then N = px + qy + r(z + t). +Therefore, to show that all sufficiently large integers can be written as the +terminus of a tribonacci sequence of length k, we just need to find linear com- +binations of pn, qn and rn combining to 0, with particular signs of the combi- +nations. This is equivalent to finding tribonacci sequences ending at 0, which +6 + +Table 1: Table for Lemma 13 +t0 +t1 +k +l +α +β +γ +x0 +x1 +x2 +0 +0.06 +0 +1 +0.6184 +0.3834 +-0.0977 +0.3410 +-0.0500 +-0.1694 +0.06 +0.16 +-1 +2 +0.2822 +0.8027 +0.4640 +0.6879 +-0.0515 +-0.2163 +0.16 +0.22 +-1 +1 +-0.3362 +0.5200 +0.8677 +0.4526 +-0.0471 +-0.2482 +0.22 +0.35 +-1 +0 +-0.9546 +0.4364 +1.6749 +0.3778 +-0.0354 +-0.0294 +0.35 +0.45 +-1 +-1 +-1.5731 +0.6360 +2.3067 +0.5517 +-0.0538 +-0.1588 +0.45 +0.56 +0 +-1 +-0.6184 +0.3834 +3.0439 +0.3410 +-0.0500 +-0.0548 +0.56 +0.66 +1 +-2 +-0.2822 +0.8027 +-2.6776 +0.6879 +-0.0515 +-0.2163 +0.66 +0.72 +1 +-1 +0.3362 +0.5200 +-2.2739 +0.4526 +-0.0471 +-0.2482 +0.72 +0.85 +1 +0 +0.9546 +0.4364 +-1.4667 +0.3778 +-0.0354 +-0.0294 +0.85 +0.95 +1 +1 +1.5731 +0.6360 +-0.8349 +0.5517 +-0.0538 +-0.1588 +0.95 +1 +0 +1 +0.6184 +0.3834 +-0.0977 +0.3745 +-0.1864 +-0.0548 +is equivalent to finding reverse-tribonacci sequences starting at 0, and hence we +can again use the expression from Lemma 6, which states that if ⟨ai⟩∞ +i=1 is a +reverse-tribonacci sequence with a1 = 0, a2 = k and a3 = l then for all n +an = (αφ−3n/2 + β cos(γ − δn))φn/2. +Note that for all but an extremely small collection of n, the term β cos(γ+δn) +dwarves αφ−3n/2. As such, for a fixed k and l, the sign of an depends only +(except for a few very rare counterexamples) on the fractional part of +δ +2πn. +Lemma 13. For each integer n ≥ 4, there exists a tribonacci sequence ⟨ai⟩n +i=1 +terminating at an = 0, with a1 > 0, 0 ≥ a2, 0 > a3 and with a1 < 0.81φn/2. +Similarly for all such n, there exists a tribonacci sequence ⟨bi⟩n +i=1 terminating +at bn = 0, with b2 > 0, 0 ≥ b1, 0 > b3 and with b2 < 0.64φn/2. +Proof. We will split into cases based on the fractional part of δn +2π = 0.3464n. See +Table 1. For each row, if t0 ≤ δn +2π − ⌊ δn +2π⌋ ≤ t1, then for the given values of k and +l, if β and γ are as defined in Lemma 6, one can confirm that β cos(γ − δn) ≥ +x0 > 0.34, while β cos(γ − δ(n − 1)) ≤ x1 < −0.035 and β cos(γ − δ(n − 2)) ≤ +x2 < −0.029. +Furthermore, for all such k, l, |α| < 1.58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.017, +from which it follows that an > 0 > an−1, an−2. Further, +an +φn/2 < β + 0.017 < +0.81. +For 4 ≤ n < 7, we can verify the sequences (1, 0, −1, 0), (2, 0, −1, 1, 0) and +(2, 0, −1, 1, 0, 0) satisfy the conditions for (a1, a2, . . . , an). +For the sequence (b1, b2, . . . , bn), see Table 2. Here, for each row, if t0 ≤ +δn +2π − ⌊ δn +2π⌋ ≤ t1, then for the given values of k and l, if β and γ are as defined in +Lemma 6, one can confirm that β cos(γ − δn) ≤ x0 < −0.071, while β cos(γ − +δ(n − 1)) ≥ x1 > 0.33 and β cos(γ − δ(n − 2)) ≤ x2 < −0.041. +Furthermore, for all such k, l, |α| < 1.58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.017, +from which it follows that an−1 > 0 > an, an−2. +For 4 ≤ n < 7, we can verify the sequences (0, 1, −1, 0), (0, 1, −1, 0, 0) and +(−1, 2, −1, 0, 1, 0) satisfy the conditions for (b1, b2, . . . , bn). +This then completes our proof. +7 + +Table 2: Other table for Lemma 13 +t0 +t1 +k +l +α +β +γ +x0 +x1 +x2 +0 +0.06 +1 +-1 +0.3362 +0.5200 +2.2739 +-0.3362 +0.4625 +-0.0678 +0.06 +0.19 +1 +0 +0.9546 +0.4364 +1.4667 +-0.1176 +0.3862 +-0.0528 +0.19 +0.29 +1 +1 +1.5731 +0.6360 +0.8349 +-0.2812 +0.5639 +-0.0791 +0.29 +0.41 +0 +1 +0.6184 +0.3834 +0.0977 +-0.1311 +0.3369 +-0.0413 +0.41 +0.56 +-1 +1 +-0.3362 +0.5200 +-0.8677 +-0.0714 +0.4625 +-0.0678 +0.56 +0.69 +-1 +0 +-0.9546 +0.4364 +-1.6749 +-0.1176 +0.3862 +-0.0528 +0.69 +0.79 +-1 +-1 +-1.5731 +0.6360 +-2.3067 +-0.2812 +0.5639 +-0.0791 +0.79 +0.91 +0 +-1 +-0.6184 +0.3834 +-3.0439 +-0.1311 +0.3369 +-0.0413 +0.91 +1 +1 +-1 +0.3362 +0.5200 +2.2739 +-0.0714 +0.4641 +-0.2528 +Proof of Theorem 4. Lemma 13 gives us tribonacci sequences ⟨ai⟩n +i=1 and ⟨bi⟩n +i=1 +terminating at an = bn = 0. It follows that a1pn + a2qn + a3rn = 0 = b1qn + +b2qn + b3rn. +Since a1, b2 > 0 > a3, b3 and 0 ≥ a2, b1, it follows that we can write +a1pn = (−a2)qn + (−a3)rn and +b2qn = (−b1)p1 + (−b3)rn +satisfying the sign requirements of Theorem 12, so it follows that every integer +N ≥ a1pn + b2qn + rn can be written in the form xpn + yqn + zrn for some +positive integers x, y and z, and hence there exists a positive tribonacci sequence +of length k ending at N. +By the bounds on a1 and b2 given in Lemma 13, we have such a tribonacci +sequence for all N ≥ 0.81φk/2uk+0.64φk/2vk+wk. Since uk ≤ vk ≤ wk < 0.11φk +and 0.81φk/2+0.64φk/2+1 < 1.74φk/2, it follows that such a tribonacci sequence +exists for all N ≥ 0.2φ3k/2 as was required. +4 +Other cubic recurrences +For non-negative a, b, c we can ask a similar question for recurrences of the form +xn = axn−1 + bxn−2 + cxn−3. Formally, let us define ka,b,c(n) to be the largest +k such there is a positive k-element solution ⟨xi⟩k +i=1 to the recurrence relation +xi = axi−1 + bxi−2 + cxi−3, and define ta,b,c(n) to be the number of positive +ka,b,c(n)-element solutions that exist. +If c = 0, this is a quadratic recurrence, and the problem is already solved. If +a = 0, b = 0 and c = 1, the recurrence is xn = xn−3, and ka,b,c(n) is not defined +for any n. +For all a, b, c ≥ 0 with c ≥ 1 and a + b + c ≥ 2, say that the recurrence +xn = axn−1 + bxn−2 + cxn−3 is congenial if there exists a finite bound B such +that for all n, ta,b,c(n) = ∞ or ta,b,c(n) ≤ B. +Firstly let us note that not all recurrences are congenial. +Lemma 14. The recurrence xn = xn−1 + xn−2 + 2xn−3 is not congenial. +Proof. Let ⟨pn⟩∞ +n=1, ⟨qn⟩∞ +n=1 and ⟨rn⟩∞ +n=1 be the solutions to the recurrence +8 + +starting with ⟨1, 0, 0⟩, ⟨0, 1, 0⟩ and ⟨0, 0, 1⟩ respectively. +Then xn = x1pn + +x2qn + x3rn. +Solutions to the recurrence can be split as the sum of two parts - a sequence of +the form ⟨x(1) +n += 2n−1k⟩ and a sequence of the form ⟨x(2) +n ⟩ which is periodic with +period 3 with x(2) +1 ++x(2) +2 +x(2) +3 += 0. It is then easy to solve for k: x1 +x2 +x3 = +x(1) +1 ++ x(1) +2 ++ x(1) +3 += 7k, so k = x1+x2+x3 +7 +. +In particular, if you let tn = 2n−1 +7 +, pn−tn is periodic with period ⟨ 6 +7, − 2 +7, − 4 +7⟩, +qn − tn with period ⟨− 1 +7, 5 +7, − 4 +7⟩ and rn − tn with period ⟨− 1 +7, − 2 +7, 3 +7⟩. +For n = 3t, xn = c(x1 + x2) + (c + 1)xn−3 and xn+1 = 2(c + 1)x1 + (2c + +1)(x2 + x3) where c = 23t−1−1 +7 +. Then xn+1 cannot be equal to (2c + 1)(2c + 3) +for positive x1, x2, x3 (x1 would have to be a multiple of 2c + 1 that is positive +but less than 2c + 1), but for all 1 ≤ i ≤ 4c + 4, if x1 = i, x2 = 4c + 5 − i and +x3 = 3, then xn = c(4c + 5) + (c + 1)3 = 4c2 + 8c + 3 = (2c + 1)(2c + 3). +The proofs in this paper can be adapted to show that many other recurrences +are congenial. Let us say a polynomial x3 − ax2 − bx − c is affable if c = 1 and +it has exactly one real root, which is more than 1. We will show that affability +leads to congeniality. +For the rest of this section, fix an affable polynomial x3 − ax2 − bx − c with +real root η1 and complex roots η2 and η3 = η2. Note that |η2| = η−1/2 +1 +. +We will make use of the following equivalent to Lemma 6. +Lemma 15. Given a sequence ⟨xi⟩n +i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi +with xn = 0, xn−1 = k and xn−2 = l, xi can be expressed as +xi = +3 +� +j=1 +(kψj + lζj)ηn−i +j +for constants ψj, ζj depending only on x3 − ax2 − bx − c, which can be rewritten +as +xi +η(n−i)/2 +1 += αη−3(n−i)/2 +1 ++ β cos(γ − δ(n − i)) +where +α = kψ1 + lζ1, +βeγi = 2(kψ2 + lζ2) and +eδi = η2 +√η1. +We will follow the steps of the proof of Theorem 2 for all recurrence relations +corresponding to affable polynomials. We will not attempt to give an actual +bound. +We note the following, which will be used in the equivalents of both Theo- +rems 3 and 4 +9 + +Lemma 16. If real numbers k and l satisfy kψ2 + lζ2 = 0, then k = l = 0. +Proof. As ψ3 = ψ2 and ζ3 = ζ2, kψ3 + lζ3 = 0 and therefore the sequence with +xn = 0, xn−1 = k and xn−2 = l can simply be expressed as xi = (kψ1+lζ1)ηn−i +1 +. +As 0 = xn = kψ1 + lζ1, it follows that xi = 0 for all i and therefore k = l = +0. +We start by following the proof of Theorem 3. +Lemma 17. There exists an absolute bound M such that for n ≥ 4 and all +non-zero integer sequences ⟨xi⟩n +i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and +xn = 0 either |x1| ≥ Mηn/2 +1 +or |x2| ≥ Mηn/2 +1 +(or both). +Proof. The set of complex numbers kψ2 + lζ2 for k, l real with |k| + |l| = 1 is +a closed subset of the complex plane (in fact a hollow parallelogram) which, by +Lemma 16 does not contain 0. As such, there exists a constant V > 0 such +that for all such k, l, |kψ2 + lζ2| > V . +Then for all real k, l it follows that +β = |kψ2 + lζ2| > V (|k| + |l|). +Clearly if U = max(|ψ1|, |ζ1|), α ≤ U(|k| + |l|). +Pick integer N such that V cos(δ/2) − Uη−3(N−1)/2 +1 +is positive. Note we +can do this because π +2 < δ < π. Then let M > 0 be such that V cos(δ/2) − +Uη−3(N−2)/2 +1 +> Mη1 and η−N/2 +1 +> M. +Now if n ≤ N, then Mηn/2 +1 +< 1 (note that η1 > 1 since 13 < a×12+b×1+c) +and x1, x2 cannot both be 0 (as then xn would have to be the same sign as x3 +and non-zero). +For n > N, we know from Lemma 8 that there exists t ∈ {1, 2} such that +| cos(γ − δ(n − t))| > cos(δ/2) > 0. +For such t, n − t ≥ N − 1 and so it follows that +|xt| +η(n−t)/2 +1 += |αη−3n/2 +1 ++ β cos(γ − δn)| +≥ |β cos(γ − δn)| − |α|η−3n/2 +1 +≥ V cos(δ/2) − Uη−3n/2 +1 +≥ Mη1 +and hence |xt| ≥ Mη(n+2−t)/2 +1 +≥ Mηn/2 +1 +. +This is enough for the equivalent of Theorem 3 +Theorem 18. There exists a fixed bound T such that for any positive integers +n, k with k ≥ 4, the number of positive sequences ⟨xi⟩k +i=1 satisfying xi+3 = +axi+2 + bxi+1 + cxi and terminating at xk = n is at most +⌈T +n +η3k/2 +1 +⌉2. +10 + +Proof. There is a fixed P such that for any positive sequence ⟨ai⟩k +i=1 satisfying +the recurrence relation with k ≥ 4, Pηk +1(a1 + a2 + a3) ≤ ak. +Thus for any such sequence terminating at n, a1 and a2 are bounded above +by +n +P ηk +1 and for any two such sequences, by Lemma 17, either the first terms or +the second terms differ by at least Mηk/2 +1 +. +Thus the number of such sequences is at most ⌈ +n +P Mη3k +1 /2⌉2. +Now we proceed to follow the proof of Theorem 4. We will need the following +Corollary to Lemma 16 +Corollary 19. Given any interval 0 ≤ x < y ≤ 2π within (0, 2π), we can pick +non-zero integers k, l for which x < γ < y. +Proof. Lemma 16 says that the set {kψ2 + lζ2 : k, l ∈ R}, when viewed geomet- +rically as a subset of the complex plane, is not of dimension 1. Thus it must +be the entire complex plane. Pick x < z < y, then there exist real k, l with +kψ2 + lζ2 = eiz. +Now let kn = ⌊nk⌋ and ln = ⌊nl⌋. +The limit as n tends to infinity of +knψ2+lnζ2 +n +is eiz and therefore for all sufficiently large n, γ (which is the argument +of knψ2+lnζ2 +n +) must be contained in the open interval (x, y). +We shelve this for the moment and focus on a simple piece of trigonometry. +Lemma 20. For all numbers π +2 < δ < π, there exists t such that cos(t) > 0 > +cos(t + δ), cos(t + 2δ) +Proof. Pick t such that π +2 − δ < t < 3π +2 − 2δ. There exists such a t because +δ < π. +Since δ < π, − pi +2 < π +2 − δ < t. Similarly since π +2 < δ, t < 3π +2 − 2δ < pi +2 . So +− pi +2 < t < pi +2 and hence cos(t) > 0. +Further π +2 < t+δ < t+2δ < 3π +2 , so cos(t+δ) and cos(t+2δ) are negative. +This leads to the following somewhat technical-seeming lemma. +Lemma 21. For all numbers π +2 < δ < π, there exists an ǫ > 0 and finitely many +intervals ⟨(xi, yi)⟩n +i=1 such that for all t there exists an interval (xi, yi) such that +for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ). +Proof. Pick a t according to Lemma 20, and let ǫ > 0 be a real number such +that cos(t) > ǫ and −ǫ > cos(t + δ), cos(t + 2δ). +Then since cos is a continuous function, there is an open region (l, u) around +t such that for all x ∈ (l, u), cos(x) > ǫ and −ǫ > cos(x + δ), cos(x + 2δ). +Let n be an integer such that +4π +n +< u − l and then define (xi, yi) to be +(i 2π +n , (i + 1) 2π +n ) for 1 ≤ i ≤ n. +For all t there is a maximum integer K such that t + K 2π +n +≤ l. +Then +l < t + (K + 1) 2π +n by maximality, but t + (K + 2) 2π +n ≤ l + 4π +n < u. +Thus if (K + 1) 2π +n < x < (K + 2) 2π +n , l < t + x < u and hence cos(t + x) > ǫ +and −ǫ > cos(t + x + δ), cos(t + x + 2δ). +11 + +Since cos is periodic with period 2π, if 1 ≤ i ≤ n and i is equivalent to K +1 +modulo n, then for all xi < x < yi, cos(t+x) > ǫ and −ǫ > cos(t+x+δ), cos(t+ +x + 2δ). +This leads to the equivalent of Lemma 13. +Lemma 22. There exists a constant C such that for all n ≥ 4, there exist +sequence ⟨ai⟩n +i=1 satisfying the recurrence relation and terminating at 0 for which +Cηn/2 +1 +> a1 > 0 > a2, a3 +Proof. Since π +2 < δ < π, we can apply Lemma 21 and get ǫ > 0 and finitely +many intervals (xi, yi) such that for all t there exists an interval (xi, yi) such +that for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ). +By Corollary 19, for each such interval (xi, yi), we can choose non-zero in- +tegers ki, li for which xi < γ(ki, li) < yi. Let A be some real number such +that |α(ki, li)| < A for all such pairs, B > 0 be some real number such that +|β(ki, li)| > B and let N be such that Aη−3N/2 +1 +< Bǫ. +Then for any j ≥ N + 3, by the statement of Lemma 21, there exists an +interval (xi, yi) such that for all x ∈ (xi, yi), cos(x − (j − 1)δ) > ǫ and −ǫ > +cos(x − (j − 2)δ), cos(x − (j − 3)δ). Since γ(ki, li) ∈ (xi, yi), it follows that +a1 +η(j−1)/2 +1 += α(ki, li)η−3(j−1)/2 +1 ++ β(ki, li) cos(γ(ki, li) − (j − 1)δ) +is the sum of a number of absolute value at most Aη−3N/2 +1 +and a number that is +at least Bǫ and so is positive. Similarly a2 and a3 are negative. +|a1| +ηj/2 +1 +is bounded +above by 2Bǫ. +For each value 4 ≤ j ≤ N +2, we can just choose any sequence satisfying the +bounds. For instance, if aj = pa1 + qa2 + ra3, we set a1 = q + r, a2 = a3 = −p. +Choose C such that C > 2Bǫ and such that for all 4 ≤ j ≤ N +2, the sequences +we have chosen satisfy a1 < Cηj/2 +1 +. +Similarly we can get the following. +Lemma 23. There exists a constant C such that for all n ≥ 4, there exist +sequence ⟨bi⟩n +i=1 satisfying the recurrence relation and terminating at 0 for which +Cηn/2 +1 +> b2 > 0 > b1, b3 +Proof. Proof entirely analagous to Lemma 22. +For Lemma 20, we need a u such that cos(u + δ) > 0 > cos(u), cos(u − 2δ). +Pick u such that π +2 − 2δ < u < −π +2 . There exists such a u because δ > π +2 . +Since δ < π, −3π +2 +< u < −π +2 and hence cos(u) < 0. Similarly π +2 < u+2δ < 3π +2 +and hence cos(u + 2δ) < 0. Finally π +2 − δ < u + δ < δ − π +2 , so − π +2 < u + δ < π +2 , +so cos(u + δ) > 0. +Then by a method equivalent to Lemma 21 there exists an ǫ′ > 0 and finitely +many intervals ⟨(x′ +i, y′ +i)⟩m +i=1 such that for all t there exists an interval (x′ +i, y′ +i) such +that for all x ∈ (x′ +i, y′ +i), cos(t + x + δ) > ǫ′ and ǫ′ > cos(t + x), cos(t + x + 2δ). +We then apply the same method as the proof of Lemma 22 +12 + +This allows us to prove the equivalent of Theorem 4. +Theorem 24. There exists a real number U such that for any positive integers +n, k with k ≥ 4 and n ≥ Uη3k/2 +1 +, there is a positive sequence ⟨xi⟩k +i=1 satisfying +xi+3 = axi+2 + bxi+1 + cxi and terminating at xk = n. +Proof. Denote by pk, qk and rk the integers such that xk = pkx1 + qkx2 + rkx3 +for all such sequences ⟨xi⟩k +i=1. +Then since there can be an integer sequence ending at xk = 1, there is no +non-trivial common divisor of pk, qk and rk. +Further, by Lemma 22 and Lemma 23 there exist integers Cηk/2 +1 +> a1 > +0 > a2, a3 and Cηk/2 +1 +> b2 > 0 > b1, b3 for which a1pk + a2qk + a3rk = +b1pk +b2qk +b3rk = 0. Hence by Theorem 12, for all n ≥ a1pk +b2qk +rk, there +is such a sequence terminating at n. +Since (2C +1)ζk/2 +1 +> a1 +b2 +1 and pk, qk, rk > T ζn +1 for some fixed constant +T , it follows that for all n ≥ (2C+1)T ζ3k/2 +1 +, there is such a sequence terminating +at n. +Finally we are able to show that all affable polynomials are congenial. +Proof of Theorem 5. For our polynomial x3 − ax2 − bx − c with c = 1 and +a + b > 1 and at most one real root, Theorem 24 has stated the existence of a +real number U uch that for any positive integers n, k with k ≥ 4 and n ≥ Uη3k/2 +1 +there is a positive sequence of length k terminating at n. +Thus if there is no positive sequence of length k + 1 terminating at n, it +follows that n < Uη3(k+1)/2 +1 +. +Then by Theorem 18 it follows that the number of sequences of length k +terminating at n is at most ⌈T +n +η3k/2 +1 +⌉2 < ⌈T Uη3/2 +1 +⌉2. +For now we leave open the following question. +Question 25. For which positive integers a, b, c with c > 0 and a + b > 0 is the +recurrence relation xn = axn−1 + bxn−2 + cxn−3 congenial? +References +[1] S. Spiro, “Problems that i would like somebody to solve,” 2020. +13 + diff --git a/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/load_file.txt b/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..659376449a822135ee34c3fd71f4ac529583ca0c --- /dev/null +++ b/3tFLT4oBgHgl3EQfsC9L/content/tmp_files/load_file.txt @@ -0,0 +1,488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf,len=487 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='12146v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='NT] 28 Jan 2023 On Tribonacci Sequences Luke Pebody Saturday 28, January 2023 Abstract Let a tribonacci sequence be a sequence of integers satisfying ak = ak−1 + ak−2 + ak−3 for all k ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any positive integers k and n, denote by fk(n) the number of tribonacci sequences with a1, a2, a3 > 0 and with ak = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all n, there is a maximum k such that fk(n) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Answering a question of Spiro [1], we show that there is a finite upper bound (we specifically prove 561001) on fk(n) for any positive integer n ≥ 3 and this maximum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We do this by showing that fk(n) has transitions in n around constant multiples of φ3k/2 (where φ is the real root of φ3 = φ2 + φ + 1): there exists a constant C such that fk(n) > 0 whenever n > Cφ3k/2 and for any constant T , the values of fk(n) with n < T φ3k/2 have an upper bound independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 1 Introduction A tribonacci sequence of length k is a sequence of integers ⟨ai⟩k i=1 such that ai = ai−1 + ai−2 + ai−3 for all 4 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We say that such a sequence terminates at ak and that it is positive if a1, a2, a3 > 0 - note that this easily implies that ai > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Denote by fk(n) the number of tribonacci sequences of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Clearly f1(n) = 1 for all n > 0, the only tribonacci sequence of length 1 terminating at n being ⟨n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Further, f2(n) = f3(n) = ∞ as we can choose any values for the proceeding terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For n ≥ 3, there exists a tribonacci sequence of length longer than 3 termi- nating at n, for example ⟨n − 2, 1, 1, n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' However for any tribonacci sequence ⟨ai⟩k i=1 of length k, and for any 4 ≤ i ≤ k, ai = ai−1 + ai−2 + ai−3 ≥ ai−1 + 2, so by induction ai ≥ 2i− 5 for all 3 ≤ i ≤ k, and hence if n < 2k − 5, fk(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let t(n) be the largest number such that ft(n)(n) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let p(n) denote the number of positive tribonacci sequences of length t(n) terminating at n, so p(n) = ft(n)(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Clearly, since t(1) = t(2) = 3 it follows that p(1) = p(2) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Spiro [1] asks Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Does there exist some absolute constant c such that for all n ≥ 3, p(n) ≤ c for all n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 1 The purpose of this paper is to give a positive answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Indeed we will show Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any integer n ≥ 3, there are at most 561001 positive tribonacci sequences of length t(n) terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' It turns out the key question for our proof is the minimum size of the vector \uf8eb \uf8ed a1 a2 a3 \uf8f6 \uf8f8 where ⟨ai⟩n i=1 is a non-zero tribonacci sequence terminating at an = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In Section 2 we will show a lower bound on such a sequence of the order of φn/2, which will allow us to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any positive integers n, k with k ≥ 4, the number of positive sequences of length k terminating at n is at most ⌈1500 n φ3k/2 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In Section 3 we turn to trying to put an upper bound on numbers that don’t have any positive tribonacci sequences of length k terminating at them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This is an instance of the Coin Problem, also known as calculating the Frobenius Number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We construct two specific tribonacci sequences terminating at an = 0 with \uf8eb \uf8ed a1 a2 a3 \uf8f6 \uf8f8 being of the order of φn/2 and with the integers a1, a2, a3 having specified signs, allowing us to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any integer n above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2φ3k/2, there exists a positive tribonacci sequence of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This will be all that is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There is no sequence of length t(n) + 1 terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Hence by Theorem 4, it follows that n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2φ3(t(n)+1)/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2φ3/2φ3t(n)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus from Theorem 3, it follows that there are at most ⌈1500 n φ3t(n)/2 ⌉2 ≤ ⌈15000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2φ3/2φ3t(n)/2 φ3t(n)/2 ⌉2 ≤ ⌈300φ3/2⌉2 = 7492 = 561001 positive tribonacci sequences of length t(n) terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In Section 4, we will investigate which recurrence relations of the form xn = axn−1+bxn−2+cxn−3 for non-negative a and b and for positive c the arguments in this paper can be carried across to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will extend the result in the earlier sections to the following case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 2 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Suppose a, b, c are non-negative integers with a + b > 0, c = 1 and such that x3 − ax2 − bx − c = 0 has exactly one real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then there is an absolute bound T such that if positive integers k ≥ 4 and n are such that there are no positive sequences ⟨ai⟩k+1 i=1 satisfying the recurrence relation ai = aai−1 + bai−2 + cai−3 of length k + 1 terminating at n, then there are at most T such sequences of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will leave open the question of which linear recurrences satisfy this prop- erty, but will at least demonstrate an example of a recurrence that does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In particular we will show the existence of positive integers k and n such that there is no positive sequence ⟨ai⟩k+1 i=1 satisfying the recurrence relation ai = ai−1 + ai−2 + 2ai−3 of length k + 1 terminating at n, but for which the number of such sequences of length k terminating at n is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 2 Lower Bound Let us say a sequence ⟨ai⟩∞ i=1 is a reverse-tribonacci sequence if for all i ≥ 0, ai = ai+1+ai+2+ai+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let us write out the expression for the reverse-tribonacci sequence starting ⟨0, k, l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Recall that φ is the real solution to φ3 = φ2 + φ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We write the complex roots as φ1 and φ2 = φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all integers k, l, if ⟨ai⟩∞ i=1 is a reverse-tribonacci sequence with a1 = 0, a2 = k and a3 = l, then for all i, ai can be expressed as ai = αφ−i + (kψ1 + lζ1)φ−i 1 + (kψ2 + lζ2)φ−i 2 = (αφ−3i/2 + β cos(γ − δi))φi/2, where ψ1 = φ3 1 + φ2 1 φ2 1 + 2φ1 + 3 ψ2 = ψ1 ζ1 = φ3 1 φ2 1 + 2φ1 + 3 ζ2 = ζ1 α = kφ2 + (k + l)φ3 φ2 + 2φ + 3 βeγi = 2(kψ1 + lζ1) and eδi = φ1 � φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Any two-way infinite tribonacci sequence ⟨ai⟩∞ −∞ can be written as ai = pφi + qφi 1 + rφi 2 for some p, q and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus any reverse-tribonacci sequence ⟨ai⟩∞ −∞ can be written as ai = pφ−i + qφ−i 1 + rφ−i 2 for some p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Solving for the p, q, r that give a1 = 0, a2 = k and a3 = l leads to the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 3 Note that in the above expressions, ψ1, ψ2, ζ1, ζ2 and δ are constants that do not depend on k and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any integers k and l, if α and β are defined as in Lemma 6, then |α| ≤ |k| + |l| and β ≥ |k|+|l| 31 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' α is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='9546k+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='6184l, which is clearly bounded above by |k|+|l|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' ψ1 is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='02267 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='217i and ζ1 is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='1908 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='0187i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' As such, if k and l are non-negative then the real part of 2(kψ1 + lζ1) (and hence β) is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='04(k + l) > |k|+|l| 31 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For k positive and l negative, the minimum value of β k−l is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='03221 > 1 31, and is achieved around k = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3653l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Finally we need a simple trigonometric property Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For any real numbers p and q with π 2 < q < π, the larger of | cos(p)| and | cos(p + q)| is at least cos(q/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Note that cos(q) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus 2(cos(p)2 + cos(p + q)2) = 2 cos(p)2 + 2 cos(p + q)2 = cos(2p) + cos(2p + 2q) + 2 = 2 cos(2p + q) cos(q) + 2 ≥ 2 + 2 cos(q) = 4 cos(q/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus either cos(p)2 ≥ cos(q/2)2 or cos(p + q)2 ≥ cos(q/2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This allows us to put a lower bound on the size of at least one of each consecutive pair of a reverse-tribonacci sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Given a non-zero integer reverse-tribonacci sequence ⟨ai⟩∞ i=1 with a1 = 0, for every integer n ≥ 2, either |an| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φn/2 or |an+1| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φ(n+1)/2 (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For n ≥ 2 if an and an+1 are both 0, then a1 is the same sign as an−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since a1 = 0, it follows that the entire series must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since the sequence is non-zero, it follows that either |an| ≥ 1 or |an+1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φn/2 for n ≤ 15, we have proved the statement for n ≤ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus we may assume n ≥ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' By Lemma 6, ai φi/2 can be written as αφ−3i/2 + β cos(γ − δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Now by Lemma 8, at least one of | cos(γ − δn)| and | cos(γ − δ(n − 1))| is at least cos(δ/2) (δ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='176 is between π 2 and π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let t be the choice from {n − 1, n} that maximises | cos(γ − δt)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' By Lemma 7, if we write α′ = α |k|+|l| and β′ = β |k|+|l| then |α′| ≤ 1 and β′ > 1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 4 Therefore | at φt/2 | = |αφ−3t/2 + β cos(γ − δt)| ≥ |α′φ−3t/2 + β′ cos(γ − δt)| ≥ |β′ cos(γ − δt)| − |α′φ−3t/2| ≥ 1 31 cos(δ/2) − φ−3t/2 ≥ cos(δ/2) 31 − φ−22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then we have a bound on the size of tribonacci sequences terminating at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For n ≥ 3, if ⟨ai⟩n i=1 is a non-zero integer tribonacci sequence terminating at 0 then either |a1| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φn/2 or |a2| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φ(n−1)/2 (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let k = an−1 and l = an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then if ⟨bi⟩∞ i=1 is the reverse-tribonacci sequence with b1 = 0, b2 = k and b3 = l, then ai = bn+1−i for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then this is just a restatement of Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This is all we need to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Partition the tribonacci sequences of length k ≥ 4 termi- nating at n ⟨ai⟩k i=1 by the pair (⌊ a1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φk/2 ⌋, ⌊ a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φk−1/2 ⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' If two sequences ⟨ai⟩k i=1 and ⟨bi⟩k i=1 have the same pair, then |a1 − b1| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φk/2 and |a2 − b2| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φ(k−1)/2 and hence, by Corollary 10, either ⟨ai − bi⟩k i=1 is zero everywhere or does not terminate at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus each distinct tribonacci sequence of length k terminating at n has a distinct pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Define tribonacci sequence by x1 = 1, x2 = 0, x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then if a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' , ak is a positive tribonacci sequence, ai ≥ xia4 for i = 2, 3 and 4 and therefore ak ≥ xka4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Now xk < φk/11 for all k ≥ 4 and hence a1 + a2 + a3 ≤ 11n φk for all tribonacci sequences of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus ⌊ a1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φk/2 ⌋ is at most 1100n φ3k/2 and ⌊ a2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='01φ(k−1)/2 ⌋ is at most 1100n φ3k−1/2 < 1500n φ3k/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' It follows that the number of Tribonacci sequences of length k ≥ 4 termi- nating at n is at most ⌈ 1500n φ3k/2 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Note we have not worked hard here to get the best bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In a previous draft we had a much more complicated proof of an upper bound which showed, in place of Corollary 10, that if ⟨ai⟩n i=1 terminated at 0 then � a2 1 + a2 2 + a2 3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='28φn/2, which led to an upper bound for the main theorem of 42875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 3 Upper Bound In this section, we turn to numbers which are not the terminus for any tribonacci sequence of length k, working towards a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 5 To that end, define three infinite tribonacci sequences ⟨pi⟩∞ i=1, ⟨qi⟩∞ i=1 and ⟨ri⟩∞ i=1 by (p1, p2, p3) = (1, 0, 0), (q1, q2, q3) = (0, 1, 0) and (r1, r2, r3) = (0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' It is clear that for any tribonacci sequence ⟨ai⟩n i=1, an = a1pn + b1qn + c1rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus we are simply looking to get an upper bound on the largest number which cannot be written as a positive integral linear combination of pn, qn and rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This is called the Frobenius Number of pn, qn and rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' First let us see that a finite bound does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all k ≥ 1, pk, qk and rk have no non-trivial common divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' If pk, qk and rk had a non-trivial common divisor t > 1 then t would be a common divisor of the terminus of every tribonacci sequence of length k, from which it would follow that t would in fact be a common divisor of pk+l for all l ≥ 0 (since ⟨pi+l⟩k 1 is a tribonacci sequence of length k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then, since pi = pi+3 − (pi+1 + pi+2), it would follow that t would be a common divisor of pk−1, pk−2 and all the way back to p0 = 1 by induction, causing a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will use the following bound, which might be originally due to Killing- bergtro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Suppose p, q and r are integers with no non-trivial common di- visor and let us suppose ap = bq + cr and dq = ep + fr where a, c, d, f > 0 and b, e ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then for every integer N ≥ ap + dq + r, N can be written in the form xp + yq + zr for some positive integers p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let x, y, z be positive integers such that px + qy + rz is equivalent to N (mod r), but for which px + qy + rz is minimal (such a triple x, y, z exist because, as is well known, if p, q and r have no non-trivial common divisor then all sufficiently large integers can be written in the form px + qy + rz, and many of these sufficiently large integers are equivalent to N (mod r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=') Since px+qy+rz is minimal, px+qy+rz−r cannot be written as a positive linear combination of x, y and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus in each of the equations px + qy + rz − r = px + qy + r(z − 1) px + qy + rz − r = p(x − a) + q(y + b) + r(z + c − 1) px + qy + rz − r = p(x + e) + q(y − d) + r(z + f − 1), it must follow that one of the coefficients must not be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Two of the coefficients in each equation are clearly positive, so it follows that x ≤ a, y ≤ d and z ≤ 1, so px + qy + rz ≤ pa + qd + r ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since N and px + qy + rz are equivalent modulo r, there exists a non-negative integer t such that N = px + qy + rz + rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then N = px + qy + r(z + t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Therefore, to show that all sufficiently large integers can be written as the terminus of a tribonacci sequence of length k, we just need to find linear com- binations of pn, qn and rn combining to 0, with particular signs of the combi- nations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This is equivalent to finding tribonacci sequences ending at 0, which 6 Table 1: Table for Lemma 13 t0 t1 k l α β γ x0 x1 x2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='06 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='6184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='0977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3410 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='0500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='1694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='16 1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2822 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='6184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='0977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='1864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='0548 is equivalent to finding reverse-tribonacci sequences starting at 0, and hence we can again use the expression from Lemma 6, which states that if ⟨ai⟩∞ i=1 is a reverse-tribonacci sequence with a1 = 0, a2 = k and a3 = l then for all n an = (αφ−3n/2 + β cos(γ − δn))φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Note that for all but an extremely small collection of n, the term β cos(γ+δn) dwarves αφ−3n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' As such, for a fixed k and l, the sign of an depends only (except for a few very rare counterexamples) on the fractional part of δ 2πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For each integer n ≥ 4, there exists a tribonacci sequence ⟨ai⟩n i=1 terminating at an = 0, with a1 > 0, 0 ≥ a2, 0 > a3 and with a1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='81φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Similarly for all such n, there exists a tribonacci sequence ⟨bi⟩n i=1 terminating at bn = 0, with b2 > 0, 0 ≥ b1, 0 > b3 and with b2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='64φn/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will split into cases based on the fractional part of δn 2π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='3464n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' See Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For each row, if t0 ≤ δn 2π − ⌊ δn 2π⌋ ≤ t1, then for the given values of k and l, if β and γ are as defined in Lemma 6, one can confirm that β cos(γ − δn) ≥ x0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='34, while β cos(γ − δ(n − 1)) ≤ x1 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='035 and β cos(γ − δ(n − 2)) ≤ x2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Furthermore, for all such k, l, |α| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='017, from which it follows that an > 0 > an−1, an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Further, an φn/2 < β + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='017 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For 4 ≤ n < 7, we can verify the sequences (1, 0, −1, 0), (2, 0, −1, 1, 0) and (2, 0, −1, 1, 0, 0) satisfy the conditions for (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For the sequence (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' , bn), see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Here, for each row, if t0 ≤ δn 2π − ⌊ δn 2π⌋ ≤ t1, then for the given values of k and l, if β and γ are as defined in Lemma 6, one can confirm that β cos(γ − δn) ≤ x0 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='071, while β cos(γ − δ(n − 1)) ≥ x1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='33 and β cos(γ − δ(n − 2)) ≤ x2 < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='041.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Furthermore, for all such k, l, |α| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='58, so if n ≥ 7, |αφ−3(n−2)/2| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='017, from which it follows that an−1 > 0 > an, an−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For 4 ≤ n < 7, we can verify the sequences (0, 1, −1, 0), (0, 1, −1, 0, 0) and (−1, 2, −1, 0, 1, 0) satisfy the conditions for (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This then completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 7 Table 2: Other table for Lemma 13 t0 t1 k l α β γ x0 x1 x2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 13 gives us tribonacci sequences ⟨ai⟩n i=1 and ⟨bi⟩n i=1 terminating at an = bn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' It follows that a1pn + a2qn + a3rn = 0 = b1qn + b2qn + b3rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since a1, b2 > 0 > a3, b3 and 0 ≥ a2, b1, it follows that we can write a1pn = (−a2)qn + (−a3)rn and b2qn = (−b1)p1 + (−b3)rn satisfying the sign requirements of Theorem 12, so it follows that every integer N ≥ a1pn + b2qn + rn can be written in the form xpn + yqn + zrn for some positive integers x, y and z, and hence there exists a positive tribonacci sequence of length k ending at N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' By the bounds on a1 and b2 given in Lemma 13, we have such a tribonacci sequence for all N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='81φk/2uk+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='64φk/2vk+wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since uk ≤ vk ≤ wk < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='11φk and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='81φk/2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='64φk/2+1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='74φk/2, it follows that such a tribonacci sequence exists for all N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content='2φ3k/2 as was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 4 Other cubic recurrences For non-negative a, b, c we can ask a similar question for recurrences of the form xn = axn−1 + bxn−2 + cxn−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Formally, let us define ka,b,c(n) to be the largest k such there is a positive k-element solution ⟨xi⟩k i=1 to the recurrence relation xi = axi−1 + bxi−2 + cxi−3, and define ta,b,c(n) to be the number of positive ka,b,c(n)-element solutions that exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' If c = 0, this is a quadratic recurrence, and the problem is already solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' If a = 0, b = 0 and c = 1, the recurrence is xn = xn−3, and ka,b,c(n) is not defined for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all a, b, c ≥ 0 with c ≥ 1 and a + b + c ≥ 2, say that the recurrence xn = axn−1 + bxn−2 + cxn−3 is congenial if there exists a finite bound B such that for all n, ta,b,c(n) = ∞ or ta,b,c(n) ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Firstly let us note that not all recurrences are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' The recurrence xn = xn−1 + xn−2 + 2xn−3 is not congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let ⟨pn⟩∞ n=1, ⟨qn⟩∞ n=1 and ⟨rn⟩∞ n=1 be the solutions to the recurrence 8 starting with ⟨1, 0, 0⟩, ⟨0, 1, 0⟩ and ⟨0, 0, 1⟩ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then xn = x1pn + x2qn + x3rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Solutions to the recurrence can be split as the sum of two parts - a sequence of the form ⟨x(1) n = 2n−1k⟩ and a sequence of the form ⟨x(2) n ⟩ which is periodic with period 3 with x(2) 1 +x(2) 2 +x(2) 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' It is then easy to solve for k: x1 +x2 +x3 = x(1) 1 + x(1) 2 + x(1) 3 = 7k, so k = x1+x2+x3 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' In particular, if you let tn = 2n−1 7 , pn−tn is periodic with period ⟨ 6 7, − 2 7, − 4 7⟩, qn − tn with period ⟨− 1 7, 5 7, − 4 7⟩ and rn − tn with period ⟨− 1 7, − 2 7, 3 7⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For n = 3t, xn = c(x1 + x2) + (c + 1)xn−3 and xn+1 = 2(c + 1)x1 + (2c + 1)(x2 + x3) where c = 23t−1−1 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then xn+1 cannot be equal to (2c + 1)(2c + 3) for positive x1, x2, x3 (x1 would have to be a multiple of 2c + 1 that is positive but less than 2c + 1), but for all 1 ≤ i ≤ 4c + 4, if x1 = i, x2 = 4c + 5 − i and x3 = 3, then xn = c(4c + 5) + (c + 1)3 = 4c2 + 8c + 3 = (2c + 1)(2c + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' The proofs in this paper can be adapted to show that many other recurrences are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let us say a polynomial x3 − ax2 − bx − c is affable if c = 1 and it has exactly one real root, which is more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will show that affability leads to congeniality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For the rest of this section, fix an affable polynomial x3 − ax2 − bx − c with real root η1 and complex roots η2 and η3 = η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Note that |η2| = η−1/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will make use of the following equivalent to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Given a sequence ⟨xi⟩n i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi with xn = 0, xn−1 = k and xn−2 = l, xi can be expressed as xi = 3 � j=1 (kψj + lζj)ηn−i j for constants ψj, ζj depending only on x3 − ax2 − bx − c, which can be rewritten as xi η(n−i)/2 1 = αη−3(n−i)/2 1 + β cos(γ − δ(n − i)) where α = kψ1 + lζ1, βeγi = 2(kψ2 + lζ2) and eδi = η2 √η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will follow the steps of the proof of Theorem 2 for all recurrence relations corresponding to affable polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will not attempt to give an actual bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We note the following, which will be used in the equivalents of both Theo- rems 3 and 4 9 Lemma 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' If real numbers k and l satisfy kψ2 + lζ2 = 0, then k = l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' As ψ3 = ψ2 and ζ3 = ζ2, kψ3 + lζ3 = 0 and therefore the sequence with xn = 0, xn−1 = k and xn−2 = l can simply be expressed as xi = (kψ1+lζ1)ηn−i 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' As 0 = xn = kψ1 + lζ1, it follows that xi = 0 for all i and therefore k = l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We start by following the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists an absolute bound M such that for n ≥ 4 and all non-zero integer sequences ⟨xi⟩n i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and xn = 0 either |x1| ≥ Mηn/2 1 or |x2| ≥ Mηn/2 1 (or both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' The set of complex numbers kψ2 + lζ2 for k, l real with |k| + |l| = 1 is a closed subset of the complex plane (in fact a hollow parallelogram) which, by Lemma 16 does not contain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' As such, there exists a constant V > 0 such that for all such k, l, |kψ2 + lζ2| > V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then for all real k, l it follows that β = |kψ2 + lζ2| > V (|k| + |l|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Clearly if U = max(|ψ1|, |ζ1|), α ≤ U(|k| + |l|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Pick integer N such that V cos(δ/2) − Uη−3(N−1)/2 1 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Note we can do this because π 2 < δ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then let M > 0 be such that V cos(δ/2) − Uη−3(N−2)/2 1 > Mη1 and η−N/2 1 > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Now if n ≤ N, then Mηn/2 1 < 1 (note that η1 > 1 since 13 < a×12+b×1+c) and x1, x2 cannot both be 0 (as then xn would have to be the same sign as x3 and non-zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For n > N, we know from Lemma 8 that there exists t ∈ {1, 2} such that | cos(γ − δ(n − t))| > cos(δ/2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For such t, n − t ≥ N − 1 and so it follows that |xt| η(n−t)/2 1 = |αη−3n/2 1 + β cos(γ − δn)| ≥ |β cos(γ − δn)| − |α|η−3n/2 1 ≥ V cos(δ/2) − Uη−3n/2 1 ≥ Mη1 and hence |xt| ≥ Mη(n+2−t)/2 1 ≥ Mηn/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This is enough for the equivalent of Theorem 3 Theorem 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists a fixed bound T such that for any positive integers n, k with k ≥ 4, the number of positive sequences ⟨xi⟩k i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and terminating at xk = n is at most ⌈T n η3k/2 1 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There is a fixed P such that for any positive sequence ⟨ai⟩k i=1 satisfying the recurrence relation with k ≥ 4, Pηk 1(a1 + a2 + a3) ≤ ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus for any such sequence terminating at n, a1 and a2 are bounded above by n P ηk 1 and for any two such sequences, by Lemma 17, either the first terms or the second terms differ by at least Mηk/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus the number of such sequences is at most ⌈ n P Mη3k 1 /2⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Now we proceed to follow the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We will need the following Corollary to Lemma 16 Corollary 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Given any interval 0 ≤ x < y ≤ 2π within (0, 2π), we can pick non-zero integers k, l for which x < γ < y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 16 says that the set {kψ2 + lζ2 : k, l ∈ R}, when viewed geomet- rically as a subset of the complex plane, is not of dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus it must be the entire complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Pick x < z < y, then there exist real k, l with kψ2 + lζ2 = eiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Now let kn = ⌊nk⌋ and ln = ⌊nl⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' The limit as n tends to infinity of knψ2+lnζ2 n is eiz and therefore for all sufficiently large n, γ (which is the argument of knψ2+lnζ2 n ) must be contained in the open interval (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We shelve this for the moment and focus on a simple piece of trigonometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all numbers π 2 < δ < π, there exists t such that cos(t) > 0 > cos(t + δ), cos(t + 2δ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Pick t such that π 2 − δ < t < 3π 2 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists such a t because δ < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since δ < π, − pi 2 < π 2 − δ < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Similarly since π 2 < δ, t < 3π 2 − 2δ < pi 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' So − pi 2 < t < pi 2 and hence cos(t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Further π 2 < t+δ < t+2δ < 3π 2 , so cos(t+δ) and cos(t+2δ) are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This leads to the following somewhat technical-seeming lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all numbers π 2 < δ < π, there exists an ǫ > 0 and finitely many intervals ⟨(xi, yi)⟩n i=1 such that for all t there exists an interval (xi, yi) such that for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Pick a t according to Lemma 20, and let ǫ > 0 be a real number such that cos(t) > ǫ and −ǫ > cos(t + δ), cos(t + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then since cos is a continuous function, there is an open region (l, u) around t such that for all x ∈ (l, u), cos(x) > ǫ and −ǫ > cos(x + δ), cos(x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let n be an integer such that 4π n < u − l and then define (xi, yi) to be (i 2π n , (i + 1) 2π n ) for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For all t there is a maximum integer K such that t + K 2π n ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then l < t + (K + 1) 2π n by maximality, but t + (K + 2) 2π n ≤ l + 4π n < u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus if (K + 1) 2π n < x < (K + 2) 2π n , l < t + x < u and hence cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 11 Since cos is periodic with period 2π, if 1 ≤ i ≤ n and i is equivalent to K +1 modulo n, then for all xi < x < yi, cos(t+x) > ǫ and −ǫ > cos(t+x+δ), cos(t+ x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' This leads to the equivalent of Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists a constant C such that for all n ≥ 4, there exist sequence ⟨ai⟩n i=1 satisfying the recurrence relation and terminating at 0 for which Cηn/2 1 > a1 > 0 > a2, a3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since π 2 < δ < π, we can apply Lemma 21 and get ǫ > 0 and finitely many intervals (xi, yi) such that for all t there exists an interval (xi, yi) such that for all x ∈ (xi, yi), cos(t + x) > ǫ and −ǫ > cos(t + x + δ), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' By Corollary 19, for each such interval (xi, yi), we can choose non-zero in- tegers ki, li for which xi < γ(ki, li) < yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Let A be some real number such that |α(ki, li)| < A for all such pairs, B > 0 be some real number such that |β(ki, li)| > B and let N be such that Aη−3N/2 1 < Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then for any j ≥ N + 3, by the statement of Lemma 21, there exists an interval (xi, yi) such that for all x ∈ (xi, yi), cos(x − (j − 1)δ) > ǫ and −ǫ > cos(x − (j − 2)δ), cos(x − (j − 3)δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since γ(ki, li) ∈ (xi, yi), it follows that a1 η(j−1)/2 1 = α(ki, li)η−3(j−1)/2 1 + β(ki, li) cos(γ(ki, li) − (j − 1)δ) is the sum of a number of absolute value at most Aη−3N/2 1 and a number that is at least Bǫ and so is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Similarly a2 and a3 are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' |a1| ηj/2 1 is bounded above by 2Bǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For each value 4 ≤ j ≤ N +2, we can just choose any sequence satisfying the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For instance, if aj = pa1 + qa2 + ra3, we set a1 = q + r, a2 = a3 = −p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Choose C such that C > 2Bǫ and such that for all 4 ≤ j ≤ N +2, the sequences we have chosen satisfy a1 < Cηj/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Similarly we can get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists a constant C such that for all n ≥ 4, there exist sequence ⟨bi⟩n i=1 satisfying the recurrence relation and terminating at 0 for which Cηn/2 1 > b2 > 0 > b1, b3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof entirely analagous to Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For Lemma 20, we need a u such that cos(u + δ) > 0 > cos(u), cos(u − 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Pick u such that π 2 − 2δ < u < −π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists such a u because δ > π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since δ < π, −3π 2 < u < −π 2 and hence cos(u) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Similarly π 2 < u+2δ < 3π 2 and hence cos(u + 2δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Finally π 2 − δ < u + δ < δ − π 2 , so − π 2 < u + δ < π 2 , so cos(u + δ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then by a method equivalent to Lemma 21 there exists an ǫ′ > 0 and finitely many intervals ⟨(x′ i, y′ i)⟩m i=1 such that for all t there exists an interval (x′ i, y′ i) such that for all x ∈ (x′ i, y′ i), cos(t + x + δ) > ǫ′ and ǫ′ > cos(t + x), cos(t + x + 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' We then apply the same method as the proof of Lemma 22 12 This allows us to prove the equivalent of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' There exists a real number U such that for any positive integers n, k with k ≥ 4 and n ≥ Uη3k/2 1 , there is a positive sequence ⟨xi⟩k i=1 satisfying xi+3 = axi+2 + bxi+1 + cxi and terminating at xk = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Denote by pk, qk and rk the integers such that xk = pkx1 + qkx2 + rkx3 for all such sequences ⟨xi⟩k i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then since there can be an integer sequence ending at xk = 1, there is no non-trivial common divisor of pk, qk and rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Further, by Lemma 22 and Lemma 23 there exist integers Cηk/2 1 > a1 > 0 > a2, a3 and Cηk/2 1 > b2 > 0 > b1, b3 for which a1pk + a2qk + a3rk = b1pk +b2qk +b3rk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Hence by Theorem 12, for all n ≥ a1pk +b2qk +rk, there is such a sequence terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Since (2C +1)ζk/2 1 > a1 +b2 +1 and pk, qk, rk > T ζn 1 for some fixed constant T , it follows that for all n ≥ (2C+1)T ζ3k/2 1 , there is such a sequence terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Finally we are able to show that all affable polynomials are congenial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For our polynomial x3 − ax2 − bx − c with c = 1 and a + b > 1 and at most one real root, Theorem 24 has stated the existence of a real number U uch that for any positive integers n, k with k ≥ 4 and n ≥ Uη3k/2 1 there is a positive sequence of length k terminating at n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Thus if there is no positive sequence of length k + 1 terminating at n, it follows that n < Uη3(k+1)/2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Then by Theorem 18 it follows that the number of sequences of length k terminating at n is at most ⌈T n η3k/2 1 ⌉2 < ⌈T Uη3/2 1 ⌉2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For now we leave open the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Question 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' For which positive integers a, b, c with c > 0 and a + b > 0 is the recurrence relation xn = axn−1 + bxn−2 + cxn−3 congenial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' Spiro, “Problems that i would like somebody to solve,” 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} +page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFLT4oBgHgl3EQfsC9L/content/2301.12146v1.pdf'} diff --git 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We address the general problem of estimating the probability that a real symmetric +tensor is close to rank–one tensors. Using Weyl’s tube formula, we turn this question into a differential +geometric one involving the study of metric invariants of the real Veronese variety. More precisely, +we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri–Weyl +metric. These results are obtained using techniques from Random Matrix theory and an explicit +description of the second fundamental form of the Veronese variety in terms of GOE matrices. Our +findings give a complete solution to the original problem, and in the case of rational normal curves +lead to some novel asymptotic results. +1. Introduction +1.1. What is the probability that a random symmetric tensor is close to rank-one? Over +the last decades, symmetric tensors have been proven to be a very flexible and valuable tool in many +different contexts. In particular, rank–one approximation and tensor decomposition found applications +in machine learning ([AGH+14]), signal processing and image analysis ([SDLF+17], [Sak16, Ch.3, 4]), +chemistry ([SBG04]), statistics ([McC87]), psychology and medical diagnostics ([Kro08, ALLF07]) and +phylogenetics ([Sak16, Ch.5], [Lan12]), to name a few. Motivated by this, in this paper we address the +following question: +“What is the probability for a real symmetric tensor to be “close” to rank–one?” +To make sense of this question, we must endow the space of tensors with a notion of distance and +with a probability distribution. We address this problem in a natural way as follows. +Observe first that the real vector space of symmetric tensors of order d on Rn+1 can be naturally +identified with the space R[x0, . . . , xn](d) of homogeneous polynomials of degree d. Under this identifi- +cation, we endow the space of real polynomials with the scalar product given by the restriction of the +real part of the Bombieri–Weyl hermitian product, defined on the space of complex polynomials by +(1) +⟨p1, p2⟩BW := +1 +πn+1 +� +Cn+1 p1(z)p2(z)e−∥z∥2dz, +where dz := (i/2)n+1dz0dz0 . . . dzndzn is the Lebesgue measure. +This defines the unique, up to +multiples, hermitian product on the space of complex polynomials which is invariant under the action +of the unitary group by change of variables. The restriction of the real part of this hermitian product to +the space of real polynomials will be still called the Bombieri–Weyl scalar product; the above unitary +invariance implies its invariance under the action of the orthogonal group by change of variables. In the +case when d = 2, the above identification is the familiar isomorphism between the space of symmetric +matrices and the space of quadratic forms, and the Bombieri–Weyl scalar product coincides with the +Frobenius inner product. +Next, we use this scalar product to turn R[x0, . . . , xn](d) into a probability space. For a Borel set +U ⊆ R[x0, . . . , xn](d) we define +P(U) := +� +U +e− +∥p∥2 +BW +2 +dµ +� +R[x0,...,xn](d) +e− +∥p∥2 +BW +2 +dµ +, +Date: September 2022. +1 + +2 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +where “dµ” denotes the integration with respect to the Lebesgue measure on the space of coefficients. +We call the resulting probability distribution Bombieri–Weyl, and sometimes the nomenclature Kostlan +is also used interchangeably. When d = 2, the Bombieri–Weyl distribution turns the space of symmetric +matrices into a gaussian space, called the Gaussian Orthogonal Ensemble, as we will describe in more +detail in Section 2.1. +Finally, we identify the set of rank–one tensors with the Veronese variety Vn,d ⊂ R[x0, . . . , xn](d) of +signed d–th powers of linear forms. At this point we are in the position of giving a precise formulation +to our question above, which therefore requires computing, for δ > 0 small enough, the quantity: +P +� +p ∈ R[x0, . . . , xn](d) +���� distBW(p, Vn,d) ≤ δ∥p∥BW +� +. +Notice that we have turned this into a conic problem that takes into account also the norm of the +tensor, as it is common procedure in numerical algebraic geometry [BC13]. In this way, we can regard +the above probability as the normalized volume of a tubular neighbourhood of the intersection of the +set of rank–one tensors with the unit sphere in the Bombieri–Weyl norm. Thus, our question becomes: +“What is the volume of a neighbourhood of the spherical Veronese variety?” +In this paper, exploiting Weyl’s Tube Formula, we derive an exact expression for the above volume, for +small enough neighbourhoods. Moreover, as a byproduct of our computations, we give a lower bound +on the size of the neighbourhood of the set of rank–one tensors that admit a unique best rank–one +approximation. +Remark 1. The properties of the Bombieri–Weyl distribution on the space of real (and complex) poly- +nomials have been intensively studied, starting from the influential works of A. Edelman, E. Kostlan, +M. Shub and S. Smale [EK95, SS93b, SS93a, SS93c]. The point of view of random tensors has been +adopted first by E. Horobet and J. Draisma in [DH16] and by P. Breiding in [Bre19] for the study of +the expected number of eigenvalues of a random symmetric tensor, with respect to the Bombieri–Weyl +distribution. Under the identification between symmetric tensors and homogeneous polynomials, eigen- +values correspond to critical values of the restriction of the polynomial to the unit sphere. Eigenvectors +correspond to critical points of the polynomial: under the Veronese embedding these critical points +give rank–one tensors that are critical points of the distance function on the Veronese variety from +the given tensor. Among these critical points (which are rank–one tensors) the closest to the original +tensor are its best rank–one approximations. In [Bre19] the average number of such critical points is +computed. In this paper we will instead give the size and estimate the probability of the set of tensors +which admit a unique best rank–one approximation. +The use of Weyl’s Tube Formula is fairly standard for results of this type [BC13, BL22]: it allows to +deduce an exact expression, for ε > 0 small enough, of the volume of an ε–neighbourhood of a smooth +submanifold W of the sphere, or the euclidean space, as a function of some differential–geometric +quantities of W, called its curvature coefficients. Our main contribution is the nontrivial computation +of the curvature coefficients of the spherical Veronese variety and the explicit quantification of the +above expression “for ε > 0 small enough” for this variety, through the computation of its reach. +One could generalize this question to higher ranks by looking at secant varieties to the Veronese, +whose geometry has been intensively studied, see [CGO14] for a survey. We propose to investigate this +in future works. +We now describe the main ingredients and state the main results of our work in more detail. +1.2. The spherical Veronese. The main object we consider in this work is the real spherical Veronese +variety Vn,d, which is the intersection of the Veronese variety in R[x0, . . . , xn](d) ≃ RN+1 with the unit +sphere for the Bombieri–Weyl norm: +Vn,d := Vn,d ∩ SN. +We regard this set as the image of the spherical Veronese embedding associated to the Bombieri– +Weyl basis, or its double copy, depending on the parity of d. This embedding is the smooth map + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +3 +�νn,d : Sn → SN given by +x +�νn,d +�−−−→ +��d +α +� 1 +2 +xα +� +α +, +where α ∈ Zn+1 +≥0 +satisfy α0 + · · · + αn = d, +�d +α +� +is the multinomial coefficient, and Sn is the euclidean +sphere in Rn+1. Denoting �νn,d(Sn) by Σn,d, we see that +Vn,d = Σn,d ∪ −Σn,d, +where Σn,d = −Σn,d if d is odd and Σn,d ∩ (−Σn,d) = ∅ if d is even. For this reason, we will call Σn,d +the spherical Veronese surface, to distinguish it from the spherical Veronese variety Vn,d, in the case d +is even. In the projective picture, the difference between the two ceases to exist: +PVn,d := P(Σn,d) = P(Vn,d) ⊂ RPN. +Recall that Σn,d parametrizes the d–th powers of norm–1 linear forms on Rn+1 and, therefore, rank– +one and norm–one tensors up to signs. Hence, the spherical Veronese surface Σn,d corresponds to an +orbit for the action of O(n + 1) on homogenous polynomials by change of variables. Even more is +true: when turning Σn,d into a Riemannian manifold with the metric induced by the Bombieri–Weyl +scalar product, the transitive action of O(n + 1) on Σn,d is through isometries induced by isometries +of SN, given the invariance property of the Bombieri–Weyl structure. The immediate, yet crucial, +consequence is that the extrinsic geometry of the isometric embedding Σn,d ֒→ SN is exactly the same +at every point. The same conclusion clearly holds for Vn,d ֒→ SN. +1.3. Weyl’s tube formula and the reach of an embedding. Let (M, g) be a Riemannian manifold +and M ֒→ M be an isometric embedding of a compact smooth submanifold. We can consider the set +of points in M at distance less than a given ε > 0 from M and call such a set a tubular neighbourhood +of M in M of radius ε, denoted as U(M, ε). +It is well known that for smooth compact embeddings M ֒→ M and small enough radii, the exponen- +tial map on the normal bundle provides a smooth parametrization of the tubular neighbourhood. This +description is what really underlies the celebrated “Weyl’s tube formula” ([Wey39]), which constitutes +one of the main tools to compute the volume of tubular neighbourhoods in a euclidean or spherical +ambient space. This formula expresses the volume as the linear combination +Vol(U(M, ε)) = +� +0≤e≤n, e even +Ks+e(M)JN,s+e(ε), +where N is the dimension of the ambient space, n is the dimension of M and s := N − n is the +codimension of the embedding. The functions J’s do not depend on the specific submanifold M and +are explicitly known in both the euclidean and spherical cases. The most remarkable aspect of the +formula is that the coefficients K’s are isometric invariants of the embedding and can be expressed +in terms of curvature, from which they are named curvature coefficients of the embedding. Remark +that nowadays Weyl’s tube formula has been re-interpreted in the more general framework of “integral +geometry”, which deals with integrals over a submanifold of polynomials in the entries of the second +fundamental form. R. Howard in [How93] showed how the above formula fits in this context and gave +a full characterization of the polynomials appearing in Weyl’s work. +In the case of the Veronese variety Vn,d ֒→ SN, the tubular neighbourhood U(Vn,d, ε) gives a +description of the norm–1 symmetric tensors that are ε–close to a rank–1 tensor in the Bombieri–Weyl +metric, in the ambient sphere. As already pointed out, it follows that asking for the probability for +a symmetric norm–1 tensor to be close to rank–1 boils down to computing the normalized volume of +this tubular neighbourhood. +For practical reasons, in the paper we will work with Σn,d instead of its “double” Vn,d. There are, +however, two technical issues to consider here. The first one is that there will be a factor of 2 to +be taken into account when switching from the Veronese surface Σn,d to the rank–one variety Vn,d, +depending on the parity of d. The second one is that the intersection of a δ–neighbourood of the set +of rank–one tensors with the unit sphere becomes an ε–neighbourhood of Vn,d in the unit sphere, with +ε = arcsin(δ). + +4 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +This is why we will use the parameter “ε” to formulate the results on the sphere and the parameter +“δ” for the results in the vector space of tensors. +1.4. The reach of the Veronese variety. Our aim is to exploit Weyl’s tube formula to compute +the volume of U(Vn,d, ε). This requires first of all the knowledge of the radii for which the above +expression holds. Since the formula is based on the parametrization through the normal exponential +map, the supremum of the radii for which this is a good parametrization, or at least a lower bound +on that, is what we need to understand to meaningfully use Weyl’s result. This quantity is usually +called the reach of the embedding M ֒→ M and in general computing it is a very difficult task, often +unfeasible since it requires to study not only how normal geodesics originating from every point of the +submanifold behave, but also how and when geodesics starting from different points cross each other, +in order to avoid overlappings in the image. +In our case, recalling the invariance property of Vn,d ֒→ SN under the action of the orthogonal group +O(n + 1), we do not need to study normal geodesics originating from any point, but it is enough to +choose a specific one and perform computations involving only geodesics originating from this chosen +one. This drastically reduces the complexity of the computation, allowing us to obtain the following +result, stated in a more detailed form in Theorem 19. +Theorem A (The reach of the spherical Veronese). The reach of the spherical Veronese variety +Vn,d ֒→ SN is given by +ρ(Vn,d) = +1 +√ +3 ++ +1 +3d +√ +3 ++ O +� 1 +d2 +� +. +The same result holds for the reach of the Veronese surface Σn,d ֒→ SN. +Given the interpretation of the neighbourhood of the Veronese variety in terms of symmetric tensors +already discussed, this theorem has an important consequence. From its proof, it follows that every +real symmetric tensor which is sufficiently close to rank–one tensors admits a unique best rank–one +approximation (see Corollary 21). In fact the reach ρ(Vn,d) equals the minimum between two quantities, +one of which estimates the size of the neighborhood of Vn,d on which the normal exponential map is +injective; we prove that this quantity equals π +4 , and this allows to deduce the following. +Corollary B (Best rank–one approximation). Every symmetric tensor p at distance less than +√ +2 +2 ∥p∥BW +from rank–one admits a unique best rank–one approximation. +The normalized volume of a neighbourhood of the Veronese variety of radius π +4 would therefore +provide a lower bound for the probability that such tensors have a unique best rank–one approximation. +Unfortunately, we are not able to compute such a volume, given that the value of the reach ρ(Vn,d) < π +4 +does not allow to use Weyl’s tube formula up to such a radius. Nevertheless, given the uniformity of +the lower bound for the reach ρ(Vn,d) ≥ +1 +√ +3 for every n, d (see Remark 20), we still get that the volume +of the neighbourhood of radius +1 +√ +3 provides a lower bound for that probability, even if not sharp. This +bound can be explicitly computed by plugging in ε = +1 +√ +3 in Theorem 24. Moreover, in the case of +tensors in two variables, which correspond to the case of rational normal curves, using the asymptotic +in Theorem 26 we get an asymptotic expression for this bound. +1.5. The curvature coefficients of the Veronese variety. The other ingredient needed in Weyl’s +formula are the curvature properties of the embedding, in particular the Weingarten operator along +normal directions, which encodes the second fundamental form. Again by the invariance of the extrinsic +geometry of Vn,d ֒→ SN, it is enough to compute this at a specific point, which we choose to be xd +0 for +simplicity of computations. +Before stating our result, recall that we have denoted by GOE(n) the Gaussian Orthogonal Ensemble, +i.e. the set Sym(n, R) endowed with the gaussian probability distribution coming from the Frobenius +scalar product, see Section 2.1 for more details. +Our main result on the extrinsic geometry of the embedding Vn,d ֒→ SN is the following, and we +refer to Theorem 22 for a more comprehensive statement. + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +5 +Theorem C (The normal bundle splitting). Let p ∈ Vn,d and denote by Lη the Weingarten operator of +Vn,d ֒→ SN along a normal direction η ∈ NpVn,d. There exists an orthogonal decomposition NpVn,d = +W ⊕ P such that the following statements hold: +(1) Lη = 0 for every η ∈ P; +(2) W with its induced Bombieri–Weyl metric is isometric to Sym(n) with the Frobenius one. +Moreover, if we pick η ∈ W Gaussian, then Lη ∼ +√ +2 +� +d−1 +d +� 1 +2 +GOE(n). +This theorem could find applications going beyond the scope of this paper. It gives a full description +of the second fundamental form in terms of GOE(n) matrices. Using this description in Weyl’s tube +formula to compute the curvature coefficients of the spherical Veronese, the consequence is that the +computation of some integrals on the normal bundle depending on the Weingarten operator boils +down to computing the expectation of a determinant involving GOE(n) matrices. This is reduced to +an easy, purely combinatorial computation (see Appendix C) and thus we obtain the following explicit +expressions for the curvature coefficients. +Theorem D (The curvature coefficients of the spherical Veronese). The curvature coefficients of the +Veronese variety Vn,d ֒→ SN are as follows: +KN−n+j(Vn,d) = (−1) +j +2 d +n +2 +�d − 1 +d +� j +2 +2n+2−jπ +N +2 Γ +� n +2 + 1 +� +Γ +� j +2 + 1 +� +Γ(n + 1 − j)Γ +� +N+j−n +2 +� +for 0 ≤ j ≤ n and j even, and KN−n+j(Vn,d) = 0 otherwise. +We remark that similar results hold true for the projective Veronese variety, using the double covering +SN −→ RPN, and for the spherical Veronese surface. +Plugging these coefficients back in Weyl’s tube formula we also obtain the explicit expression of the +volume of the tubular neighbourhood for radii smaller than the reach (see Theorem 24), in particular +giving an answer to question stated at the beginning of the paper. +Theorem E (The probability of being close to rank–one). Let p be a random Bombieri–Weyl symmet- +ric tensor of order d on Rn+1 and Vn,d ⊂ R[x0, . . . , xn](d) ≃ RN+1 be the Veronese variety of rank–one +tensors. For every δ such that 0 ≤ arcsin(δ) < +1 +√ +3 + +1 +3d +√ +3 + O +� 1 +d2 +� +we have +P +� +distBW(p, Vn,d) ≤ δ∥p∥BW +� += +� +0≤j≤n +j even +(−1) +j +2 d +n +2 +�d − 1 +d +� j +2 +2n−j+1π− 1 +2 +· +Γ +� n +2 + 1 +� +Γ +� N+1 +2 +� +Γ +� j +2 + 1 +� +Γ(n + 1 − j)Γ +� +N+j−n +2 +� +� +δ +√ +1−δ2 +0 +tN−n+j−1 +(1 + t2) +N+1 +2 +dt. +We remark that the above expression, even if unpleasant, gives an exact formula for our probability. +In the last section, we present an asymptotic expression, based on Laplace’s method, for such a +probability in the case of rational normal curves, corresponding to the case of tensors in two variables, +when the degree goes to infinity (see Theorem 26). We stress that it is possible to obtain a meaningful +asymptotic since the reach is uniformly bounded below. We also remark that the decay is exponential +in d, as one might expect looking at the codimension of V1,d ֒→ Sd, which is d − 1. More generally, the +probability has an exponential decay in the codimension of Vn,d ֒→ SN for any n, d. +2. Preliminaries +2.1. The Gaussian Orthogonal Ensemble and the Bombieri–Weyl distribution. In this sec- +tion, we point out the correspondence between the Gaussian Orthogonal Ensemble on the space of +symmetric matrices and the Bombieri–Weyl distribution on the space of homogeneous polynomials. + +6 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +Let Sym(n, R) be the space of symmetric n × n matrices with real entries and denote by Eij the +elementary matrix having all entries 0 except for the ij-th one being 1. Consider a random matrix +Q = +n +� +i=1 +ηiiEii + +n +� +i 0. We also call normal exponential map the +restriction of the exponential map of M to NM. +Definition 4. If there exists an ε > 0 such that +exp |N εM : N εM −→ M +is a diffeomorphism on its image, where exp denotes the exponential map of M, we call N εM a tubular +neighbourhood of M in M. +Recall the definition of distance of a point x ∈ M from the submanifold M, given by dg(x, M) := +inf{dg(x, y) | y ∈ M} where dg(x, y) is the Riemannian distance between x, y. +We introduce the +following set +(5) +U(M, ε) := {x ∈ M | dg(x, M) < ε}, +consisting of points at distance less than ε > 0 from M. The description of this set for submanifolds of +R3 is quite easy: the distance of a point from a surface or a curve will always be given by the length of +a segment starting from the point and meeting the submanifold orthogonally, given that segments are +geodesics. This situation can be generalized, as the following theorem shows. Even though this result +is well known, we were not able to find a full explicit reference for this general setting. We, therefore, +provide a full proof in Appendix A, filling in the details of the outline given in [CdS01, Theorem 6.6]. +Theorem 5 (Tubular neighbourhood theorem). Let M be a compact isometrically embedded +submanifold of a Riemannian manifold (M, g). Then there exists an ε > 0 small enough such that +exp |N εM : N εM −→ M is a diffeomorphism on its image and exp(N εM) = U(M, ε). +Theorem 5 can be seen as an existence result for tubular neighbourhoods of compact submanifolds and +as a characterization of the set U(M, ε) for ε small enough. For this reason, in the following, we will +refer also to U(M, ε) as a tubular neighbourhood. +Remark 6. The compactness assumption in theorem (5) is crucial. If we remove compactness, we +can only prove the existence of a smooth function ε(·) : M −→ R>0 such that the restriction of the +exponential map to N ε(·)M is an embedding, where N ε(·)M = {v ∈ NxM | x ∈ M, ∥v∥ < ε(x)}, i.e. +the ε is not uniform anymore but it depends on the point. +Definition 7. Let M be an isometrically embedded submanifold of a Riemannian manifold (M, g). +We define the reach of M ֒→ M as +ρ(M) = sup{ε ≥ 0 | N εM is a tubular neighbourhood of M}. +We can restate theorem (5) by saying that the reach of a compact submanifold is always positive. +Remark that even if for brevity we write ρ(M), the reach is not an intrinsic property of M but it +depends on the way M is embedded into M. From the very definition it follows that ρ(M) can be +expressed as the minimum between ρ1(M) = sup{ε ≥ 0 | exp |N εM is an immersion} and ρ2(M) = +sup{ε ≥ 0 | exp |N εM is injective}. +The points where the differential of the normal exponential map is not injective are called focal points. +Their existence is linked to the presence of particular Jacobi fields, see [dC92, Section 10.4]. +For +compact submanifolds of round spheres, one obtains the following expression +(ρ1(M))−1 = sup +x∈M +sup{∥ +..γ(0)∥ s.t. γ : (−δ, δ) −→ M +(6) +arclength geodesic in M with γ(0) = x}. +If ε > 0 is the first value for which we lose injectivity of the restriction of the normal exponential, it +means that there is a point in M that is reached by two length–ε normal geodesics starting from two +different points of M. Using the Generalized Gauss Lemma for geodesic variations, see [Gra04, Lemma +2.11], and uniqueness of geodesics, one can show that the union of these two normal geodesics is again + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +9 +a geodesic meeting M orthogonally at its endpoints. Therefore we have the following expression for +ρ2(M) +ρ2(M) = 1 +2 inf{l(γ) | γ : [a, b] −→ M geodesic s.t. γ(a), γ(b) ∈ M, +(7) +˙γ(a) ∈ Nγ(a)M, ˙γ(b) ∈ Nγ(b)M}. +We will apply these expressions in chapter 4 to compute the reach of the Veronese variety. +In [Wey39] Weyl presented a fundamental work, answering a question posed by Harold Hotelling: +how can we compute the volume of a “tube”, i.e. a tubular neighbourhood, of fixed radius around a +closed n–dimensional manifold in RN or SN? Hotelling himself answered the case of curves, both in +the euclidean and in the spherical setting. Weyl extended these results to any dimension n as follows. +Theorem 8 (Weyl’s tube formula). Let M be a smooth, n–dimensional, compact submanifold +isometrically embedded in RN (or SN) with their standard metrics. Then, for ε < ρ(M), the following +formula holds: +Vol +� +U(M, ε) +� += +� +0≤e≤n, e even +Ks+e(M)JN,s+e(ε), +(8) +where s := N − n is the codimension of M and JN,s+e are linearly independent functions of ε only. In +the euclidean case, the universal functions J have the following form: +JN,k(ε) := εk, +(9) +while in the spherical one, they are given by +JN,k(ε) := +� ε +0 +(sin ρ)k−1(cos ρ)N−k dρ = +� tan ε +0 +tk−1 +(1 + t2) +N+1 +2 +dt. +(10) +Moreover, the coefficients Kj(M) are isometric invariants of M. +The coefficients Kj(M) are called the curvature coefficients of M. The motivation for this comes +from the fact that they are integrals of functions on the second fundamental form of M. +Notice +that in [Wey39] Weyl uses different normalization constants for (9) and (10). We chose to follow the +normalization introduced by Nijenhuis in [Nij74], which proves to be handier for applications, see for +instance [Bue06]. Remark that the ε’s for which the formula holds depend on the reach and therefore on +the embedding, while isometric embeddings will give the same curvature coefficients. The dependence +of the validity of the formula on the reach is due to the proof relying on parametrizing the tubular +neighbourhood through the normal exponential map. In [How93] Howard contextualizes Weyl’s result +in the framework of “Integral Geometry”, where he considers more general integrals of polynomials on +the second fundamental forms of submanifolds of homogeneous spaces. +There are explicit integral versions of formula (8) for both the euclidean and spherical cases. For the +latter, with the same notations above, this reads as +Vol +� +U(M, ε) +� += +� +p∈M +� tan ε +t=0 +� +S(NpM) +tm−1 det +� +In − tLη +� +(1 + t2) +N+1 +2 +volM dη dt, +(11) +where S(NpM) denotes the unit sphere in NpM, In is the n × n identity matrix, Lη is the Weingarten +operator of M ֒→ M along the unit normal vector η, and dη is a short notation for the volume form +on S(NpM). If one explicitly develops the determinant, it is easy to get back formula (8). +2.3. A lemma on integration on spheres. Consider a sphere Sm for some m ≥ 2 and fix k ∈ +{1, . . ., m − 1}. Denote by ι : Sk ֒→ Rk+1 the inclusion map and consider the map +Sk × +◦ +Dm−k +ϕ +−→ Sm ⊂ Rm+1 = Rk+1 × Rm−k +(12) +(σ, z) +�−→ +( +� +1 − |z|2 ι(σ), z), +giving a smooth parametrization of Sm \ +� +{0} × Sm−k−1� +⊂ Rk+1 × Rm−k. For every l ∈ N, consider +Rl endowed with a non–degenerate scalar product and coordinate functions x1, . . . , xl with respect to +an orthonormal basis. Then we have the standard volume form volRl = dx1 ∧ · · · ∧ dxl which induces + +10 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +a volume form vol ◦ +Dl on the open norm–1 disc +◦ +Dl and through the pullback of the inclusion also a +volume form volSl−1 on Sl−1. Notice that with respect to volSm, the part of Sm not parametrized by +ϕ in (12) has measure 0. We have the following result about integration on spheres, see Appendix B +for a proof. +Lemma 9. With the same notations above, the pullback of volSm through ϕ is given by +ϕ∗(volSm) = +� +1 − |z|2� k−1 +2 +volSk ∧ vol ◦ +Dm−k. +In particular, this implies that +� +Sm f(p) volSm = +� +Sk +� +◦ +Dm−k f +�� +1 − |z|2 ι(σ), z +�� +1 − |z|2� k−1 +2 +volSk ∧ vol ◦ +Dm−k, +(13) +for any measurable function f on Sm. +2.4. Laplace’s method. One of the most important asymptotic methods for computing integrals +depending on one large parameter is the so-called “Laplace’s method”. For a proof of this result and +more details on asymptotic methods for integrals, we refer to [Won01]. +Theorem 10 (Laplace’s method). Consider the following integral depending on one parameter +λ > 0: +I(λ) := +� t2 +t1 +e−λa(t)b(t) dt, +where a, b : [t1, t2] −→ R are functions satisfying the following conditions: +(1) a is smooth in a neighbourhood of t1, and there exist µ > 0 and a0 ̸= 0 such that, for t −→ t1, +we have: +a(t) = a(t1) + a0(t − t1)µ + O +� +|t − t1|µ+1� +; +(2) b is smooth in a neighbourhood of t1, and there exist ν ≥ 1 and b0 ̸= 0 such that, for t −→ t1, +we have: +b(t) = b0(t − t1)ν−1 + O +� +|t − t1|ν� +; +(3) t1 is a global minimum for a on [t1, t2], i.e. a(t) > a(t1) for any t ∈]t1, t2[. Moreover for all +ε > 0 we have: +inf +t∈[t1+ε,t2[{a(t) − a(t1)} > 0; +(4) the integral I(λ) converges absolutely for sufficiently large λ. +Then, as λ −→ +∞, we have: +I(λ) = e−λa(t1) · +Γ +� ν +µ +� +λ +ν +µ +· +b0 +µ a +ν +µ +0 +· +� +1 + O +� +λ− 1+ν +µ +�� +. +Remark that if the minimum of a(t) is attained at the extremum t2, the theorem holds with the roles +of t1 and t2 reversed. The idea behind the statement is that the major contribution to I(λ) will be +given by the behaviour of the integrand around the minimum point of a, which can be assumed to +be one of the endpoints of the integration domain. It can be proven that some of the smoothness +hypotheses can be relaxed, even if some regularity is still needed. The statement of Theorem 10 is +the standard form for the Laplace’s method. For statements with weaker regularity assumptions and +generalizations to larger classes of integrals, we refer to [Olv97] and [Nem20]. + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +11 +3. The Veronese variety +Consider the space of homogeneous polynomials of degree d in n + 1 variables R[x0, . . . , xn](d) ∼= +RN+1, where N := +�n+d +d +� +− 1, with the basis described in (3). +Definition 11. For n ≥ 1 and d ≥ 1, the real Bombieri–Weyl Veronese embedding is the map +νn,d : RPn −→ +RPN +[a] +�−→ +��d +α +�1/2 +aα +� +and it is the Veronese projective embedding associated to the Bombieri–Weyl basis. The Bombieri– +Weyl Veronese variety is the image of this embedding, denoted by PVn,d := im(νn,d). +The main object we will consider in what follows is the spherical counterpart of PVn,d. +Definition 12. The spherical (Bombieri–Weyl) Veronese map is the map +�νn,d : Sn −→ SN +a +�−→ +��d +α +�1/2 +aα +� +. +The spherical (Bombieri–Weyl) Veronese surface is the image of this map, denoted by Σn,d := im(�νn,d). +It is worth stressing in the definition of �νn,d that Sn is the sphere with respect to the standard +euclidean product in Rn+1 while SN is the sphere with respect to the Bombieri–Weyl product in +R[x0, . . . , xn](d) and that �νn,d is well defined, as one can check by an explicit computation. +The objects we just introduced have a particularly useful description. Recall that to each b = +(b0, . . . , bn) ∈ Rn+1 we can associate the linear form on Rn+1 given by lb(x0, . . . , xn) = b0x0+· · ·+bnxn. +It is known that PVn,d parametrizes projective classes of d−th powers of linear forms +PVn,d = +� +[d–th powers of linear forms on Rn+1] +� += +� +[aα] ∈ RPN | ∃ b = (b0, . . . , bn) ∈ Rn+1 s.t. aα0,...,αn = +�d +α +�1/2 +bα0 +0 . . . bαn +n +� +as one can prove by showing that νn,d([b0, . . . , bn]) = [(b0x0 + · · ·+ bnxn)d]. This also leads to the well- +known description of the Veronese variety PVn,d as the variety of symmetric decomposable d−tensors +on Rn+1 and is one of the main reasons Veronese varieties have been so intensively studied. A similar +description holds for the spherical Veronese surface +Σn,d = {d–th powers of norm–1 linear forms on Rn+1} += { (aα) ∈ SN | ∃ b = (b0, . . . , bn) ∈ Sn s.t. aα0,...,αn = +�d +α +�1/2 +bα0 +0 . . . bαn +n }. +Using these descriptions of PVn,d and Σn,d, it is immediate to prove the following. +Proposition 13. PVn,d is an orbit for the action of the orthogonal group O(n + 1) on RPN = +P(R[x0, . . . , xn](d)) by change of variables. +Similarly Σn,d is an orbit for the same action of the +orthogonal group O(n + 1) on SN = S(R[x0, . . . , xn](d)). +Recall the two-fold covering map πN : SN −→ RPN given by the identification of antipodal points. +Its restriction to the spherical Veronese Vn,d := Vn,d ∩ SN gives a covering map �πn,d : Vn,d −→ PVn,d +whose degree depends on the parity of d: if d is odd �πn,d is a 2 : 1 covering, while if d is even it is 1 : 1, +since in this case for b ∈ Sn we have �νn,d(b) = �νn,d(−b). +We now turn to metric properties of Veronese manifolds. Consider on Rn+1 the standard euclidean +metric and on RN+1 = R[x0, . . . , xn](d) the Bombieri–Weyl one. The metrics induced on Sn and SN +respectively are invariant under the antipodal map and therefore induce metrics on the corresponding +projective spaces RPn and RPN. We denote the metrics on the spheres by gSn and gSN and those on + +12 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +the projective spaces by gRPn and gRPN . Since the covering maps πn, πN are Riemannian coverings +with these metrics, any relation between gSn and gSN will also hold between gRPn and gRPN and +viceversa. Through a direct computation, one can prove the following result. +Proposition 14. Pulling back the Bombieri–Weyl metric through the Veronese embedding νn,d : +RPn −→ RPN, for any n ≥ 1, d ≥ 1 we have +ν∗ +n,d gRPN = +√ +d gRPn. +(14) +Corollary 15. For every n, d ∈ N and any smooth n–dimensional submanifold C ֒→ Σn,d we have +VolBW +n +(C) = + + + + + +1 +2d +n +2 Voln +� +�ν−1 +n,d(C) +� +for d even +d +n +2 Voln +� +�ν−1 +n,d(C) +� +for d odd +, +(15) +where Voln is the n–dimensional volume with respect to gSn and VolBW +n +is the n–dimensional volume +with respect to the metric induced by gSN on Σn,d. In particular +VolBW +n +(Σn,d) = + + + + + + + + + +d +n +2 +π +n+1 +2 +Γ( n+1 +2 +) +for d even +2d +n +2 +π +n+1 +2 +Γ( n+1 +2 +) +for d odd +. +(16) +In section 5.2 we will need the explicit expression of VolBW +n +(Vn,d). This easily follows from formula +(16), since Vn,d = Σn,d ∪ −Σn,d where Σn,d = −Σn,d for d odd and Σn,d ∩ (−Σn,d) = ∅ for d even. +Therefore +VolBW +n +(Vn,d) = 2d +n +2 +π +n+1 +2 +Γ( n+1 +2 ) = d +n +2 Vol(Sn). +(17) +Remark 16. For every orthogonal matrix R ∈ O(n + 1) we have the following commutative diagram +(RPn, gRPn) +(RPn, gRPn) +PVn,d +PVn,d +νn,d +R +νn,d +ρ(R)|PVn,d +, +(18) +since PVn,d is an orbit for the action ρ and is therefore preserved under ρ(R). +Moreover, by the +invariance of the Bombieri–Weyl scalar product, ρ(R) is an isometry of (RPN, gRPN ), and its restriction +to PVn,d defines an isometry of PVn,d. Therefore PVn,d is an orbit for an isometric action of O(n+1) over +R[x0, . . . , xn](d) and these isometries of PVn,d are induced by isometries of the ambient space. The same +property also holds for Σn,d considering the action on the sphere SN. This simple observation, which +is essentially due to PVn,d and Σn,d being orbits, will allow us to drastically simplify the computations +we will carry out in Chapter 4 and Section 5.1. +4. The reach of the spherical Veronese variety +In this chapter we provide an explicit computation for the reach of Σn,d ֒→ SN. The interest in this +quantity relies on the fact it provides a lower bound for the ε’s of validity for Weyl’s tube formula, as +theorem 8 shows. Remark that, since Σn,d is compact, by theorem 5 we have ρ(Σn,d) > 0. +Recall that ρ(Σn,d) = min{ρ1(Σn,d), ρ2(Σn,d)} and the expressions (6) and (7). Using remark 16 +we can prove the following. + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +13 +Lemma 17. The formula in (6) simplifies to +� +ρ1(Σn,d) +�−1 = sup{ ∥ +..γ(0)∥ | γ : (−δ, δ) −→ Σn,d arclength geodesic in Σn,d, γ(0) = xd +0 }, +(19) +that is to say the inner supremum in (6) does not depend on x ∈ Σn,d. Moreover (19) does not depend +on the direction of +.γ(0). Similarly, the formula in (7) simplifies to +ρ2(Σn,d) = 1 +2 inf{l(γ) +�� γ : [a, b] −→ SN geodesic s.t. γ(a) = xd +0, γ(b) ∈ Σn,d, +(20) +˙γ(a) ∈ Nxd +0Σn,d, ˙γ(b) ∈ Nγ(b)Σn,d}. +Proof. Consider an arclength geodesic γ : (−δ, δ) −→ Σn,d with γ(0) = p1 and pick another point +p2 ∈ Σn,d. By remark 16 there exists R ∈ O(n + 1) such that ρ(R)p1 = p2. Recall that the image of +a geodesic through an isometry is still a geodesic, hence ˜γ := ρ(R)(γ) is an arclength geodesic with +˜γ(0) = ρ(R)(γ(0)) = ρ(R)p1 = p2. Since ρ(R) is also an isometry of the ambient space SN, we have +∥ +..γ(0)∥ = ∥ +.. +˜γ(0)∥. We just proved that given any two points in Σn,d, using the isometries ρ(R) for +R ∈ O(n + 1) we can transport any arclength geodesic passing through the first point into another +arclength geodesic passing through the second point, preserving the norm of second derivatives. It +follows that the expression in (6) is independent of the specific point x ∈ Σn,d. Now observe that given +any arclength geodesic γ with γ(0) = xd +0 and +.γ(0) = v, we can change the direction of +.γ(0) through +ρ(R) for some R ∈ O(n + 1) with xd +0 a fixed point (it is sufficient to choose a rotation R such that +(1, 0, . . . , 0) ∈ Sn is in the axis of rotation), obtaining any other possible direction in Txd +0Σn,d without +changing ∥ +··γ(0)∥, for the same reason as above. It follows that (19) does not depend on the specific +direction of +.γ(0). Since isometries preserve orthogonality and lengths, the second part of the lemma +also follows in a similar way. +□ +The choice of xd +0 in formulae (19) and (20) is motivated by convenience for computations only and we +could have chosen any other point. +The first step to compute (19) and (20) for Σn,d ֒→ SN is to understand tangent and normal spaces. +Lemma 18. For p ∈ Σn,d with p = ld, where l is a norm-1 linear form, we have +TpΣn,d = +� +{ld−1λ | λ is a linear form orthogonal to l} +� +. +Proof. Write l(x) = a0x0 +· · ·+anxn and set a = (a0, . . . , an) ∈ Sn. Recalling that Σn,d = im(�νn,d), a +curve on Σn,d can be expressed as the image of a curve on Sn through �νn,d. Consider b = (b0, . . . , bn) ∈ +Sn such that ⟨a, b⟩ = 0. Then γ(t) = (cos t(a0x0 + · · · + anxn) + sin t(b0x0 + · · · + bnxn))d is a curve in +Σn,d with γ(0) = p. Remark that the orthogonality condition is needed to ensure we are taking d–th +power of a norm–1 form. We have +d +dtγ(t) +�� +t=0 = d ld−1(b0x0 + · · · + bnxn), +therefore +� +{ld−1λ | λ is a linear form orthogonal to l} +� +⊂ TpΣn,d. By dimension count, equality follows. +□ +Now we have the ingredients we need to perform the computation of ρ(Σn,d). +Theorem 19. For the reach of Σn,d ֒→ SN we have +ρ1(Σn,d) = +1 +√ +3 + +1 +3d +√ +3 + O +� 1 +d2 +� +and +ρ2(Σn,d) = π +4 . +Therefore, for d ≥ 2, the reach of Σn,d is given by +ρ(Σn,d) = min{ρ1(Σn,d), ρ2(Σn,d)} = +1 +√ +3 + +1 +3d +√ +3 + O +� 1 +d2 +� +. +(21) + +14 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +Proof. We begin with the computation of ρ1(Σn,d). +By Proposition 14 geodesics in Σn,d can be +realized as images through �νn,d of geodesics in Sn. +Moreover, thanks to Lemma 17, it is enough +to consider geodesics passing through xd +0 = (1, 0, . . . , 0) = �νn,d((1, 0, . . . , 0)) at time 0 and their +direction plays no role, hence it is enough to consider the image of the geodesic in Sn given by +α(t) = x0 cos(td− 1 +2 ) + x1 sin(td− 1 +2 ) with x0 being the point of coordinates (1, 0, . . . , 0) and x1 that +of coordinates (0, 1, 0, . . ., 0). Explicitly we have α(t) = (cos(td− 1 +2 ), sin(td− 1 +2 ), 0, . . . , 0) and the corre- +sponding geodesic in Σn,d is given by +γ(t) := (�νn,d ◦ α)(t) = +� +cosd(td− 1 +2 ), +√ +d cosd−1(td− 1 +2 ) sin(td− 1 +2 ), . . . , sind(td− 1 +2 ) +� +, +with γ(0) = xd +0 = (1, 0, . . . , 0) and ∥ ˙γ(0)∥ = 1. Notice that the only components of γ(t) which are +not constantly zero are those corresponding to multi–indices (β0, β1, 0, . . . , 0) with β0 + β1 = d. To +compute ∥ +..γ(0)∥, we need the second derivatives of the non–constantly zero components of γ(t). These +have the following expression for k = 0, . . . , d: +�d +k +� 1 +2 +cosk(td− 1 +2 ) sind−k(td− 1 +2 ). +Computing second derivatives and evaluating at t = 0 we find +..γ(0) = (−1, 0, +√ +2 +� +d(d − 1) +d +, 0, . . . , 0). +Using the Taylor-MacLaurin expansion for √1 + x we obtain +∥ +..γ(0)∥ = +� +1 + 2(d − 1) +d += +√ +3 +� +1 − 2 +3d = +√ +3 − +1 +√ +3d + O +� 1 +d2 +� +. +Applying again a Taylor-MacLaurin expansion for +1 +1−x, we get the final expression for ρ1(Σn,d): +ρ1(Σn,d) = +1 +∥ +..γ(0)∥ = +1 +√ +3 + +1 +3 +√ +3d + O +� 1 +d2 +� +. +Remark 20. Notice that, since ρ1(Σn,d) = +�√ +3 +� +1 − 2 +3d +�−1 +, then ρ1(Σn,d) > +1 +√ +3 for all n, d. +For ρ2(Σn,d), by Lemma 17 it is enough to consider geodesics γ(θ) in SN starting at xd +0 at time θ = 0. +By Lemma 18 and recalling that the normal space at a point p ∈ Σn,d is the orthogonal complement of +TpΣn,d inside TpSN, we have Nxd +0Σn,d = +���d +α +� 1 +2 xα0 +0 . . . xαn +n +| α0 < d − 1 +�� +. Pick a vector w ∈ Nxd +0Σn,d +and let γw(θ) be the geodesic in SN with γw(0) = xd +0 and ˙γw(0) = w, i.e. +γw(θ) = xd +0 cos +� +θ∥w∥ +� ++ +w +∥w∥ sin +� +θ∥w∥ +� +. +The goal now is to understand when γw meets again Σn,d orthogonally. The first step is to find for +which b = (b0, . . . , bn) ∈ Sn and θ we at least have a solution to the equation +(b0x0 + · · · + bnxn)d = xd +0 cos +� +θ∥w∥ +� ++ +w +∥w∥ sin +� +θ∥w∥ +� +. +On the right hand side, we expand w as w = � +α0 2 we parametrize S(W ⊕P) as in (12), where here we use m = N −n−1 and k = n(n+1) +2 +−1. With +the same notation of section 2.3, for σ ∈ S(W) and z ∈ +◦ +D(P), if ϕ(σ, z) = η ∈ S(W ⊕ P), we have +that +� +1 − |z|2ι(σ) will be the component of η along W, while z itself will be the component along P. +We also apply the linear isometry discussed in the previous section to change variable from σ ∈ S(W) +to Q ∈ S(Sym(n, R)) = S +n(n+1) +2 +−1. +It is clear by its definition that the Weingarten operator is linear in the normal vector argument: +given an isometric embedding M ֒→ M, for every p ∈ M, η, ξ ∈ NpM and a, b ∈ R, we have +Laη+bξ = aLη + bLξ. Therefore for η = ϕ(Q, z) ∈ S(W ⊕ P) we have +Lη = L√ +1−|z|2Q+z = +� +1 − |z|2LQ + Lz = +� +1 − |z|2LQ. +(33) +Applying Lemma 9 to (32) and using (33) the integral becomes +Vol +� +U(Σn,d, ε) +� +=Vol(Σn,d) +� tan ε +t=0 +� +S +n(n+1) +2 +−1 +� +DN−n− n(n+1) +2 +� +tN−n−1 +(1 + t2) +N+1 +2 +× +(34) +× det(In − t +� +1 − |z|2LQ)(1 − |z|2) +n(n+1) +4 +−1 +� +dz dS(Q) dt, +where dz is a short notation for vol +DN−n− n(n+1) +2 +and dS(Q) is a short notation for vol +S +n(n+1) +2 +−1, with +the convention that for d = 2 the integral over D0 is set to 1. The only non-explicit term in (34) is +the one involving the determinant. Recall that by Theorem 22, if Q ∈ Sym(n, R) is a random GOE(n) +matrix, then LQ is a random matrix distributed as +√ +2 +� d−1 +d +� 1 +2 GOE(n). Set τ := t +√ +2 +� d−1 +d +� 1 +2 . We have +the expansion +det +� +In − τ +� +1 − |z|2Q +� += +n +� +j=0 +(−1)jτ j(1 − |z|2) +j +2 gj(Q), +(35) + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +19 +where gj(Q) are homogeneous polynomials of degree j in the coefficients of Q for j = 1, . . . , n and +g0(Q) = 1. Substituting (35) into (34) in the integral splits as +Vol(U(Σn,d, ε)) = Vol(Σn,d) +n +� +j=0 +(−1)j2 +j +2 +�d − 1 +d +� j +2 �� tan ε +0 +tN−n−1+j +(1 + t2) +N+1 +2 +dt +� +× +(36) +× +�� +DN−n− n(n+1) +2 +(1 − |z|2) +n(n+1) +4 +−1+ j +2 dz +� +× +× +�� +S +n(n+1) +2 +−1 gj(Q) dS(Q) +� +, +where the first term is the integral of a rational function in t, while the second one is a “polynomial” +in |z|. +Remark that since gj are homogeneous polynomials, we have gj(Q) = ∥Q∥jgj( Q +∥Q∥). Recalling expres- +sion (2) we have +E +Q∈GOE(n)gj(Q) = +1 +(2π) +n(n+1) +4 +� +Sym(n,R) +∥Q∥jgj +� Q +∥Q∥ +� +e− ∥Q∥2 +2 +dQ = +(37) += +1 +(2π) +n(n+1) +4 +�� +∞ +0 +ρ +n(n+1) +2 +−1+je− ρ2 +2 dρ +��� +S +n(n+1) +2 +−1 gj( ˜Q) dS +� ˜Q +�� +. +From (37) we obtain +� +S +n(n+1) +2 +−1 gj(Q) dS(Q) = +E +Q∈GOE(n)[gj(Q)] (2π) +n(n+1) +4 +� +∞ +0 +ρ +n(n+1) +2 +−1+je− ρ2 +2 dρ +. +(38) +By linearity of expectation and the expansion det(In − λQ) = �n +j=0(−1)jλjgj(Q), to compute the +expectation of gj(Q) it is enough to compute that of det(In − λQ) for Q ∈ GOE(n) and look at the +homogeneous part of degree j in λ. This procedure gives us the explicit expression for (38) +� +S +n(n+1) +2 +−1 gj(Q) dS(Q) = +(−1) +j +2 (2π) +n(n+1) +4 +j! +( j +2 )! +�n +j +� +2j � +∞ +0 +ρ +n(n+1) +2 +−1+je− ρ2 +2 dρ +if 0 ≤ j ≤ n, j even +(39) +and 0 otherwise, see Appendix C for a proof of this result. By standard computations involving Gamma +and Beta functions, one can show that the following identities hold +� +∞ +0 +ρ +n(n+1) +2 ++j−1e− ρ2 +2 dρ = 2 +n(n+1) +4 ++ j +2 −1 Γ +�1 +4(n2 + n + 2j) +� +, +(40) +� +DN−n− n(n+1) +2 +� +1 − |z|2� n(n+1) +4 +−1+ j +2 dz = π +2N−n2−3n +4 +Γ +� 1 +4(n2 + n + 2j) +� +Γ +� 1 +2(N − n + j) +� , +(41) +where we notice that for d = 2 (41) gives 1, agreeing with our convention. Substituting (39), (40) +and (41) into (36) and using the duplication formula for the gamma function, we obtain the explicit +expression of Vol +� +U(Σn,d, ε) +� +. Finally, recalling that Vn,d = Σn,d ∪ −Σn,d and using formula (16) to +express Vol(Σn,d), the proof of Theorem 24 is complete. +5.3. Asymptotics for rational normal curves. Recall the interpretation of the Veronese variety +Vn,d as the set of rank–1, norm–1 symmetric tensors of order d on Rn+1, while the sphere SN can be +thought of as the space of all norm–1 such tensors. Then the quantity +Vol +� +U(Vn,d, arcsinδ) +� +Vol(SN) +(42) +expresses the probability for a symmetric tensor p to be (δ∥p∥BW)–close to rank–1 one with respect +to the Bombieri–Weyl distribution. We will focus on the case n = 1, d −→ +∞, which corresponds to +the so called “spherical” rational normal curves V1,d. Notice that in this case N = d. Our asymptotic +analysis will be based on Laplace’s method, as described in Theorem 10. + +20 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +Theorem 26. For spherical rational normal curves V1,d, the following asymptotic expansion of (42) +holds: +Vol +� +U(V1,d, arcsinδ) +� +Vol(Sd) += +√ +d δd−1� +1 + O +� +d−1�� +(43) +as d −→ +∞, for arcsinδ ≤ +1 +√ +3. +From this theorem and the bound (24), we immediately obtain the following. +Corollary 27. Denote by Ad the set of symmetric tensors of order d on R2 that admit a unique best +rank–one approximation. Then, as d −→ +∞, we have +P +� +Ad +� +> +√ +d +� +sin 1 +√ +3 +�d−1� +1 + O +� +d−1�� +≈ +√ +d (0.546)d−1� +1 + O +� +d−1�� +. +Proof of Theorem 26. Set ε := arcsinδ. We start by noticing that, since we are looking at d −→ +∞, +by Theorem 19 the asymptotic analysis makes sense only for ε ≤ +1 +√ +3. Instead of using the implicit +form of Theorem 24, to express Vol(U +� +Σn,d, ε) +� +we will use (36), substituting (39) in it. This gives the +following expression +Vol +� +U(Σ1,d, ε) +� +Vol(Sd) += Vol(Σ1,d) +V ol(Sd) · +(2π) +1 +2 +� +∞ +0 +e− ρ2 +2 dρ +· +�� tan ε +0 +td−2 +(1 + t2) +d+1 +2 +dt +� +· +�� +Dd−2 +� +1 − |z|2�− 1 +2 dz +� +. +(44) +For n = 1 (16) reads as +Vol +� +Σ1,d +� += +� +2 +√ +d π +for d odd +√ +d π +for d even , +while it is known that Vol(Sd) = 2π +d+1 +2 +Γ( d+1 +2 +) and +� +∞ +0 +e− ρ2 +2 dρ = � π +2 . With easy algebraic manipulations, +we can rewrite the integral in t as +� tan ε +0 +td−2 +(1 + t2) +d+1 +2 +dt = +� tan ε +0 +1 +t2(1 + t2) +1 +2 exp +� +− d +� +− log +� +t +(1 + t2) +1 +2 +��� +dt = += +� tan ε +0 +e−d a(t)b(t) dt, +where we have set +a(t) = − log +� +t +(1 + t2) +1 +2 +� +, +b(t) = +1 +t2(1 + t2) +1 +2 . +One can show that the hypotheses of Laplace’s theorem are satisfied and that the minimum of a(t) in +(0, tan ε] is attained at tan(ε). Taking Taylor expansions of a(t) and b(t) around tan(ε) and applying +Theorem 10 we obtain +� tan ε +0 +td−2 +(1 + t2) +d+1 +2 +dt = 1 +d +� +sin ε)d−1 +� +1 + O +� +d−2�� +. +For the last integral in (44), we pass to spherical coordinates and reduce it to a Beta function (and +therefore to Gamma functions), obtaining +� +Dd−2 +� +1 − |z|2�− 1 +2 dz = +π +d−1 +2 +Γ +� d−1 +2 +�. +Plugging all the expressions we found in (44), and recalling that Vn,d = Σn,d ∪ −Σn,d, we obtain the +desired asymptotic. +□ + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +21 +Appendix A. +Proof of the Tubular Neighbourhood Theorem. We will use the same notation of Section 2.2. +Throughout the proof, we will identify M with the zero section in NM. We start by computing the +differential d(x,0)(exp|NM) : T(x,0)(NM) −→ TxM of exp|NM at (x, 0) ∈ NM for any x ∈ M. Notice +that dim(T(x,0)(NM)) = dim(TxM), hence surjectivity is enough to have a linear isomorphism. Denote +by γ(p,v) the unique geodesic on M such that γ(p,v)(0) = p and ˙γ(p,v)(0) = v. Let y ∈ TxM. Since +TxM = TxM ⊕ NxM, we can decompose y as y = y1 + y2 with y1 ∈ TxM and y2 ∈ NxM. Then there +exists σ1 : (−δ, δ) −→ M such that σ1(0) = x and ˙σ1(0) = y1. Define a curve σ : (−δ, δ) −→ NM by +σ(t) = (σ1(t), 0) ∈ NM. We have σ(0) = (x, 0) and ˙σ(0) = (y1, 0) ∈ T(x,0)(NM) and it follows that +d(x,0)(exp|NM)(y1, 0) = d +dtexp +� +σ(t) +����� +t=0 += y1, +proving that TxM is contained in the image of d(x,0)(exp|NM). Now take y2 ∈ NxM and define a curve +α : (−δ, δ) −→ NM by α(t) = (x, ty2). Then α(0) = (x, 0) and ˙α(0) = (0, y2) and it follows that +d(x,0)(exp|NM)(0, y2)) = d +dtexp +� +α(t) +����� +t=0 += y2, +proving that also NxM is contained in the image of d(x,0)(exp|NM). By linearity of the differential, we +obtain surjectivity and therefore d(x,0)(exp|NM) is an isomorphism. +As a consequence for every x ∈ M there exists an open neighbourhood Wx of (x, 0) in NM such +that the rank of the differential d(q,v)(exp|NM) is maximal for every (q, v) ∈ Wx. Up to shrinking the +neighbourhood, we can assume that Wx = +� +Ux × B(0, εx) +� +∩ NM where Ux is an open neighbourhood +of x ∈ M, B(0, εx) denotes the ball of radius εx centered at the origin in TxM and exp|Wx is an +embedding. By compactness we have a finite covering of M {Ux1, . . . , Uxr} for some x1, . . . , xr ∈ M. +Choosing ε := min{εx1, . . . , εxr} we get that +exp|N εM : N εM −→ M +is an immersion and a local embedding. Notice that for every ˜ε ≤ ε also exp|N ˜ +εM is an immersion and a +local embedding. We claim that there exists an ˜ε < ε such that this restriction is also globally injective. +If this is the case, then the restriction to the closure of the +˜ε +2–small normal bundle is an embedding, +since injective immersions with compact domain are embeddings. It follows that any number less than +˜ε +2 satisfies the statement of the theorem. +To prove the claim we argue by contradiction: suppose that for every n ∈ N there exist (xn, vn), +(yn, wn) ∈ N +1 +n M such that exp(xn, vn) = exp(yn, wn). +Since M is compact, up to restricting to +subsequences we can assume that xn converges to x ∈ M and yn converges to y ∈ M, while vn and +wn both converge to 0 since vn, wn ∈ B(0, 1 +n) for every n ∈ N. By compactness of M there exists +δ > 0 such that for every p ∈ M the map expp : B(0, δ) ⊂ TpM −→ M is a diffeomorphism on its +image, where expp(z) = exp(p, z). It follows that since vn −→ 0, for n large enough γ(xn,vn) will be the +unique geodesic joining xn with expxn(vn) = exp(xn, vn) = γ(xn,vn)(1) and dg(xn, exp(xn, vn)) = ∥vn∥. +Analogously, for n large enough we will also have dg(yn, exp(yn, wn)) = ∥wn∥, where we stress that +the uniformity of δ is crucial. Since by hypothesis exp(xn, vn) = exp(yn, wn), we have that +dg(xn, yn) ≤ dg(xn, exp(xn, vn)) + dg(yn, exp(yn, wn)) = ∥vn∥ + ∥wn∥ −→ 0, +and this forces x = y. Then for n sufficiently large, we have that (xn, vn), (yn, wn) ∈ Wp for some +p ∈ M, but on every Wp we have a local embedding, leading to a contradiction. The proof is concluded. +Appendix B. +Proof of lemma (9). We will use the same notations as in section 2.3. We want to prove that +ϕ∗(volSm) = +� +1 − |z|2� k−1 +2 +volSk ∧ vol ◦ +Dm−k, + +22 +ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA +where ϕ is given by (12). +By the usual formula for the pullback of a differential form through a +diffeomorphism, we have +ϕ∗(volSm) = |det(JϕT · Jϕ))| +1 +2 vol ◦ +Dk ∧ vol ◦ +Dm−k, +where Jϕ denotes the (m + 1) × m Jacobian matrix of ϕ and JϕT is its transpose. Denote by Jι the +Jacobian matrix of the inclusion ι : Sk ֒→ Rk+1. Then Jϕ is the following block matrix +Jϕ(σ, z) = + + +� +1 − |z|2Jι(σ) +� +−zj +√ +1−|z|2 ι(σ) +� +0 +Im−k + + , +where Im−k denotes the (m − k) × (m − k) identity matrix. Since JιT (σ) · ι(σ) = ι(σ) · Jι(σ) = 0 and +ι(σ)T · ι(σ) = 1, we obtain +(JϕT · Jϕ)(σ, z) = +� +(1 − |z|2)(JιT · Jι)(σ) +0 +0 +� +Im−k + +z·zT +1−|z|2 +� +� +. +For every z ∈ Rm−k consider R ∈ O(m − k) such that z = Re1|z|, where e1 = (1, 0, . . . , 0). Then we +can compute the determinant of the lower right block as +det +� +Im−k + z · zT +1 − |z|2 +� += det R +� +Im−k + +|z|2 +1 − |z|2 E11 +� +RT = +1 +1 − |z|2 , +where E11 = e1eT +1 has all zero entries except for the (1, 1)–th one which is 1. +Recalling that the +determinant of a diagonal block matrix is given by the product of the determinants of its blocks, we +find the following expression +|det(JϕT · Jϕ)| = +� +1 − |z|2�k |det(JιT · Jι)| +1 +1 − |z|2 = +� +1 − |z|2�k−1 |det(JιT · Jι)|. +Finally, applying again the formula for the pullback of a differential form, we can conclude that +ϕ∗(volSm) = +� +1 − |z|2� k−1 +2 +|det(JιT · Jι)| +1 +2 vol ◦ +Dk ∧ vol ◦ +Dm−k = +� +1 − |z|2� k−1 +2 +volSk ∧ vol ◦ +Dm−k. +Appendix C. +Proof of formula (39). We will use the same notations as Section 5.2. By linearity of the expectation, +in order to prove formula (39) all we have to do is computing +E +Q∈GOE(n)[det(In − λQ)] = +n +� +j=0 +(−1)jλj +E +Q∈GOE(n)[gj(Q)], +since the expectation of gj(Q) can then be deduced by looking at the degree j coefficient in above +polynomial expression in λ. First, we write the determinant according to its very definition +det(In − λQ) = +� +σ∈Sn +sgn(σ) +n +� +i=1 +� +δiσ(i) − λQiσ(i) +� +, +where Sn is the group of permutations on {1, . . ., n} and sgn(σ) is the signature of the permutation +σ ∈ Sn. Recall that for Q ∈ GOE(n) we have Qii ∼ N(0, 1) and Qij ∼ N(0, 1 +2) for i ̸= j and, apart +from the obvious symmetry conditions, the entries are independent. By linearity +E +Q∈GOE(n)[det(In − λQ)] = +� +σ∈Sn +sgn(σ) +E +Q∈GOE(n) +� +n +� +i=1 +� +δiσ(i) − λQiσ(i) +�� +. +(45) +Given σ ∈ Sn, suppose that it contains a cycle of length at least 3, i.e. there exists i ∈ {1, . . . , n} such +that σ(i) ̸= i and σ2(i) ̸= i. Then, by independence of the entries, in the term of (45) corresponding +to σ, we can split the expectation into a product of expectations, separating the term corresponding +to such i. Since δiσ(i) = 0 and Qiσ(i) is centered, this expectation is 0 and σ gives no contribution +to (45). It follows that the only permutations contributing to (45) are those formed by transpositions + +WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE? +23 +and fixed points only. +For such a σ ∈ Sn, denote by fix(σ) = {i ∈ {1, . . . , n} | σ(i) = i} the set of fixed points of σ and by +s(σ) the number of disjoint transpositions in σ. Then we have +E +Q∈GOE(n) +� n +� +i=1 +� +δiσ(i) − λQiσ(i) +�� += E +� � +i∈fix(σ) +� +1 − λξi +�� +· E +�s(σ) +� +k=1 +1 +2λ2γ2 +k +� += += +� +� +i∈fix(σ) +E +�� +1 − λξi +��� +· +�s(σ) +� +k=1 +E +�1 +2λ2γ2 +k +�� +, +where ξi ∼ N(0, 1) and γk ∼ N(0, 1). For these terms we have +E +�� +1 − λξi +�� += 1, +E +�1 +2λ2γ2 +k +� += 1 +2λ2E +� +γ2 +k +� += 1 +2λ2. +The contribution of such σ ∈ Sn in (45) is +sgn(σ) +E +Q∈GOE(n) +� n +� +i=1 +� +δiσ(i) − λQiσ(i) +�� += sgn(σ) +�1 +2λ2 +�s(σ) +, +(46) +and notice it depends only on s(σ). To conclude the computation we, therefore, have to count how +many permutations in Sn are given by exactly k disjoint transpositions for every k = 0, . . . , ⌊ n +2 ⌋. Denote +this number by N(k). To construct a permutation with exactly k disjoint transpositions we proceed +as follows: choose two elements in {1, . . . , n} forming the first transposition, then choose another 2 +among the remaining ones to form the second transposition and so on until the k–th one is formed. +Moreover, since the supports of the transpositions are disjoint, the order in which they are picked is +not relevant. It follows that +N(k) = +�n +2 +��n−2 +2 +� +. . . +�n−2k+2 +2 +� +k! += +n! +2k(n − 2k)! k! +and using this and (46) in (45) gives +E +Q∈GOE(n)[det(In − τQ)] = +⌊ n +2 ⌋ +� +k=0 +(−1)kN(k) +�1 +2λ2 +�k += +⌊ n +2 ⌋ +� +k=0 +(−1)kλ2k (2k)! +22kk! +� n +2k +� +. +(47) +Since the expectation of gj(Q) is given by the degree j term in (47) multiplied by (−1)j, we obtain +E +Q∈GOE(n) +� +gj(Q) +� += + + + + + + + +0 +if j odd +(−1) +j +2 +2j +j! +( j +2 )! +�n +j +� +if 0 ≤ j ≤ n, j even +. +(48) +Plugging (48) into (38), we finally get (39). +References +[AGH+14] +Animashree Anandkumar, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. Tensor decompo- +sitions for learning latent variable models. J. Mach. Learn. Res., 15:2773–2832, 2014. +[ALLF07] +Andrew Alexander, Jee Lee, Mariana Lazar, and Aaron Field. Diffusion tensor imaging of the brain. 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Corrected reprint of the 1989 original. + diff --git a/5dE5T4oBgHgl3EQfPA60/content/tmp_files/load_file.txt b/5dE5T4oBgHgl3EQfPA60/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..537d8db2b43b616b67c827b96cd1d5e5ed2860fb --- /dev/null +++ b/5dE5T4oBgHgl3EQfPA60/content/tmp_files/load_file.txt @@ -0,0 +1,1095 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf,len=1094 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='05502v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='AG] 13 Jan 2023 WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We address the general problem of estimating the probability that a real symmetric tensor is close to rank–one tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Using Weyl’s tube formula, we turn this question into a differential geometric one involving the study of metric invariants of the real Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' More precisely, we give an explicit formula for its reach and curvature coefficients with respect to the Bombieri–Weyl metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' These results are obtained using techniques from Random Matrix theory and an explicit description of the second fundamental form of the Veronese variety in terms of GOE matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Our findings give a complete solution to the original problem, and in the case of rational normal curves lead to some novel asymptotic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' What is the probability that a random symmetric tensor is close to rank-one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Over the last decades, symmetric tensors have been proven to be a very flexible and valuable tool in many different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In particular, rank–one approximation and tensor decomposition found applications in machine learning ([AGH+14]), signal processing and image analysis ([SDLF+17], [Sak16, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='3, 4]), chemistry ([SBG04]), statistics ([McC87]), psychology and medical diagnostics ([Kro08, ALLF07]) and phylogenetics ([Sak16, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='5], [Lan12]), to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Motivated by this, in this paper we address the following question: “What is the probability for a real symmetric tensor to be “close” to rank–one?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' To make sense of this question, we must endow the space of tensors with a notion of distance and with a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We address this problem in a natural way as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Observe first that the real vector space of symmetric tensors of order d on Rn+1 can be naturally identified with the space R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) of homogeneous polynomials of degree d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Under this identifi- cation, we endow the space of real polynomials with the scalar product given by the restriction of the real part of the Bombieri–Weyl hermitian product, defined on the space of complex polynomials by (1) ⟨p1, p2⟩BW := 1 πn+1 � Cn+1 p1(z)p2(z)e−∥z∥2dz, where dz := (i/2)n+1dz0dz0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' dzndzn is the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This defines the unique, up to multiples, hermitian product on the space of complex polynomials which is invariant under the action of the unitary group by change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The restriction of the real part of this hermitian product to the space of real polynomials will be still called the Bombieri–Weyl scalar product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' the above unitary invariance implies its invariance under the action of the orthogonal group by change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In the case when d = 2, the above identification is the familiar isomorphism between the space of symmetric matrices and the space of quadratic forms, and the Bombieri–Weyl scalar product coincides with the Frobenius inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Next, we use this scalar product to turn R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) into a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' For a Borel set U ⊆ R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) we define P(U) := � U e− ∥p∥2 BW 2 dµ � R[x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=',xn](d) e− ∥p∥2 BW 2 dµ , Date: September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1 2 ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA where “dµ” denotes the integration with respect to the Lebesgue measure on the space of coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We call the resulting probability distribution Bombieri–Weyl, and sometimes the nomenclature Kostlan is also used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' When d = 2, the Bombieri–Weyl distribution turns the space of symmetric matrices into a gaussian space, called the Gaussian Orthogonal Ensemble, as we will describe in more detail in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Finally, we identify the set of rank–one tensors with the Veronese variety Vn,d ⊂ R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) of signed d–th powers of linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' At this point we are in the position of giving a precise formulation to our question above, which therefore requires computing, for δ > 0 small enough, the quantity: P � p ∈ R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) ���� distBW(p, Vn,d) ≤ δ∥p∥BW � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Notice that we have turned this into a conic problem that takes into account also the norm of the tensor, as it is common procedure in numerical algebraic geometry [BC13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In this way, we can regard the above probability as the normalized volume of a tubular neighbourhood of the intersection of the set of rank–one tensors with the unit sphere in the Bombieri–Weyl norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Thus, our question becomes: “What is the volume of a neighbourhood of the spherical Veronese variety?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In this paper, exploiting Weyl’s Tube Formula, we derive an exact expression for the above volume, for small enough neighbourhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Moreover, as a byproduct of our computations, we give a lower bound on the size of the neighbourhood of the set of rank–one tensors that admit a unique best rank–one approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The properties of the Bombieri–Weyl distribution on the space of real (and complex) poly- nomials have been intensively studied, starting from the influential works of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Edelman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Kostlan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Shub and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Smale [EK95, SS93b, SS93a, SS93c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The point of view of random tensors has been adopted first by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Horobet and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Draisma in [DH16] and by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Breiding in [Bre19] for the study of the expected number of eigenvalues of a random symmetric tensor, with respect to the Bombieri–Weyl distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Under the identification between symmetric tensors and homogeneous polynomials, eigen- values correspond to critical values of the restriction of the polynomial to the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Eigenvectors correspond to critical points of the polynomial: under the Veronese embedding these critical points give rank–one tensors that are critical points of the distance function on the Veronese variety from the given tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Among these critical points (which are rank–one tensors) the closest to the original tensor are its best rank–one approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In [Bre19] the average number of such critical points is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In this paper we will instead give the size and estimate the probability of the set of tensors which admit a unique best rank–one approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The use of Weyl’s Tube Formula is fairly standard for results of this type [BC13, BL22]: it allows to deduce an exact expression, for ε > 0 small enough, of the volume of an ε–neighbourhood of a smooth submanifold W of the sphere, or the euclidean space, as a function of some differential–geometric quantities of W, called its curvature coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Our main contribution is the nontrivial computation of the curvature coefficients of the spherical Veronese variety and the explicit quantification of the above expression “for ε > 0 small enough” for this variety, through the computation of its reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' One could generalize this question to higher ranks by looking at secant varieties to the Veronese, whose geometry has been intensively studied, see [CGO14] for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We propose to investigate this in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We now describe the main ingredients and state the main results of our work in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The spherical Veronese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The main object we consider in this work is the real spherical Veronese variety Vn,d, which is the intersection of the Veronese variety in R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) ≃ RN+1 with the unit sphere for the Bombieri–Weyl norm: Vn,d := Vn,d ∩ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We regard this set as the image of the spherical Veronese embedding associated to the Bombieri– Weyl basis, or its double copy, depending on the parity of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This embedding is the smooth map WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 3 �νn,d : Sn → SN given by x �νn,d �−−−→ ��d α � 1 2 xα � α , where α ∈ Zn+1 ≥0 satisfy α0 + · · · + αn = d, �d α � is the multinomial coefficient, and Sn is the euclidean sphere in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Denoting �νn,d(Sn) by Σn,d, we see that Vn,d = Σn,d ∪ −Σn,d, where Σn,d = −Σn,d if d is odd and Σn,d ∩ (−Σn,d) = ∅ if d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' For this reason, we will call Σn,d the spherical Veronese surface, to distinguish it from the spherical Veronese variety Vn,d, in the case d is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In the projective picture, the difference between the two ceases to exist: PVn,d := P(Σn,d) = P(Vn,d) ⊂ RPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Recall that Σn,d parametrizes the d–th powers of norm–1 linear forms on Rn+1 and, therefore, rank– one and norm–one tensors up to signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Hence, the spherical Veronese surface Σn,d corresponds to an orbit for the action of O(n + 1) on homogenous polynomials by change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Even more is true: when turning Σn,d into a Riemannian manifold with the metric induced by the Bombieri–Weyl scalar product, the transitive action of O(n + 1) on Σn,d is through isometries induced by isometries of SN, given the invariance property of the Bombieri–Weyl structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The immediate, yet crucial, consequence is that the extrinsic geometry of the isometric embedding Σn,d ֒→ SN is exactly the same at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The same conclusion clearly holds for Vn,d ֒→ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Weyl’s tube formula and the reach of an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Let (M, g) be a Riemannian manifold and M ֒→ M be an isometric embedding of a compact smooth submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We can consider the set of points in M at distance less than a given ε > 0 from M and call such a set a tubular neighbourhood of M in M of radius ε, denoted as U(M, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' It is well known that for smooth compact embeddings M ֒→ M and small enough radii, the exponen- tial map on the normal bundle provides a smooth parametrization of the tubular neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This description is what really underlies the celebrated “Weyl’s tube formula” ([Wey39]), which constitutes one of the main tools to compute the volume of tubular neighbourhoods in a euclidean or spherical ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This formula expresses the volume as the linear combination Vol(U(M, ε)) = � 0≤e≤n, e even Ks+e(M)JN,s+e(ε), where N is the dimension of the ambient space, n is the dimension of M and s := N − n is the codimension of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The functions J’s do not depend on the specific submanifold M and are explicitly known in both the euclidean and spherical cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The most remarkable aspect of the formula is that the coefficients K’s are isometric invariants of the embedding and can be expressed in terms of curvature, from which they are named curvature coefficients of the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Remark that nowadays Weyl’s tube formula has been re-interpreted in the more general framework of “integral geometry”, which deals with integrals over a submanifold of polynomials in the entries of the second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Howard in [How93] showed how the above formula fits in this context and gave a full characterization of the polynomials appearing in Weyl’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In the case of the Veronese variety Vn,d ֒→ SN, the tubular neighbourhood U(Vn,d, ε) gives a description of the norm–1 symmetric tensors that are ε–close to a rank–1 tensor in the Bombieri–Weyl metric, in the ambient sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' As already pointed out, it follows that asking for the probability for a symmetric norm–1 tensor to be close to rank–1 boils down to computing the normalized volume of this tubular neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' For practical reasons, in the paper we will work with Σn,d instead of its “double” Vn,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' There are, however, two technical issues to consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The first one is that there will be a factor of 2 to be taken into account when switching from the Veronese surface Σn,d to the rank–one variety Vn,d, depending on the parity of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The second one is that the intersection of a δ–neighbourood of the set of rank–one tensors with the unit sphere becomes an ε–neighbourhood of Vn,d in the unit sphere, with ε = arcsin(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 4 ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA This is why we will use the parameter “ε” to formulate the results on the sphere and the parameter “δ” for the results in the vector space of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The reach of the Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Our aim is to exploit Weyl’s tube formula to compute the volume of U(Vn,d, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This requires first of all the knowledge of the radii for which the above expression holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Since the formula is based on the parametrization through the normal exponential map, the supremum of the radii for which this is a good parametrization, or at least a lower bound on that, is what we need to understand to meaningfully use Weyl’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This quantity is usually called the reach of the embedding M ֒→ M and in general computing it is a very difficult task, often unfeasible since it requires to study not only how normal geodesics originating from every point of the submanifold behave, but also how and when geodesics starting from different points cross each other, in order to avoid overlappings in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In our case, recalling the invariance property of Vn,d ֒→ SN under the action of the orthogonal group O(n + 1), we do not need to study normal geodesics originating from any point, but it is enough to choose a specific one and perform computations involving only geodesics originating from this chosen one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This drastically reduces the complexity of the computation, allowing us to obtain the following result, stated in a more detailed form in Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Theorem A (The reach of the spherical Veronese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The reach of the spherical Veronese variety Vn,d ֒→ SN is given by ρ(Vn,d) = 1 √ 3 + 1 3d √ 3 + O � 1 d2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The same result holds for the reach of the Veronese surface Σn,d ֒→ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Given the interpretation of the neighbourhood of the Veronese variety in terms of symmetric tensors already discussed, this theorem has an important consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' From its proof, it follows that every real symmetric tensor which is sufficiently close to rank–one tensors admits a unique best rank–one approximation (see Corollary 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In fact the reach ρ(Vn,d) equals the minimum between two quantities, one of which estimates the size of the neighborhood of Vn,d on which the normal exponential map is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' we prove that this quantity equals π 4 , and this allows to deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Corollary B (Best rank–one approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Every symmetric tensor p at distance less than √ 2 2 ∥p∥BW from rank–one admits a unique best rank–one approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The normalized volume of a neighbourhood of the Veronese variety of radius π 4 would therefore provide a lower bound for the probability that such tensors have a unique best rank–one approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Unfortunately, we are not able to compute such a volume, given that the value of the reach ρ(Vn,d) < π 4 does not allow to use Weyl’s tube formula up to such a radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Nevertheless, given the uniformity of the lower bound for the reach ρ(Vn,d) ≥ 1 √ 3 for every n, d (see Remark 20), we still get that the volume of the neighbourhood of radius 1 √ 3 provides a lower bound for that probability, even if not sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This bound can be explicitly computed by plugging in ε = 1 √ 3 in Theorem 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Moreover, in the case of tensors in two variables, which correspond to the case of rational normal curves, using the asymptotic in Theorem 26 we get an asymptotic expression for this bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The curvature coefficients of the Veronese variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The other ingredient needed in Weyl’s formula are the curvature properties of the embedding, in particular the Weingarten operator along normal directions, which encodes the second fundamental form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Again by the invariance of the extrinsic geometry of Vn,d ֒→ SN, it is enough to compute this at a specific point, which we choose to be xd 0 for simplicity of computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Before stating our result, recall that we have denoted by GOE(n) the Gaussian Orthogonal Ensemble, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' the set Sym(n, R) endowed with the gaussian probability distribution coming from the Frobenius scalar product, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Our main result on the extrinsic geometry of the embedding Vn,d ֒→ SN is the following, and we refer to Theorem 22 for a more comprehensive statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' WHAT IS THE PROBABILITY THAT A RANDOM SYMMETRIC TENSOR IS CLOSE TO RANK-ONE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 5 Theorem C (The normal bundle splitting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Let p ∈ Vn,d and denote by Lη the Weingarten operator of Vn,d ֒→ SN along a normal direction η ∈ NpVn,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' There exists an orthogonal decomposition NpVn,d = W ⊕ P such that the following statements hold: (1) Lη = 0 for every η ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' (2) W with its induced Bombieri–Weyl metric is isometric to Sym(n) with the Frobenius one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Moreover, if we pick η ∈ W Gaussian, then Lη ∼ √ 2 � d−1 d � 1 2 GOE(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This theorem could find applications going beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' It gives a full description of the second fundamental form in terms of GOE(n) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Using this description in Weyl’s tube formula to compute the curvature coefficients of the spherical Veronese, the consequence is that the computation of some integrals on the normal bundle depending on the Weingarten operator boils down to computing the expectation of a determinant involving GOE(n) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' This is reduced to an easy, purely combinatorial computation (see Appendix C) and thus we obtain the following explicit expressions for the curvature coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Theorem D (The curvature coefficients of the spherical Veronese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The curvature coefficients of the Veronese variety Vn,d ֒→ SN are as follows: KN−n+j(Vn,d) = (−1) j 2 d n 2 �d − 1 d � j 2 2n+2−jπ N 2 Γ � n 2 + 1 � Γ � j 2 + 1 � Γ(n + 1 − j)Γ � N+j−n 2 � for 0 ≤ j ≤ n and j even, and KN−n+j(Vn,d) = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We remark that similar results hold true for the projective Veronese variety, using the double covering SN −→ RPN, and for the spherical Veronese surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Plugging these coefficients back in Weyl’s tube formula we also obtain the explicit expression of the volume of the tubular neighbourhood for radii smaller than the reach (see Theorem 24), in particular giving an answer to question stated at the beginning of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Theorem E (The probability of being close to rank–one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Let p be a random Bombieri–Weyl symmet- ric tensor of order d on Rn+1 and Vn,d ⊂ R[x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' , xn](d) ≃ RN+1 be the Veronese variety of rank–one tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' For every δ such that 0 ≤ arcsin(δ) < 1 √ 3 + 1 3d √ 3 + O � 1 d2 � we have P � distBW(p, Vn,d) ≤ δ∥p∥BW � = � 0≤j≤n j even (−1) j 2 d n 2 �d − 1 d � j 2 2n−j+1π− 1 2 Γ � n 2 + 1 � Γ � N+1 2 � Γ � j 2 + 1 � Γ(n + 1 − j)Γ � N+j−n 2 � � δ √ 1−δ2 0 tN−n+j−1 (1 + t2) N+1 2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We remark that the above expression, even if unpleasant, gives an exact formula for our probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In the last section, we present an asymptotic expression, based on Laplace’s method, for such a probability in the case of rational normal curves, corresponding to the case of tensors in two variables, when the degree goes to infinity (see Theorem 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We stress that it is possible to obtain a meaningful asymptotic since the reach is uniformly bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' We also remark that the decay is exponential in d, as one might expect looking at the codimension of V1,d ֒→ Sd, which is d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' More generally, the probability has an exponential decay in the codimension of Vn,d ֒→ SN for any n, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' The Gaussian Orthogonal Ensemble and the Bombieri–Weyl distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' In this sec- tion, we point out the correspondence between the Gaussian Orthogonal Ensemble on the space of symmetric matrices and the Bombieri–Weyl distribution on the space of homogeneous polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' 6 ALBERTO CAZZANIGA, ANTONIO LERARIO, ANDREA ROSANA Let Sym(n, R) be the space of symmetric n × n matrices with real entries and denote by Eij the elementary matrix having all entries 0 except for the ij-th one being 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE5T4oBgHgl3EQfPA60/content/2301.05502v1.pdf'} +page_content=' Consider a random matrix Q = n � i=1 ηiiEii + n � i Fth which are again in good +agreement with the measured durations of a high reflectivity stage corresponding to molten +germanium [10] (Fig. 1b). It should be mentioned that different values are reported for the +thermal conductivity of liquid Ge. In modeling, we use the value of 29.7 W/(m·K) [43] while +in Ref. [44], κ = 43 W/(m·K) was measured. The calculations performed for various pulse +durations demonstrate now a stronger τL dependence than that for silicon, close to the  τL0.5 +dependence (Table 1). +3.3. Gallium Arsenide +For simulations of laser heating of GaAs, we used the same values of the thermophysical +properties as in Ref. [8]. For the temperature dependence of cp, the data from [45] were used, +which are also in a good agreement with the data reported in [46]. Optical properties were taken +from measurements [47], which are also in a good agreement with [48]. The absorption and +reflection coefficients were calculated from the refractive index and the extinction coefficient +and taken as temperature independent. The material properties used in the simulations are +presented in Tables A9–A12 of Appendix. +Several regimes of laser irradiation of GaAs corresponding to available experimental +and theoretical data were investigated in our modelling with the laser wavelength ranging from +308 to 694 nm and pulse duration ranging from 15 to 70 ns. For all the conditions, the melting +thresholds calculated here with our unified model are in good agreement with the values +reported in the literature (Table 1). Below we discuss each irradiation regime in more details. + +λ = 308 nm, τ = 30 ns. Our model implements the same optical and temperature- +dependent material properties as in the model presented by Kim et al. [8]. For the solid state +reflectivity and the optical absorption, the data used for simulations in [8] are in agreement with +the measured data for solid GaAs [46]. As the parameters of the model [8] and ours are very +similar, we take this comparison as a validation for our model that gives a deviation of only +~6% (see Table 1). + λ = 694 nm, τ = 15 ns. García et al. [12] carried out simulations for a ruby laser with a +15 ns pulse duration using an explicit numerical scheme. The melting threshold was identified +at a laser fluence of 300 mJ/cm2 that corresponded to the situation when a ~65-nm-thick surface +layer was molten [12]. In our simulations, this fluence of 300 mJ/cm2 results in the melting +depth of 13 nm while the melting threshold corresponding to reaching the melting point on the +sample surface is 265 mJ/cm2 (see Fig. 2 for comparison). The difference in the melting depth +can be attributed to two factors. First, the authors [12] extrapolated the temperature-dependent +absorption coefficient for the solid state GaAs from the room temperature till the melting point +that appears to be questionable. In our simulations, we use constant but reliable data on optical +absorption and reflectivity of molten GaAs at the wavelength of the ruby laser [41]. The +reflectivity coefficient of liquid GaAs in both Ref. [12] and this work was adopted from [13], +R = 0.67. The second factor may be related to the using an explicit numerical scheme whose +approximation to the initial equations often represent a challenge. +λ = 694 nm, τ = 20 ns. Pospieszczyk et al. [13] presented two sets of measurements. +Using a HeNe probe laser, the temperature-dependent reflectivity was investigated. The second +set of the data gives time-of-flight measurements of particles evaporated from the GaAs surface +(Fig. 3a). Comparison of their experimental data and our simulations is given in Table 1, which +are in a reasonable agreement. The simulated damage threshold associated with achieving the +melting temperature (Fig. 3b) is somewhat higher than in the experiments [13] but is still in the +range of fluences where a transient uneven melting is observed (Fig. 3). This discrepancy, +although relatively small, can be related to the effect of decreasing the melting temperature due +to depletion of the target surface by a more volatile component [23,24,49] that is not taken into +account in our model. +3.4. Cadmium Telluride +We have applied our model to CdTe irradiated by a KrF excimer laser (248 nm) for the +conditions of Gnatyuk et al. [30] where TRR measurements and numerical simulations of +pulsed laser heating of CdTe were performed. For this material, reliable physical and optical +properties are extensively reported in the literature. In our simulations, the value of thermal +conductivity was taken from Ref. [50] and [51] for solid and liquid state, respectively. The +specific heat for both solid and liquid state was taken from Ref. [52]. The same thermophysical +properties were also used in simulations [24,30]. Measurements of optical properties of CdTe +using spectroscopic ellipsometry and modeling were performed in [53] for a wide range of +wavelengths. Reflectivity and absorption are the same for solid and liquid state and independent +of temperature. +The authors [30] identified a laser fluence of 50 mJ/cm2 as the melting threshold. In +their simulations, this value corresponds to the molten layer with a thickness of the laser +absorption depth. Their TRR measurements detected an abrupt although small rise of the +reflectivity at a laser fluence of 48-50 mJ/cm2. These results are in excellent agreement with + +our simulations (Table 1). Indeed, for F = 50 mJ/cm2, our model gives the depth of the molten +layer of 7 nm, very close to the absorption depth of CdTe at 248 nm (~9nm). According to our +definition of the melting threshold, achieving the melting temperature at the very surface of the +irradiated sample, the calculated threshold is slightly lower, 46 mJ/cm2 (Table 1). +We would also note that in Refs. [25,49], an effect of enhanced evaporation of Cd atoms +with enriching the surface by tellurium upon laser heating was studies. It was shown that this +effect can have an impact on the melting and ablation processes. This effect was not taken into +account in this work, nor in [30]. +3.5. Indium Phosphide +We have applied our model for the conditions of experiments [23] where InP was +irradiated by a nanosecond laser at λ = 532 nm. In this paper, a laser fluence of 97 mJ/cm2 was +identified as the damage threshold. In our simulations, we have obtained a threshold value of +106 mJ/cm2 which can be considered as a good agreement taking into account that there is no +any fitting parameters in our model. It should be mentioned that, although laser processing of +InP is a common technique in its industrial applications, the thermophysical parameters at +enhanced temperatures are still not well studied. Thus, several sets of data are available for the +heat capacity of solid InP, see e.g. [54]. The major problem is that measurements of the +thermophysical properties at enhanced temperatures are affected by a high vapor pressure of +phosphorous due to its high volatility. The thermal conductivity and the specific heat of molten +InP are given in [46]. The reflectivity and absorption are calculated form data provided in [41]. +Optical properties are taken as temperature independent and considered the same for both solid +and liquid state. In reference article [23], ablation of compound semiconductors is studied and +a model that takes into account evaporation of their components gives the melting threshold. +Our result, that disregards this effect, gives Fth that is about 10% higher. +3.6. Generalization of the damage threshold data into a predictive dependence +A wide set of data on the damage thresholds of five semiconductors under various ns-laser +irradiation conditions are obtained in our calculations in the frames of a unified thermal model +and all the thresholds are in good agreement with available literature data, both experimental +and theoretical ones. The obtained threshold values vary in a wide range depending on material, +from ~ 50 mJ/cm2 for CdTe to almost 1 J/cm2 for Si (Table 1). The irradiation conditions also +affect the threshold values which are generally smaller for shorter laser wavelengths and pulse +durations. It is very attractive to generalize the obtained results in terms of a unified parameter +combining the basic material properties (thermophysical and optical) in order to be able to +predict the ns-laser-induced melting thresholds, at least approximately, without performing +detailed simulations. + +D. Bäuerle [22] considered “optimal” melting conditions during ns-laser-induced +thermal surface melting, when minimal laser energy is required for a certain melt depth. +Assuming that such conditions are fulfilled when the melt depth is equal to the heat-diffusion +length, he estimated the optimal laser fluence as +𝑃B = +2𝜌∆𝐻 +1−𝑅 ( +𝐷 +𝜏L) +1 +2 𝜏L (8) + +where D = κ/cp is the thermal diffusivity and H = Lm + cp(Tm-300) is the total energy needed +to heat the sample to the complete melting state from room temperature, A similar parameter +was introduced in [39] as an evaporation threshold under ns-laser ablation (assuming naturally +by H in Eq. (8) the specific heat for evaporation instead of that for melting and omitting the +2/(1-R) factor). +Figure 4 shows the calculated melting threshold values plotted as a function of the PB +parameter, Eq. (8), evaluated for all the studied materials using their room-temperature +properties. All the data are nicely groupped around a streight line in the logarithmic plot. This +clear correlation is rather surprising for such a simplified generalization approach when the +material absorption coefficient and temperature dependencies of thermophysical properties are +not taken into account. The least square fitting line in Fig. 4 is described by a power law Fth ≈ +0.05PB1.16 whcih can be used for rough estimation of the melting threshold of semiconductors +based on their basic room-temperatureproperties. +The parameter PB predicts a growth of the melting threshold with the laser pulse duration +as τL0.5. However, as was noticed above, this is not always the case according to our simulations. +Some semiconductors (Ge, CdTe, InP) follow closely the τL0.5 dependence while others (Si, +GaAs) demonstrate weaker dependencies (Table 1 and Fig. 4). This is probably mainly due to +a difference in the thermal diffusivity D of the materials. Thus, at room temperature, D ≈ 0.8 +cm2/s for Si and it is around 0.35 cm2/s for Ge, InP and CdTe. A higher thermal diffusivity +results in a higher heat diffusion length and smaller in-depth temperature gradients and thus in +a lower heat flow from the surface at an increased pulse duration. The temperature dependencies +of materials parameters (included to our model simulations) can additionally affect the pulse +duration dependence of the melting threshold. +4. Conclusions +In this work, based on the classical thermal model, we have developed a numerical approach +to investigate the continuous solid-liquid phase change in solid targets heated by nanosecond +laser pulses. The model is applied to a number of semiconductors and various irradiation +conditions and the obtained results on the melting thresholds, melt duration and melt depth are +compared with experimental and theoretical data available in the literature. The comparison is +not always straightforward as the value presented as melting threshold fluence is not always +describing the same state of the studied material. However, in most cases, good agreement with +the literature data is obtained. The simulations predict also the dependence of the melting +thresholds on the laser pulse duration which is found to be material dependent and weaker than +that expected from simple heat-flow considerations. A good correlation of all the calculated +melting threshold values with a parameter combining material thermophysical properties and +surface reflectivity is obtained. The correlation can be used as a simple method for estimation +of the melting thresholds of ns-laser irradiated semiconductors based on their room-temperature +properties. +Acknowledgements +This work was supported by the European Regional Development Fund and the state budget of +the Czech Republic (project BIATRI: No. CZ.02.1.01/0.0/0.0/15_003/0000445). J. B. +acknowledges funding of the Grant Agency of the Czech Technical University in Prague No. +SGS22/182/OHK4/3T/14. + +Declarations +Conflict of interest. The authors declare no conflict of interests. +Appendix +Here we provide all the parameters for semiconductors, which were selected after a thorough +literature analysis and used in our modeling. Some reliable data, which are widely cited in +literature and web-sites, are given without references. +Silicon +Table A1. c-Si – thermophysical properties +Property +Value +Ref. +ρ,g/cm3 +2.328 + +Tm, K +1688 + +Lm, J/kg +1.826 × 106 +[55] +cp, J/kg K +847.05 + 118.1 × 10-3 T – 155.6 × 105 T -2 +[35] +κ, W/mK +97269 T -1.165 (300 0, λ : A → At +est une Z× +(p)-polarisation, ι : OF → End(A) une action d’un anneau d’entiers d’un +corps de nombre totalement réel F/Q sur A tel que [F : Q] = d, et κ une structure +de niveau en dehors de p. Lorsque l’extension F/Q est ramifiée, le modèle ShPEL n’est +plus lisse sur Spec Zp. Pappas et Rapoport ont défini dans [PR02] un modèle lisse +ShPR sur Spec OK où K est une extension contenant les clôtures galoisiennes de toutes +1 + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +2 +les extensions Fv où v|p. Ce modèle est défini via une résolution des singularités du +modèle local. C’est un espace de module classifiant les 5-uplés (A, λ, ι, κ, Fil(ω)) où +(A, λ, ι, κ) est un quadruplé de ShPEL et Fil(ω) est une filtration du faisceau conormal +satisfaisant certaines propriétés (voir 5.8 pour plus de détails). L’un des objectifs dans +l’étude de la géométrie de la réduction modulo p des variétés de Shimura est de définir +des stratifications, dont les strates possèdent de bonnes propriétés. On peut trouver +dans la littérature de nombreux travaux sur les stratifications de la réduction modulo +p du modèle ShPEL (voir par exemple [VW13]). Concernant le modèle ShPR plusieurs +stratifications ont été définies dans [DK22], [RX14]. Dans cet article nous allons nous +intéresser à la stratification de Hodge de ShPR. Celle-ci est définie via le polygone de +Hodge du faisceau conormal. +1.2. Principaux résultats. Pour ne pas alourdir les notations nous allons énoncer +les principaux résultats de cet article dans le cadre d’une extension L/Qp totalement +ramifiée (l’extension L/Qp jouant le rôle de Fv/Qp). Mis à part le Théorème 1.3, les +résultats qui suivent sont valables et sont démontrés dans le cadre générale d’une ex- +tension totalement réelle F/Q sans condition sur la ramification en p. +Soit L/Qp une extension totalement ramifiée de degré e. On note ShPEL la fibre spé- +ciale du modèle de Kottwitz et ShPR celle du modèle de Pappas-Rapoport des variétés +modulaires de Hilbert (voir les Définitions 5.1, 5.3.4). L’oubli de la filtration induit un +morphisme : +π : ShPR −→ ShPEL +On dispose d’une stratification appelée stratification de Kottwitz-Rapoport (KR) du +modèle PEL : +ShPEL = +� +λ∈Adm(µ)K +ShPEL +λ +où Adm(µ)K désigne l’ensemble µ-admissible (voir la Remarque 4.5). Dans le cas Hilbert +la stratification (KR) de ShPEL coïncide avec la stratification par le polygone de Hodge +(voir Remarque 6.1). Il est naturel de se demander si cette stratification induit une +(bonne) stratification (voir Définition 2.8) de ShPR. On prouve dans cet article que la +réponse est oui : +Théorème 1.1. +(1) Les strates (ShPR +λ )λ∈Adm(µ)K forment une bonne stratification de ShPR. Autre- +ment dit pour tout λ ∈ Adm(µ)K on a la relation d’adhérence +ShPR +λ += +� +λ′≤λ +ShPR +λ′ +(2) Pour tout λ ∈ Adm(µ)K la strate ShPR +λ +est quasi-projective lisse de dimension +⟨ρ, |µ•| + λ⟩. +Ce théorème est surprenant car il est faux dans des cas plus généraux que le cas +Hilbert (voir [BH22b]). La preuve du point (2) repose sur les travaux de Haines ([Hai06]) +sur la géométrie du morphisme de convolution dans le cadre des Grassmanniennes + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +3 +affines. Ses travaux nous permettent également de prouver le théorème suivant qui +concerne la géométrie du morphisme π : +Théoreme 1.2. Pour tout λ ∈ Adm(µ)K la restriction +π : ShPR +λ +→ ShPEL +λ +est un morphisme plat. +Les travaux de Haines n’ayant pas de restriction sur le groupe réductif, le théorème +ci-dessus est en fait valable dans le cas Hilbert-Siegel plus général. +Dans [RX14] les auteurs ont défini des invariants de Hasse partiels (mi)1≤i≤e et dans +[DK22] il est prouvé que les sous schémas localement fermés définis comme les lieux +d’annulation de ces sections définissent une bonne stratification de ShPR : +ShPR = +� +T⊂{1,...,e} +ShPR +T +où ShPR +T +est le sous schéma localement fermé défini par l’annulation des (mi)i∈T et +l’inversibilité des (mi)i/∈T. Il est alors naturel d’étudier l’interaction entre la stratification +par le polygone de Hodge, et celle définit par ces invariants de Hasse partiels. Pour tout +λ ∈ Adm(µ) et tout T ⊂ {1, . . ., e} on note ShPR +(λ,T) l’intersection de la strate ShPR +λ +et +de la strate ShPR +T . On montre le résultat suivant dans le cas e = 4 : +Théoreme 1.3. Pour L/Qp totalement ramifiée de degré e = 4 la stratification : +ShPR = +� +(λ,T)∈Aµ• +ShPR +(λ,T) +est une bonne stratification où les relations d’adhérences sont données par la relation +d’ordre naïve sur Aµ•. +Dans le théorème ci-dessus l’ensemble Aµ• désigne le sous ensemble des couples (λ, T) +tels que la strate ShPR +(λ,T) soit non vide. Il semble difficile de calculer ce sous ensemble +en général. +Remerciements. Je tiens à remercier Stéphane Bijakowski de m’avoir encouragé à +écrire cet article et de m’avoir expliqué les résultats de déformations de la première +section. Je tiens également à remercier Benoit Stroh, Thibault Alexandre et Arnaud +Eteve pour toutes les discussions qui m’ont aidé à écrire cet article. Enfin je tiens à +remercier Xu Shen pour ses commentaires et remarques. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +4 +2. Préliminaires +Dans tout ce qui suit on fixe un nombre premier p > 0. +Cette première partie est composée essentiellement de rappels sur la théorie de Dieu- +donné cristalline. Nous présentons tous les outils nécessaires pour pouvoir déformer un +groupe p-divisible le long d’un morphisme k[[t]] → k où k est un corps de caractéristique +p. Nous aurons besoin de la théorie de Dieudonné cristalline [BBM82], de la théorie des +display de Zink et Lau [Zin02], [Lau09], [Lau14] et de Grothendieck-Messing [Mes72]. +En fait, nous pourrions nous contenter de la théorie des display car elle englobe les +anneaux locaux complet de corps résiduels parfait comme kperf[[t]] (où kperf désigne la +perfection de k), ce qui nous suffit amplement pour nos problèmes de déformations. +2.1. Théorie de Dieudonné cristalline. +2.1.1. +Soit k un corps parfait de caractéristique p et S un schéma sur k. Soit A → S +un schéma abélien. On note G = A[p∞] le groupe p-divisible associé. On note Σ = +Spec (W(k)), et G le faisceau sur le site cristallin Cris(S/Σ) induit par G. En suivant +les notations de [BBM82] on note : +E (G) := E xt1 +S/Σ(G, OS/Σ) +le cristal de Dieudonné contravariant de G. C’est un OS/Σ-cristal localement libre de +rang h où h désigne la hauteur de G ([BBM82], Corollaire 1.4.7). Notez qu’avec cette +convention, les applications : +F : E (G)(p) → E (G), +V : E (G) → E (G)(p) +(où ( · )(p) désigne le twist par le Frobenius) sont induites respectivement par : +F : G → G(p), +V : G(p) → G +Si l’on évalue ce cristal sur l’épaississement (S +id +−→ S) on obtient une filtration de OS- +modules localement libres ([BBM82], Corollaire 3.3.5) +0 −→ ωG −→ E (G) +(S +id−→S) −→ ω∨ +GD −→ 0 +(2.1) +appelée filtration de Hodge. D’après [BBM82] Proposition 3.3.7 on dispose d’un iso- +morphisme +E xt1 +S/Σ(A, OS/Σ) ≃ E xt1 +S/Σ(G, OS/Σ) +reliant le cristal de A/S et celui de son groupe p-divisible, compatible aux filtrations de +Hodge respectives. En combinant maintenant avec l’isomorphisme ([BBM82], Proposi- +tion 2.5.8) : +E xt1 +S/Σ(A, OS/Σ)(S→S) ≃ H1 +dR(A/S) +on retrouve la filtration de Hodge induit par la suite spectrale de Hodge bien connue : +0 −→ ωA −→ H1 +dR(A/S) −→ ω∨ +At −→ 0 + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +5 +2.1.2. +Plaçons nous maintenant dans le cas où S = Spec(k) est le spectre d’un corps +k parfait de caractéristique p. On note : +D(G) := HomS(G, CW) +le module de Dieudonné contravariant au sens de Fontaine ([Fon77] ou [BBM82] Section +4.2). En évaluant notre cristal le long de l’épaississement (W(k) ։ k) on obtient un +isomorphisme de W(k) module compatible avec F et V des deux cotés : +E (G)(W (k)։k) ≃ D(G)(p) +(voir [BBM82] Théorème 4.2.14). La réduction modulo p de cet isomorphisme permet +l’identification de la p-torsion : +E xt1 +S/Σ(G[p], OS/Σ) ≃ D(G[p])(p) ≃ (D(G)/pD(G))(p) +où E xt1 +S/Σ(G[p], OS/Σ) est le cristal associé au groupe fini plat de p-torsion G[p]. Par +définition E (G) étant un cristal, il commute aux changement de bases et par conséquent +la réduction modulo p ci dessus redonne l’identification bien connue entre le module de +Dieudonné de la p-torsion et le premier groupe de cohomologie de De Rham : +E (G)(W (k)։k) ⊗W (k) k ≃ E (G)(k→k) ≃ H1 +dR(A/k) +Toujours d’après [BBM82] (Proposition 4.3.10) on dispose d’un isomorphisme : +ω(p) +G ≃ E (G)(S→S)/F(E (G)(p) +(S→S)) +2.1.3. +On s’intéresse au cas où A/S est muni d’une action de OL où L/Qp est une +extension de degré fini. D’après [Far06] on dispose du résultat suivant : +Proposition 2.1 ([Far06], Lemme B.1). Soit E un F-cristal en OS/Σ-modules loca- +lement libre de rang fini sur Cris(S/Σ) muni d’une action de OL. Alors E est un +OS/Σ ⊗Zp OL-module localement libre sur Cris(S/Σ). +Remarque 2.2. D’après [BBM82] on dispose d’un isomorphisme : +ωG ≃ E xt1(G, JS/Σ)(S→S) +où JS/Σ est le faisceau d’idéal à puissances divisées. Or E xt1(G, JS/Σ) n’est à priori pas +un cristal et donc la proposition ci-dessus ne s’applique pas. +2.1.4. +Les morphismes F et V de G induisent un diagramme commutatif aux lignes +horizontales exactes : +0 +ωG +E (G)(S→S) +ω∨ +GD +0 +0 +ω(p) +G +E (G)(p) +(S→S) +(ω∨ +GD)(p) +0 +0 +ωG +E (G)(S→S) +ω∨ +GD +0 +V +V +V +F +F +F + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +6 +Les composées verticales sont nulles car égales à p. Puisque F est nul sur ω(p) +G le mor- +phisme F se factorise en : +F : (ω∨ +GD)(p) −→ E (G)(S→S) +De même V est nul sur ω∨ +GD (utiliser que VG = F D +GD) donc on obtient une factorisation : +V : E (G)(S→S) → ω(p) +G +Nous aurons besoin de cette factorisation pour définir l’invariant de Hasse primitif dans +la section 7.1.2. +2.2. Déformations. +2.2.1. Grothendieck-Messing. Nous allons maintenant rappeler quelques résultats sur +les déformations des groupes de Barsotti-Tate. +Soit S0 ֒→ S une immersion fermée nilpotente munie d’une structure de puissances +divisées (PD-structure), avec p localement nilpotent. On dispose d’un morphisme sur +le site Cris(S0/Σ) : +(S0 → S0, γ0) −→ (S0 ֒→ S, γ) +Si F est un cristal en OS0/Σ-module alors par définition on dispose d’un isomorphisme +de OS0-modules canonique : +F(S0֒→S) ⊗OS OS0 ≃ F(S0→S0) +On note BT(S) la catégorie des groupes de Barsotti-Tate sur S. Soit G ∈ BT(S) un tel +groupe. On note E (G) le cristal associé et G|S0 = G ×S S0 le changement de base. On +a alors un isomorphisme canonique de OS-module : +E (G)(S→S) ≃ E (G|S0)(S0֒→S) +En effet d’un coté on dispose d’un isomorphisme : +E (G|S0)(S0֒→S) ≃ E (G)(S0֒→S) +provenant essentiellement de la définition du cristal E (G) ([BM79], (2.3)). De l’autre +coté puisque E (G) est un cristal, le morphisme (S0 → S) → (S → S) dans Cris(S/Σ) +induit un isomorphisme : +E (G)(S0֒→S) ⊗OS OS = E (G)(S→S) +et le résultat en découle. Cet isomorphisme est compatible aux filtrations de Hodge : +ωG +E (G)(S→S) +E (G|S0)(S0֒→S) +ωG|S0 +E (G|S0)(S0→S0) +E (G|S0)(S0֒→S) ⊗OS OS0 +≃ +≃ +dans le sens où le diagramme est commutatif et la ligne du bas correspond à la réduction +de la ligne du haut. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +7 +Définition 2.3. Soit G0 ∈ BT(S0) un groupe de Barsotti-Tate sur S0. Une filtration +Fil1 ⊂ E (G0)(S0֒→S) est dite admissible si c’est un OS-module localement facteur direct +de E (G0)(S0֒→S) qui relève ωG0 ⊂ E (G0)(S0→S0). +D’après ce qui précède si G ∈ BT(S) relève G0 ∈ BT(S0) alors ωG ⊂ E (G)(S→S) ≃ +E (G|S0)(S0֒→S) est une filtration admissible. On note DefBT(S0 ֒→ S) la catégorie dont +les objets sont les couples (G0, Fil1) où G0 ∈ BT(S0), Fil1 ⊂ E (G0)(S0֒→S) est une +filtration admissible et où les morphismes (G0, Fil1) → (G′ +0, Fil1′) sont les morphismes +G0 → G′ +0 compatibles avec les filtrations respectives. On peut maintenant énoncer le +théorème de Grothendieck-Messing : +Théoreme 2.4 ([Mes72] V.1.6). Le foncteur +BT(S) +−→ +DefBT(S0 ֒→ S) +G +�−→ +� +G|S0, ωG ⊂ E (G|S0)(S0֒→S) +� +est une équivalence de catégorie. +2.2.2. Applications. Dans cette section nous allons voir comment appliquer le théorème +de déformation de Grothendieck-Messing le long de l’immersion fermée k[[t]] → k, qui +n’est pas munie de PD-structure. Cela nous sera utile lorsque nous voudrons calculer +des relations d’adhérences entres strates de nos variétés de Shimura (voir Proposition +2.9). +Soit R un anneaux de caractéristique p et I un idéal tel que I2 = 0. On peut munir +I d’une PD-structure faisant du morphisme Spec R/I → Spec R une immersion fermée +nilpotente avec PD-structure. En effet il suffit de poser γ1(x) = x et γn(x) = 0 pour +tout n ≥ 2 et tout x ∈ I. Nous allons appliquer cette remarque à la suite d’immersions +fermées : +· · · → k[t]/(tn+1) → k[t]/(tn) → k[t]/(tn−1) → · · · → k[t]/(t2) → k +Pour tout n ≥ 1 on note Rn = k[t]/(tn), In = Ker +� +k[t]/(tn) → k[t]/(tn−1) +� +et Sn = +Spec Rn. Chaque immersion fermée Sn−1 ֒→ Sn est définie par un idéal In = (tn−1)/(tn) +satisfaisant I2 +n = 0. D’après ce qui précède on peut donc munir chacune de ces immer- +sions fermées d’une PD-structure. Le résultat suivant fonctionne pour tout anneau R +local complet de corps résiduel parfait. +Proposition 2.5 ([Lau09], Lemme 2.10). Soit G1 ∈ BT(k) et Fil1 ⊂ E (G1)(k→k) ⊗k k[[t]] +un relèvement de ωG1 ⊂ E (G1)(k→k) qui est localement un facteur direct. Alors il existe +G ∈ BT(k[[t]]) tel que : +(1) G est un relèvement de G1 le long de Spec k → Spec k[[t]] +(2) E (G)(k[[t]]→k[[t]]) ≃ E (G)(k→k) ⊗k k[[t]] +(3) ωG ≃ Fil1 (via l’identification ci dessus) +Démonstration. Nous allons utiliser la théorie des display de Zink et Lau. Posons R = +k[[t]] et S = Spec R. Notez que se donner un groupe p-divisible G ∈ BT(S) c’est se +donner un système compatible de groupes p-divisibles (Gn)n≥1 où Gn ∈ BT(Sn) ([Lau14] + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +8 +Lemme 2.15). Nous allons donc construire pour tout n ≥ 2 un groupe p-divisible Gn ∈ +BT(Sn) satisfaisant les propriétés souhaitées. Puisque pour tout n ≥ 2 l’anneau Rn +est un anneau local artinien sur lequel p est nilpotent et de corps résiduel parfait, +on dispose donc d’une équivalence de catégorie BT(Sn) ∼= Disp(Rn). On note P1 = +(P1, Q1, F1, V −1 +1 +) le display associé à G1 par Lau ([Lau14], Théorème A). Toujours +d’après loc cit, Théorème A, on dispose d’un isomorphisme canonique entre le cristal +de Dieudonné de G1 et le cristal associé à P1 par Zink ([Zin02] ou [Lau14]) : +E (G1) ≃ D(P1) +Le cristal D(P1) étant défini sur le site Crisadm(R) ⊂ Cris(R) des épaississements avec +pd-structure (B → A, δ) avec A admissible (i.e. si le nilradical NR est « bounded nil- +potent » et que Rred = R/NR est un anneau parfait de caractéristique p), l’isomorphisme +ci-dessus est un isomorphisme de cristaux sur le site Crisadm(R) et on fait ici l’abus de +notation d’également noter E (G1) la restriction du cristal E (G1) au site Crisadm(R). +En particulier en évaluant sur l’épaississement tautologique (S1 → S1) on obtient un +isomorphisme canonique de R1-modules : +E (G1)(S1→S1) ≃ P1 ⊗W(R1) R1 +(2.2) +D’après la remarque précédente, R2 → R1 est muni d’une PD-structure nilpotente, +et par conséquent se donner un relèvement P2 ∈ Disp(R2) de P1 c’est se donner un +relèvement Fil1 +(2) ⊂ D(P1)(R2։R1) (voir [Lau14], Corollaire 2.10, ou [Lau09] Lemme 4.2). +On pose P2 = P1 ⊗W(R1) W(R2) (notez qu’on dispose pour tout n ≥ 2 d’un morphisme +R1 → Rn). D’après [Lau14], Section 2.6, on a par construction du cristal D(P1) un +isomorphisme : +D(P1)(R2։R1) ≃ P2 ⊗W(R2) R2 = P1 ⊗W(R1) R2 +Via l’isomorphisme (2.2) on obtient : +E (G1)(S1→S1) ⊗R1 R2 ≃ D(P1)(R2։R1) +On définit le relèvement de la filtration de Hodge de P1 comme étant : +Fil1 +(2) := Fil1 ⊗R R2 ⊂ E (G1)(S1→S1) ⊗R1 R2 ≃ D(P1)(R2։R1) +Ce relèvement définit une display P2 ∈ Disp(R2) et donc un groupe p-divisible G2 ∈ +BT(S2). On dispose de nouveau d’une identification entre cristaux E (G2) ≃ D(P2). +Puisque G2 est un relèvement de G1 le long de S1 ֒→ S2 on a : +E (G2)(S2→S2) ≃ E (G1)(S1֒→S2) +≃ D(P1)(R2։R1) +≃ E (G1)(S1→S1) ⊗R1 R2 +≃ E ⊗R R2 +où l’on a posé E := E (G1)(S1→S1) ⊗R1 R. On continue le processus par induction en +posant pour tout n ≥ 3 : +Pn := P1 ⊗W(R1) W(Rn), +Fil1 +(n) := Fil1 ⊗R Rn ⊂ D(Pn−1)(Rn։Rn−1) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +9 +ce qui nous fournit à chaque étape un display Pn ∈ Disp(Rn) et un groupe p-divisible +Gn ∈ BT(Sn) satisfaisant les équations : +ωGn ≃ Fil1 +(n), +E (Gn)(Sn→Sn) ≃ E ⊗R Rn +Par passage à la limite on obtient un groupe p-divisible G ∈ BT(S) satisfaisant par +construction : +E (G)(S→S) ≃ lim +←− +n +E ⊗R Rn ≃ E +ωG ≃ lim +←− +n +Fil1 +(n) ≃ lim +←− +n +Fil1 ⊗R Rn ≃ Fil1 +ce qui démontre les points (1), (2), (3). +□ +Remarque 2.6. +(1) En fait la proposition ci-dessus est une reformulation en termes de cristaux de +Dieudonné des lemmes 2.15 et 2.16 de [Lau14]. Cette reformulation est possible +grâce à l’identification entre le cristal de Dieudonné E (G) et le cristal associé +à un display D(P). Nous aurions très bien pu nous passer de la théorie de +Dieudonné cristalline et simplement utiliser la théorie des displays. +(2) L’un des apports de la théorie des displays dans la preuve ci-dessus est qu’elle +rend explicite le calcul du faisceau E (Gn)(Rn+1։Rn) : il suffit de prendre n’importe +quel Pn+1 qui relève le display Pn (voir [Zin02] Théorème 3). Cela découle du +fait que le morphisme de « frame »(DRn+1/Rn → DRn) est cristallin (voir [Lau14] +Proposition 2.8). Dans la preuve ci-dessus à chaque étape nous avons défini la +display Pn+1 comme étant le changement de base de P1 le long de la section +R1 → Rn+1. La proposition ci-dessus n’est donc qu’un passage à la limite d’un +fait bien connu. +Énonçons pour finir un lemme dont nous aurons besoin dans la section 7.1.2 : +Lemme 2.7. Soit S0 = Spec R/I ֒→ S = Spec R une immersion fermée telle que +I2 = 0. Soit G0 ∈ BT(S0) un groupe p-divisible sur S0. Soit Fil1 +(1), Fil1 +(2) ⊂ E (G0)(S0֒→S) +deux filtrations admissibles. Dans E (G0)(p) +(S0֒→S) on dispose de l’égalité : +(Fil1 +(1))(p) = (Fil1 +(2))(p) +En particulier le faisceau ω(p) +G ne dépend pas du relèvement G ∈ BT(S). +Démonstration. C’est un simple résultat d’algèbre commutative. On pose M = E (G0)(S0֒→S). +C’est un R-module. Puisque E est un cristal on dispose de l’identification E (G0)(S0→S0) = +M/IM. Soit donc N1, N2 ⊂ M tels que N1/IM = N2/IM. On veut montrer que +N(p) +1 += N(p) +2 +⊂ M ⊗R,σ R où σ : R → R désigne le Frobenius. Mais c’est clair puisque +0 = IM(p) ⊂ M ⊗R,σ R (voir que im ⊗ 1 = m ⊗ σ(i) et que σ(i) = 0 car par hypothèse +I2 = 0). +□ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +10 +2.3. Relation d’adhérence. +Définition 2.8. Soit S un espace topologique. Une stratification de S par rapport à un +ensemble partiellement ordonné (I, ≤) est une décomposition : +S = +� +i∈I +Si +telle que pour tout i ∈ I on ait la relation d’adhérence : +Si ⊂ +� +j≤i +Sj +Une stratification est appelée bonne stratification si de plus elle satisfait pour tout i ∈ I : +Si = +� +j≤i +Sj +Dans cette partie nous allons énoncer un résultat dont nous aurons besoin par la +suite lorsque nous nous intéresserons aux relations d’adhérences entre strates de nos +variétés de Shimura. Ce résultat est certainement bien connu et est notamment utiliser +dans [BH22a] (Preuve du théorème 4.11) mais à défaut d’avoir trouvé une preuve, nous +allons en proposer une ci-dessous. +Proposition 2.9. Soit X un un schéma noethérien de caractéristique p > 0 et Y ⊂ X +un sous schéma localement fermé. Les conditions suivantes sont équivalentes : +(1) x ∈ Y +(2) Il existe y ∈ Y tel que y ⇝ x +(3) Il existe une extension de corps k/κ(x) et un morphisme de schémas Spec k[[t]] → +X qui envoi le point fermé sur x et le point générique sur y. +Démonstration. (3) ⇒ (1) découle de la continuité du morphisme. (1) ⇒ (2) découle +du lemme 2.10 ci dessous et du fait que l’espace topologique sous-jacent à un schéma +soit sobre et qu’un espace topologique noethérien sobre soit spectral (dans un espace +topologique noethérien tout sous-ensemble est quasi-compact). Montrons que (2) ⇒ (3). +D’après le lemme 2.11 ci dessous, il existe un anneau de valuation discrète R satisfaisant +les propriétés du point (3). Le théorème de structure de Cohen nous assure que �R ≃ k[[t]] +où �R désigne la complétion le long de l’idéal maximal m ⊂ R et k = R/m désigne le +corps résiduel. Le morphisme Spec �R → X satisfait les propriétés souhaitées. +□ +Lemme 2.10. Soit X un espace topologique spectral, et Y ⊂ X un sous ensemble +constructible. Alors +Y = +� +y∈Y +{y} +Démonstration. Voir [Sta18] Lemme 5.23.6. +□ +Lemme 2.11. Soit X un schéma noethérien et y ⇝ x une spécialisation. Alors il existe +un anneau de valuation discrète R et un morphisme Spec R → X qui envoie le point +fermé sur x et le point générique sur y. +Démonstration. Voir [Sta18] Lemme 28.5.10. +□ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +11 +3. Grassmanniennes Affines +3.1. Grassmanienne affine. Le début de cette section est constitué essentiellement +de rappels sur la géométrie des grassmanniennes affines que l’on peut retrouver dans +[Zhu16] par exemple. +Soit G un groupe réductif connexe déployé lisse sur un corps k. On fixe un tore +déployé T ⊂ G sur k et un borel B le contenant On notera ⟨, ⟩ : X∗(T) × X∗(T) → Z +le produit scalaire, X∗(T)+ l’ensemble des cocaractères B-dominants et 2ρ ∈ X∗(T) la +somme des racines positives. On note Algk la catégorie des k-algèbres. On définit le +groupes de lacets LG et le groupe d’arcs L+G comme les foncteurs Algk → Sets qui à +une k algèbre R associent : +LG(R) = G(R((u))), +L+G(R) = G(R[[u]]) +On définit la grassmannienne affine pour le groupe G comme le quotient (pour la topo- +logie étale ou fppf puisque G est lisse) +GrG = LG/L+G +On montre que ce quotient est ind-représentable par un schéma propre sur Spec k. +En utilisant le fait que G soit supposé lisse on peut montrer que l’on a en fait une +description explicite de ce quotient : +GrG(R) = +� +(E, β) +����� +E est un G torseur sur DR, +β : E|D∗ +R ≃ E0|D∗ +R est une trivialisation +� +où DR = Spec R[[u]] désigne le disque unité, D∗ +R = Spec R((u)) le disque unité épointé et +E0 le G-torseur trivial sur DR. Dans la définition ci dessus E est un G-torseur pour la +topologie fppf ou étale (encore une fois puisque G est supposé lisse). Par la suite nous +noterons E ��� E0 la trivialisation β : E|D∗ +R ≃ E0|D∗ +R. +Par la suite nous aurons besoin du lemme facile suivant : +Lemme 3.1. Soit G1, G2 deux groupes réductifs connexes déployés lisses sur k. On +dispose d’un isomorphisme canonique +GrG1×G2 ≃ GrG1 × GrG2 +Démonstration. Il suffit de voir que le résultat est vrai au niveau des foncteurs : +L(G1 × G2)(R) = (G1 × G2)(R((u))) = G1(R((u))) × G2(R((u))) +□ +On dispose d’une action à gauche : +L+G × GrG +−→ +GrG +(g, (E, β)) +�−→ +(E, g · β) +dont le quotient est appelé champs de Hecke +HeckeG = [L+G\LG/L+G] + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +12 +Par construction il associe à une k algèbre R +HeckeG(R) = +� +(E, E′β) +����� +E, E′ sont des G torseurs sur DR, +β : E|D∗ +R ≃ E′|D∗ +R est un isomorphisme +� +Désormais nous ferons l’abus de notation de retirer l’indice G et de noter Hecke et Gr. +On rappelle que par la décomposition de Cartan, on peut associer à une modification +β : E ��� E′, sa position relative Inv(β) ∈ X∗(T)+ via la bijection +G(k[[u]])\G(k((u)))/G(k[[u]]) +−→ +X∗(T)+ +[g] +�−→ +Inv(g) +[uλ] +←−� +λ +où uλ = λ(u) ∈ T(k((u))). Pour tout élément λ ∈ X∗(T)+ on définit +Grλ := {(E, β) ∈ Gr | Inv(β) = λ } , +Gr≤λ := {(E, β) ∈ Gr | Inv(β) ≤ λ } +De la même manière on définit Heckeλ et Hecke≤λ. La proposition suivante résume la +plupart des propriétés de la décomposition de Gr en L+G-orbites. +Proposition 3.2. +(1) Pour tout λ ∈ X∗(T)+ on a Grλ = L+G · uλ est une L+G- +orbite. +(2) Grλ est une variété quasi-projective lisse de dimension ⟨2ρ, λ⟩ +(3) On dispose d’une décomposition de Gr en L+G-orbites +Gr = +� +λ∈X∗(T)+ +Grλ +(4) Pour tout λ ∈ X∗(T)+ on a la relation d’adhérence +Grλ = +� +λ′≤λ +Grλ′ +(5) L’ouvert dense Grλ ⊂ Gr≤λ coïncide avec le lieu lisse de Gr≤λ. +Démonstration. Les points (1), (2), (3), (4) sont démontrés dans [Zhu16] (Proposition. +2.1.5). Pour (5) on pourra trouver une preuve dans [MOV03] (Corollary B) +□ +Remarque 3.3. La proposition précédente nous fournit une description de l’espace +topologique sous-jacent au champs de Hecke borné : +|Hecke≤λ| ≃ {λ′ ∈ X∗(T)+ | λ′ ≤ λ} +L’identification ci-dessus est un homéomorphisme où la topologie du membre de droite +est celle induite par la relation d’ordre sur X∗(T)+ +Remarque 3.4. Si µ ∈ X∗(T)+ est minuscule alors on dispose d’une identification +Grµ ≃ G/Pµ +où Pµ désigne le sous groupe parabolique associé à µ. En effet on dispose de deux +morphismes dont on montre qu’ils sont réciproques l’un de l’autre : +Grµ +−→ +G/Pµ +g · uµ +�−→ +[ev(g)] , +G/Pµ +−→ +Grµ +[g] +�−→ +g · uµ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +13 +où ev : L+G → G, g �→ g (mod u) et G ֒→ L+G désigne le groupe des lacets +« constants ». +3.2. Produit de convolution. On définit le produit de n-convolution Gr˜× . . . ˜×Gr +comme étant le champ paramétrant les modifications (βi : Ei ��� Ei−1)i=1,...n : +En ��� En−1 ��� . . . ��� E1 ��� E0 +On dispose d’un morphisme : +Gr˜× . . . ˜×Gr +−→ +Gr +(En ��� . . . ��� E0) +�−→ +(En ��� E0) +appelé morphisme de convolution. Ce procédé fournit également pour tout 1 ≤ i ≤ n +un morphisme : +mi : Gr˜× . . . ˜×Gr +−→ +Gr +(En ��� . . . ��� E0) +�−→ +(Ei ��� E0) +Ces morphismes mis ensemble nous donnent un isomorphisme : +n +� +i=1 +mi : Gr˜× . . . ˜×Gr ≃ Gr × · · · × Gr +En particulier le produit de convolution Gr˜× . . . ˜×Gr est ind-représentable. De la même +manière que pour Gr, on dispose d’une uniformisation du produit de convolution : +Gr˜× . . . ˜×Gr ≃ LG ×L+G · · · ×L+G LG ×L+G Gr +Via cet isomorphisme le morphisme de convolution devient le morphisme de multipli- +cation : +LG ×L+G · · · ×L+G LG ×L+G Gr +−→ +LG/L+G +(gn, . . . , [g1]) +�−→ +[gn . . . g1] +Par la suite pour alléger les notations et lorsque le nombre n est explicite nous note- +rons le produit de n-convolution � +Gr = Gr˜× . . . ˜×Gr. Pour tout n-uplet de cocaractères +λ• = (λ1, . . . , λn) on définit les sous schémas localement fermés du produit de convolu- +tion : +� +Grλ• := +� +(Ei, βi) ∈ � +Gr | Inv(βi) = λi +� +, +� +Gr≤λ• := +� +(Ei, βi) ∈ � +Gr | Inv(βi) ≤ λi +� +(3.1) +En particulier � +Grλ• est représentable par un schéma. Il n’est pas difficile de montrer +qu’on dispose alors d’une bonne stratification pour tout µ• ∈ (X∗(T)+)n : +� +Gr≤µ• = +� +λ•≤µ• +� +Grλ• +où λ• ≤ µ• ⇔ λi ≤ µi ∀i = 1, . . . n. Notez que si chacun des µi est minuscule cette +stratification est constituée d’une seule strate à savoir � +Gr≤µ• = � +Grµ•. Par la suite si +λ• = (λ1, . . . , λn) est un n-uplet de cocaractères on notera |λ•| := λ1 + · · · + λn. +Proposition 3.5. � +Grµ• est lisse. En particulier si µi est minuscule pour tout i = +1, . . .n, alors � +Gr≤µ• est lisse. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +14 +Démonstration. On commence par regarder le morphisme de projection +� +Gr(µ1,µ2) +−→ +Grµ1 +(E2 ��� E1 ��� E0) +�−→ +(E1 ��� E0) +En termes de groupes de lacets, ce morphisme correspond à la projection sur la première +coordonnée : +LGµ1 ×L+G Grµ2 → Grµ1 +où LGµ1 = p−1(Grµ1) ⊂ LG avec p : LG → Gr la projection. Par définition ce mor- +phisme est une fibration avec pour fibre Grµ2 (identification non canonique) et est donc +lisse. Enfin puisque Grµ1 est lisse, il s’en suit que � +Gr(µ1,µ2) l’est également. Le résultat +s’en déduit par récurrence. +□ +Remarque 3.6. En particulier si µ• = (µ1, . . . µn) avec chacun des µi minuscule, alors +le morphisme de convolution +mµ• : � +Gr≤µ• → Gr≤|µ•| +est une résolution des singularités. Elle est parfois appelé résolution de Demazure en +référence à [Dem74]. Dans le cas G = GL2 et µi = (1, 0), la preuve précédente nous +dit que le produit de convolution est obtenu par des fibrations successives en Grµi = +G/Pµi ≃ P1 +k. +Le théorème suivant décrit la géométrie du morphisme de convolution : +Théoreme 3.7 (T.Haines, [Hai06]). Soit µ• = (µ1, . . . µn) un n-uplet de cocaractères +quelconques. Alors +(1) Le morphisme mµ• : � +Gr≤µ• → Gr≤|µ•| est localement trivial en restriction à +Grλ ⊂ Gr≤|µ•| pour tout λ ≤ |µ•|. +(2) Si chacun des µi est minuscule alors pour tout y ∈ Grλ la fibre m−1 +µ• (y) est +équidimensionnelle de dimension ⟨ρ, |µ•| − λ⟩. +Démonstration. Le point (1) correspond au Lemme 2.1 de [Hai06] et le point (2) cor- +respond au Théorème 1.1 de loc cit. +□ +Remarque 3.8. Le point (1) découle directement d’un fait plus général : si p : X → Y +est un morphisme G-équivariant tel que G agit transitivement sur Y , alors en choisissant +un point de base y0 ∈ Y on obtient un isomorphisme G-équivariant : +G ×H p−1(y0) ≃ X +où H = StabG(y0). Dans notre situation le morphisme m : m−1(Grλ) → Grλ est bien +L+G équivariant et Grλ est une L+G-orbite par définition. +3.2.1. Exemples. En utilisant le fait qu’un GLn-torseur E sur DR (où R est une k- +algèbre) correspond à un R[[u]]-module Λ localement libre de rang n, on obtient une +description de la grassmannienne affine pour GLn en termes de réseaux : +Gr(R) = +� +Λ ⊂ R((u))n +����� +Λ R[[u]]-module localement libre, +Λ ⊗R[[u]] R((u)) ≃ R((u))n +� + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +15 +En prenant T = Gn +m le tore des matrices diagonales la décomposition de Cartan prend +la forme : +(Zn)+ +−→ +GLn(k[[u]])\GLn(k((u)))/GLn(k[[u]]) +λ = (λ1, . . . , λn) +�−→ +uλ = diag(uλ1, . . . uλn) +En termes de réseaux cela se traduit comme suit : pour tout réseau Λ ⊂ k((u))n il +existe une base (e1, . . . , en) de Λ0 = k[[u]]n et λ = (λ1, . . . , λn) ∈ (Zn)+ tels que +(uλ1e1, . . ., uλnen) soit une base de Λ. Ici on a utilisé la notation +(Zn)+ = {(λ1, . . . , λn) | λ1 ≥ · · · ≥ λn } +Désormais nous noterons Λ0 := k[[u]]n le réseau associé au GLn-torseur trivial E0. +Exemple 3.9. Dans tout ce qui suit R désigne une k-algèbre. Voici quelques exemples : +(1) Pour µ = (1d, 0n−d) on obtient en termes de réseaux : +Gr(1d,0n−d)(R) = {Λ ⊂ Λ0 := R[[u]]n | uΛ0 ⊂ Λ ⊂ Λ0, dimkΛ0/Λ = d } +On retrouve bien la variété GLn/Pµ = Grass(n − d, d) via : +Λ �→ (Rn = Λ0/uΛ0 → Λ0/Λ) +(2) Pour GL2 et µ = (e, 0) on trouve : +Gr≤(e,0)(R) = {Λ | ueΛ0 ⊂ Λ ⊂ Λ0, dimkΛ0/Λ = e } +Soit Λ ∈ Gr≤(e,0)(k). Il existe un plus petit entier i ≤ e tel que uiΛ ⊂ ueΛ0. +Notons N(Λ) ce nombre (notation non standard). On obtient alors la description +suivante des différentes strates +Gr(i,e−i)(R) = {Λ | N(Λ) = i, dimkΛ0/Λ = e } +(3) On s’intéresse au produit de e-convolution pour le groupe GL2 . Pour des coca- +ractères µi = (1, 0) on a la description suivante +Gr(1,0) ˜× . . . ˜×Gr(1,0)(R) = {Λe ⊂ · · · ⊂ Λ1 ⊂ Λ0 | uΛi ⊂ Λi−1, dimkΛi/Λi−1 = 1 } +Le morphisme de convolution prend la forme +m : +Gr(1,0) ˜× . . . ˜×Gr(1,0) +−→ +Gr≤(e,0) +(Λe ⊂ . . . Λ1 ⊂ Λ0) +�−→ +Λe +(4) On peut donner une autre interprétation du produit de convolution précédent. +On définit +M(R) = +� +Λ1 ⊂ · · · ⊂ Λe ⊂ Λ0 +����� +∀ i < e : uΛi ⊂ Λi−1, dimkΛi/Λi−1 = 1, +Λe ∈ Gr≤(e,0) +� +(3.2) +On dispose d’un isomorphisme +Gr(1,0) ˜× . . . ˜×Gr(1,0) +−→ +M +(Λe ⊂ · · · ⊂ Λ1 ⊂ Λ0) +�−→ +(ue−1Λ1 ⊂ · · · ⊂ uΛe−1 ⊂ Λe) +(3.3) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +16 +Cet isomorphisme s’insère dans le diagramme commutatif suivant +Gr(1,0) ˜× . . . ˜×Gr(1,0) +M +Gr≤(e,0) +π +m +(3.4) +où π : (Λ1 ⊂ · · · ⊂ Λe ⊂ Λ0) �→ Λe. +Remarque 3.10. Dans les exemples ci-dessus nous avons fait quelques abus de no- +tations. Si R ∈ Algk est une k-algèbre et (E, β) ∈ Gr(R), alors Inv(β) n’est pas bien +défini : la position relative n’est définie qu’en un point x ∈ Spec R. En particulier nous +aurions dû adopter la notation plus rigoureuse : +Gr≤µ(R) = {(E, β) | Invx(β) ≤ µ pour tout x ∈ Spec R} +Nous n’avons donc pas nécessairement Invx(β) = Invx′(β) pour x ̸= x′ ∈ Spec R +(prendre par exemple Spec R non connexe). +Définition 3.11. (Non standard) Soit Λ ⊂ k((u))n un réseau défini par un point x ∈ Gr. +On définit : +Hodge(x) = Hodge(Λ) := Inv(Λ, Λ0) ∈ X•(T)+ +Remarque 3.12. Concrètement pour GLn si Λ ⊂ k((u))n est un réseau alors pour N +assez grand uNΛ0 ⊂ Λ et l’invariant Hodge(Λ) = (a1, . . . , an) est caractérisé par : +Λ/uNΛ0 ≃ +n +� +i=1 +k[u]/(uN−ai) +Remarque 3.13. Si (Λ1 ⊂ · · · ⊂ Λe) est un point de � +Grµ• alors on notera Hodge(Λk) = +Inv(Λk, Λ0) ∈ X•(T)+. +Exemple 3.14. Les invariants de Hodge de la filtration : +Λ1 = ⟨u2e1, u3e2⟩ ⊂ Λ2 = ⟨u2e1, u2e2⟩ ⊂ Λ3 = ⟨u2e1, ue2⟩ +sont : +Hodge(Λ1) = (3, 2), +Hodge(Λ2) = (2, 2), +Hodge(Λ3) = (2, 1) +Remarque 3.15. Cette notation prendra sens lorsque nous aurons relié la stratification +de Hodge de notre variété de Shimura à celle de la Grassmannienne affine (voir 6.1.1). +3.3. Résultats. On en vient maintenant au principal résultat de cet article : +Proposition 3.16. Si µi = (1, 0) pour tout i = 1, ..., e, alors {m−1(Grλ)}λ≤|µ| défi- +nit une bonne stratification de Grµ1 ˜× . . . ˜×Grµe. En d’autres termes si on note Xλ := +m−1(Grλ) alors pour tout λ ≤ |µ•| +Xλ = +� +λ′≤λ +Xλ′ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +17 +Démonstration. Nous allons démontrer l’assertion pour l’espace de module M de l’exemple +(3.9) (équation (3.2)) car c’est cet espace de module que nous allons considérer par la +suite dans le cadre des modèles entiers des variétés de Shimura. Bien sûr pour obtenir +le résultat pour Grµ1 ˜× . . . ˜×Grµe il suffit de réécrire la preuve ci dessous en appliquant +l’isomorphisme (3.3) et d’utiliser la commutativité du diagramme (3.4). +Soit λ = (i, j) < (e, 0) = |µ•| (pour λ = (e, 0) il n’y a rien à démontrer). Soit x ∈ Xλ +un point de corps résiduel k. Notons Λ1 ⊂ · · · ⊂ Λe la filtration associée. Soit (vk)k⩾1 des +éléments tels que Λk = Λk−1 ⊕(k · vk) (somme directe en tant que k-espaces vectoriels). +On définit des entiers (sk)k pour tout k ≥ 1 +sk = min{s | usΛk ⊂ ueΛ0} = N(Λk) +où N(Λk) est l’entier définit en (3.9) et Λ0 = k[[u]]2. On a +Hodge(Λk) = (e − k + sk, e − sk) +On note k0 = max{k|sk = sk−1}. Notez que k0 ̸= 0 car on a supposé (i, j) < (e, 0). No- +tons (a, b) = Hodge(Λk0) avec a ⩾ b. On a alors par hypothèse (a+1, b) = Hodge(Λk0−1). +Soit (e1, e2) une base de Λ0 telle que : +Λk0−1 = ⟨ua+1e1, ube2⟩ +Dans cette base on peut écrire +vk0 = xuae1 + yub−1e2 +On a nécessairement y = 0 car sinon on aurait usk0 ·vk0 /∈ ueΛ0. Par conséquent on peut +écrire +vk0 = uae1, +Λk0 = Λk0−1 ⊕ (k · uae1) = ⟨uae1, ube2⟩ +Ensuite par définition pour tout n ≥ 1 on a : +u · vk0+n ∈ Λk0+n−1 = Λk0−1 +n−1 +� +ℓ=0 +(k · vk0+ℓ) +On peut donc fixer une décomposition par récurrence : +vk0+n = wn +u + +n−1 +� +ℓ=0 +xn,ℓ +vk0+ℓ +u +, +wn ∈ Λk0−1, xn,ℓ ∈ k +Nous allons maintenant définir une déformation sur R = k�t� de notre filtration initiale. +Soit J := { n | xn,n−1 = 0 }. On travaille dans (k�t�⊗k�u�)2. On commence par déformer +trivialement la filtration pour tout ℓ ≤ k0 − 1 : +˜Λℓ := Λℓ ⊗ k�t� +∀ℓ ≤ k0 − 1 +Pour ℓ = k0 on définit +˜vk0 = uae1 + tub−1e2 = vk0 + tub−1e2 +et on pose +˜Λk0 = ˜Λk0−1 ⊕ (k · ˜vk0) +Ensuite on déforme par récurrence sur n en fonction de si n ∈ J ou n /∈ J : + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +18 +(1) (n ∈ J) Dans ce cas on définit +˜vk0+n = vk0+n + t˜vk0+(n−1) +u +(2) (n /∈ J) Dans ce cas on définit +˜vk0+n = wn +u + +n−1 +� +ℓ=0 +xn,ℓ +˜vk0+ℓ +u +Dans les deux situations on définit la déformation de Λk0+n comme étant : +˜Λk0+n = ˜Λk0+(n−1) ⊕ (k · ˜vk0+n) +Il faut vérifier que l’équation u · ˜vk0+n ∈ ˜Λk0+(n−1) est bien satisfaite. Pour la situation +(2) c’est évident. Pour la situation (1) il faut voir que par hypothèse on a +u · ˜vk0+n = wn + +n−2 +� +ℓ=0 +xn,ℓ˜vk0+ℓ + t˜vk0+(n−1) ∈ ˜Λk0+(n−2) ⊕ (k · ˜vk0+(n−1)) = ˜Λk0+(n−1) +Au point fermé t = 0 on a ˜vk0 = vk0 et par suite ˜vk0+n = vk0+n pour tout n ≥ 1. +Par conséquent cette filtration correspond bien à une déformation de notre filtration +initiale. Pour calculer Hodge(˜Λe ⊗ k((t))) il faut voir que par construction on a +˜sk0+n = ˜sk0+n + 1 ∀n ≥ 1 +et que par conséquent puisque ˜sk0 = sk0 + 1 on trouve +˜se = ˜sk0+e−k0 = ˜sk0 + e − k0 = sk0 + 1 + e − k0 = se + 1 +On a donc bien en fibre générique : +Hodge(˜Λe ⊗ k((t))) = (i + 1, j − 1) +□ +Remarque 3.17. L’idée de la preuve est la suivante. +(1) Si Λe = ⟨uie1, uje2⟩ alors on aimerait déformer sur k�t� en prenant l’élément ˜ve = +uie1 + tuj−1e2. Le problème est que cet élément ne satisfait pas nécessairement +u · ˜ve ∈ Λe−1. Il faut donc déformer Λe−1 également. Le problème est que cette +déformation doit de nouveau satisfaire l’équation u · ˜Λe−1 ⊂ Λe−2... +(2) Il existe un rang k0 tel que là déformation ˜Λk0 existe. Autrement dit on peut +trouver un élément ˜vk0 de la « bonne valuation », c’est-à-dire celle de vk0 moins +1. +(3) Ensuite on déforme par récurrence les vk0+n en à « divisant par u »à chaque +étape de sorte à faire apparaître du uj−1 dans la décomposition de ˜ve. +Donnons un exemple explicite de déformation. On considère la filtration +Λ1 = ⟨u3e1, u2e2⟩, +Λ2 = ⟨u2e2, u2e1⟩, Λ3 = ⟨ue2, u2e1⟩ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +19 +On représente cette filtration par une matrice + + +u2e1 +ue1 +e1 +u2e2 +ue2 +e2 +v1 +∗ +v2 +∗ +v3 +∗ + + +La multiplication par u consiste à décaler les colonnes vers la gauche. La déformation +construit dans la preuve précédente est donnée par la matrice : + + +u2e1 +ue1 +e1 +u2e2 +ue2 +e2 +v1 +∗ +v2 +∗ +t2 +v3 +t3 +∗ +t3t2 + + +Remarque 3.18. Le théorème ci-dessus n’est pas vrai dans le cas général (voir [BH22b] +Proposition 3.9 pour un contre exemple). En fait dans la preuve ci dessus on utilise un +fait spécifique au cas G = GL2 : si Λ ∈ Gr≤(e,0) alors avec les notations de 3.9 : +Λ ∈ Gr(i,j) ⇔ N(Λ) = i +L’invariant N(Λ) est égal à l’indice de nilpotence de u ∈ End(Λ/ueΛ0) ce qui rend +le calcul de Hodge(Λ) = Inv(Λ, Λ0) beaucoup plus simple à calculer en pratique. La +preuve du Théorème 3.16 consiste simplement à déformer une filtration Λ1 ⊂ · · · ⊂ Λe +de sorte à faire apparaître le bon indice de nilpotence en fibre générique. +4. Modèles locaux +4.1. Notations. Soit F un corps totalement réel de degré d > 1 sur Q. On note OF +son anneau d’entiers. Pour tout v|p on note ev l’indice de ramification et fv le degré +résiduel. On note Fv la complétion de F en v et Ov son anneau d’entiers. On note F nr +v +la sous extension maximale non ramifiée et Onr +v son anneau d’entiers. Soit K/Qp une +extension qui contient tous les plongement Fv → Qp pour tout v|p. On note OK son +anneau d’entiers, k son corps résiduel, et on fixe une uniformisante ̟ ∈ OK. On dispose +d’une décomposition +OF ⊗Z OK ∼= +� +v|p +� +τ∈Σnr +v +Ov ⊗Onr +v ,τ OK +(4.1) +où Σnr +v = Hom(F nr +v , Qp). Si on fixe une uniformisante ̟v de Ov alors on peut identifier +Ov ⊗Onr +v ,τ k ≃ k[u]/(uev) +(4.2) +On obtient donc une décomposition non canonique : +OF ⊗Z k ≃ +� +v|p +� +τ∈Σnr +v +k[u]/(uev) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +20 +4.2. Modèle local PEL. Le groupe réductif qui nous intéresse est : +G = ResF/QGL2 +Lorsqu’on le change de base à K on obtient une décomposition +G ⊗Q K = +� +v|p +� +τ∈Σnr +v +(ResFv/F nr +v GL2) ⊗Qp K +Pour simplifier les notations nous allons dans un premier temps décrire le modèle +local PEL pour le groupe G = ResL/LnrGL2 avec L/Qp une extension finie d’indice de +ramification e (jouant le rôle de Fv/Qp). On fixe un plongement τ : Lnr ֒→ Qp et on +note Στ = HomLnr(L, Qp) l’ensemble des plongements qui prolongent τ. On fixe une +numérotation Στ ≃ {ϕ1, . . ., ϕe}. +Soit V un L-espace vectoriel de dimension 2. On fixe une base (e1, e2) de V . On +note Λ le OL-module libre de base (e1, e2). Le modèle local PEL pour le groupe G noté +MPEL, est le schéma sur Spec Onr +L représentant le foncteur qui à (S → Spec Onr +L ) associe +l’ensemble MPEL(S) des OL ⊗Onr +L OS-sous modules F ⊂ ΛS := Λ ⊗Onr +L OS tels que +• F est localement sur S (pour la topologie Zariski) un OS-facteur direct de ΛS +de rang e. +• Pour tout a ∈ OL on a l’égalité polynomiale suivante +det(a | F) = +� +ϕi∈Στ +ϕi(a) +On note G = AutOL(Λ) le schéma en groupe sur Spec Onr +L des automorphismes de Λ +compatibles avec l’action de OL. On a alors la proposition suivante : +Proposition 4.1. G est lisse sur Spec Onr +L +Démonstration. Voir [RZ96] (Proposition A.4). +□ +En fait la proposition ci dessus est démontrée pour G = AutOL((Λ)i∈I) où (Λ)i∈I est +une chaîne périodique de réseaux (voir [RZ96]). Cette situation apparaît lorsque l’on +autorise du niveau en p, ce qui n’est pas notre cas ici. Dans notre situation ce groupe +est en fait explicite : +G = ResOL/Onr +L GL2 +et est bien sûr lisse. +4.3. Modèle local de Pappas-Rapoport. On considère maintenant le foncteur MPR +qui à un schéma (S → Spec OK) associe l’ensemble MPR(S) des filtrations (F (i))i=1,...,e +de OL ⊗OLnr OS-sous modules de ΛS : +0 = F (0) ⊂ F (1) ⊂ · · · ⊂ F (e) ⊂ ΛS +telles que : +• Les F (i) sont Zariski-localement des OS-facteurs directs de ΛS de rang i +• Pour tout a ∈ L et pour tout i = 1, ..., e +(a ⊗ 1 − 1 ⊗ ϕi(a)) · F (i) ⊂ F (i−1) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +21 +Proposition 4.2. Ce foncteur est représentable par uns schéma projectif sur Spec OK. +Démonstration. Il suffit de voir que l’on peut le plonger dans un produit de Grassman- +niennes convenables. Plus précisément, on peut utiliser 4.7 et le fait que le produit de +convolution soit représentable (voir 3.1). +□ +On dispose d’un morphisme d’oubli +π : MPR +−→ +MPEL ⊗Onr +L OK +(F (i)) +�−→ +F (e) +Désormais pour alléger les notations nous noterons de la même manière MPEL = +MPEL ⊗Onr +L OK. +4.4. Plongement dans les grassmanniennes affines. Nous allons maintenant dé- +crire les plongements des fibres spéciales des modèles MPR et MPEL dans certaines grass- +manniennes affines. On suit presque à la lettre [PR02]. On note M +PEL = MPEL ⊗OK k +et M +PR = MPR ⊗OK k les fibres spéciales de ces deux modèles. Comme en (4.2) le choix +d’une uniformisante ̟ ∈ OL nous fournit une identification : +OL ⊗Zp k ≃ k[[u]]/(ue), +̟ ⊗ 1 �→ u +Cela induit un isomorphisme de OL ⊗Zp k-modules : +Λ ⊗Zp k ≃ Λ0 ⊗k[[u]] k[[u]]/(ue) +(on rappelle que Λ ⊂ V est un OL-réseau fixé (4.2) et que Λ0 = k[[u]]2). On note p la +projection : +p : Λ0 → Λ0 ⊗k[[u]] k[[u]]/(ue) +Soit (S → Spec k) un schéma et F ∈ M +PEL(S). L’identification précédente permet de +voir F comme un sous module de Λ0 ⊗k[[u]] OS[[u]]/(ue). On définit le OS[[u]]-module ΛF +comme étant +ΛF := p−1 +� +F ⊂ Λ0 ⊗k[[u]] OS[[u]]/(ue) +� +(4.3) +On obtient donc finalement par ce procédé une immersion fermée +M +PEL ֒↛ Gr +Remarque 4.3. Par construction on dispose des inclusions de réseaux +ueΛ0,S ⊂ ΛF ⊂ Λ0,S +dont les gradués sont des OS-modules localement libres de rang e. Notez que l’image +de cette immersion est entièrement caractérisée par les inclusions ci-dessus et le rang +des gradués. +Proposition 4.4. L’immersion fermée ι est équivariante pour l’action de G⊗k à gauche +et L+G à droite. Elle induit un isomorphisme +M +PEL ≃ +� +λ⩽(e,0) +Grλ +Démonstration. Découle de la remarque précédente et de l’exemple 3.9. +□ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +22 +Remarque 4.5. En d’autres termes l’ensemble des copoids µ-admissibles (voir [Goe01], +section 4.3) est ici tout à fait explicite +Adm(µ)K = {λ ⩽ (e, 0)} +(Ici on est dans le cas parahorique maximal K = G(Zp)) +De la même manière on dispose d’une immersion fermée pour le modèle de Pappas- +Rapoport dans un produit de grassmanniennes affines : +ιPR : M +PR +−→ +Gr × ... × Gr +(F (1), ..., F (e)) +�−→ +(ΛF (1), . . . , ΛF (e)) +(4.4) +Remarque 4.6. Par construction on dispose des inclusions +ueΛ0 ⊂ ΛF (1) ⊂ ... ⊂ ΛF (e) ⊂ Λ0 +dont les gradués sont localement libres de rang 1 pour j = 1, ..., e − 1 et de rang e pour +j = e. D’après l’exemple 3.9 cette immersion fermée induit un isomorphisme M +PR ≃ M +où M est l’espace de module décrit en (3.2). +Proposition 4.7. L’immersion fermée ιPR induit un isomorphisme équivariant pour +l’action de G à gauche et L+G à droite : +M +PR ≃ Gr(1,0) ˜× . . . ˜×Gr(1,0) +où le produit de convolution est pris e fois. +Démonstration. Découle de la remarque précédente en composant avec l’isomorphisme +(3.3). +□ +Proposition 4.8. Le carré suivant est cartésien +M +PR +Gr(1,0) ˜× . . . ˜×Gr(1,0) +M +PEL +Gr≤(e,0) +π +m +Démonstration. Il s’agit essentiellement de montrer que le diagramme suivant est com- +mutatif. Cela découle du fait que le diagramme (3.4) est commutatif. +□ +4.5. Cas général. Revenons maintenant au cas général où l’on considère une exten- +sion F/Q de degré d. On note G = ResF/QGL2. Pour tout v|p et tout τ ∈ Σnr +v , on note +MPEL +v,τ +le modèle local pour le groupe ResFv/F nr +v GL2 définit dans la section précédente. +4.5.1. Modèles locaux. Soit V un F-espace vectoriel de dimension 2. On fixe une base +(e1, e2) de V . On note Λ le OF-module libre de base (e1, e2). On fixe (à conjugaison près) +un cocaractère µ : Gm,C → GC, et on suppose que ce dernier induit une décomposition +en espaces propres : +V ⊗ C = V0 ⊕ V1 + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +23 +Enfin on suppose que le corps de définition de µ est Q (ces hypothèses seront vérifiées +dans le cadre des variétés de Shimura de type Hilbert). On définit le modèle local +MPEL comme étant le foncteur qui à (S → Spec Zp) associe l’ensemble MPEL(S) des +OF ⊗Zp OS-sous modules F ⊂ ΛS := Λ ⊗Zp OS tels que +• F est Zariski-localement sur S un OS-facteur direct de ΛS de rang [F : Q] = d ; +• Pour tout a ∈ OF on a l’égalité polynomiale suivante (condition de Kottwitz) : +det(a | F) = det(a | V0) +On note G = AutOL(Λ) le schéma en groupe sur Spec Zp des automorphismes de Λ +compatibles avec l’action de OF. On a alors la proposition suivante : +Proposition 4.9. Après changement de base à OK, on dispose d’un isomorphisme +canonique +MPEL ⊗Zp OK ≃ +� +v|p +� +τ∈Σnr +v +MPEL +v,τ +Démonstration. Découle de (4.1). +□ +Comme précédemment on note G = AutOF (Λ). On a alors pour les mêmes raisons +une décomposition +G ⊗ Zp ≃ +� +v|p +ResOv/ZpGL2 +En particulier G ⊗ Zp est bien lisse. +Cette décomposition suggère la définition suivante pour le modèle de Pappas-Rapoport +dans le cas général d’une extension F/Q +Définition 4.10. Le modèle local de Pappas-Rapoport pour le groupe G = ResF/QGL2 +est : +MPR = +� +v|p +� +τ∈Σnr +v +MPR +v,τ +4.5.2. Plongements dans les grassmanniennes affines. La « compatibilité »des grass- +mannienne affines avec le produit vu au Lemme 3.1 combinée aux décompositions des +modèles locaux de la section précédente nous donne sans trop d’efforts les isomorphismes +non canoniques (dépend des différents choix d’uniformisantes) : +M +PEL ≃ +� +v|p +� +τ∈Σnr +v +Gr≤(ev,0), +M +PR ≃ +� +v|p +� +τ∈Σnr +v +� +Gr(ev,0) +où � +Gr(ev,0) désigne le produit de convolution de ev copie de Gr(1,0). Finalement le carré +cartésien de la proposition 4.8 devient +M +PR +� +v|p +� +τ∈Σnr +v +� +Gr(ev,0) +M +PEL +� +v|p +� +τ∈Σnr +v Gr≤(ev,0) +π +(mv,τ )v,τ +(4.5) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +24 +5. Modèles entiers +5.1. Modèle entier PEL. On garde les notations de 4.1 et 4.5.1. On considère l’espace +de module ShPEL sur Spec Zp qui à un schéma localement noethérien (S → Spec Zp) +associe l’ensemble des quadruplés (A, λ, ι, η) à Z× +(p)-isogénie où : +(1) A → S est un schéma abélien de dimension g = d = [F : Q]. +(2) λ : A → A∨ est une Z× +(p)-polarisation +(3) η est une structure de niveau en dehors de p (voir [Lan13], section 1.4.1) +(4) ι : OF ֒→ End(A) ⊗Z Z(p) morphisme respectant les involutions des deux cotés. +(5) (A, λ, ι, η) satisfait la condition de déterminant de Kottwitz : +det(a | ωA/S) = det(a | V0) +∀ a ∈ OF +(voir [Lan13] Définition 1.3.4.1 pour plus de détails). +On a alors le résultat suivant dû à Mumford puis Kottwitz : +Proposition 5.1 ([MFK94], [Kot92] section 5). ShPEL est représentable par un schéma +quasi-projectif sur Spec Zp. +Remarque 5.2. L’espace de module définit ci-dessus est celui définit par Kottwitz. On +consultera l’article de I.Vollaard [Vol03] pour plus de détails concernant l’équivalence +des différents espaces de modules considérés dans le cas Hilbert. +Nous allons maintenant expliquer le lien entre le modèle entier ShPEL est le modèle +local MPEL. On considère le foncteur ShPEL,□ qui à un schéma localement noethérien +(S → Spec Zp) associe les 5-uplés (A, λ, ι, η, γ) où +γ : H1 +dR(A/S) ≃ Λ ⊗Zp OS +est une trivialisation du premier groupe de cohomologie de deRham (en tant que OF ⊗Zp +OS-module). Notez que ShPEL,□ est muni d’une action de G = AutOF (Λ) où ce dernier +agit sur la trivialisation γ. L’oubli de la trivialisation γ fournit un morphisme +ϕ : ShPEL,□ → ShPEL +L’un des résultats majeurs de [RZ96] est le suivant +Proposition 5.3 ([RZ96] Théorème 3.16). ϕ : ShPEL,□ → ShPEL est un G-torseur. +La trivialisation γ nous fournit un morphisme vers le modèle local PEL +ψ : +ShPEL,□ +−→ +MPEL +(A, λ, ι, γ) +�−→ +γ(ωA/S) ⊂ ΛS +Le fait que ce morphisme soit bien défini découle essentiellement de la définition de +l’espace de module ShPEL. On a alors la proposition suivante qui résulte de [dJ91] et +[RZ96] : +Proposition 5.4. ψ : ShPEL,□ −→ MPEL est un morphisme lisse G-équivariant de +dimension relative dim G. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +25 +Démonstration. Le fait que le morphisme soit G-équivariant est évident. La lissité dé- +coule de Grothendieck-Messing, voir [dJ91] (Proposition 4.5 du document .dvi du même +nom sur sa page web). +□ +En d’autres termes on dispose d’un diagramme de modèle local au sens de [RZ96] : +ShPEL,□ +ShPEL +MPEL +ϕ +ψ +Ce qui correspond à un morphisme de champs algébriques +ShPEL → +� +MPEL/G +� +(5.1) +lisse de dimension relative dim G. +5.2. Modèle entier de Pappas-Rapoport. On définit le modèle entier de Pappas- +Rapoport ShPEL comme le produit cartésien : +ShPR +� +MPR/GOK +� +ShPEL +OK +� +MPEL +OK /GOK +� +(5.2) +où l’indice (·)OK désigne le changement de base ⊗ZpOK. +Avant de rendre la définition de ce modèle explicite, mentionnons un corollaire direct +de la Proposition 5.1 : +Proposition 5.5. ShPR est représentable par un schéma quasi projectif lisse sur Spec OK. +Démonstration. Puisque le carré (5.2) est cartésien il suffit de montrer que MPR est +représentable. Cela découle de la proposition 4.2. Le même raisonnement et le fait que +le morphisme de convolution soit propre montrent que ShPR est quasi projectif puisque +ShPEL l’est. Pour la lissité, il suffit de voir que le modèle local MPR est lisse. D’après +[PR02] Théorème 5.3 le modèle local est plat par conséquent il suffit de montrer que la +fibre spéciale est lisse (la fibre générique étant toujours lisse). La lissité de MPR découle +de la Proposition 3.5. +□ +Remarque 5.6. En fait la preuve du théorème 5.3 de [PR02] repose sur un diagramme +de torseurs reliant le modèle local MPR à un produit de modèles locaux non ramifiés. +Dans le cas PEL sans niveau en p, ces derniers sont lisses (et pas seulement plats) ce +qui donne directement la lissité sur Spec OK, sans passer par la fibre spéciale. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +26 +5.3. Donnée de Pappas-Rapoport. Nous allons maintenant donner une définition +plus explicite du modèle entier de Pappas-Rapoport. Pour ce faire nous allons devoir +définir la notion de « donnée de Pappas-Rapoport »pour un groupe p-divisible. +Pour simplifier les notations, comme pour le §4.2 nous allons ici travailler avec une +seule place v|p. Soient L/Qp une extension finie de degré d > 1 (jouant le rôle de +Fv/Qp), K/Qp une extension contenant la clôture Galoisienne de L, et S un schéma +sur Spec OK. On note Lnr l’extension maximale non ramifiée dans L. On note e l’indice +de ramification. Soit ̟ une uniformisante de L. On fixe un plongement τ : Lnr ֒→ K. +On définit l’ensemble Στ = HomLnr(L, Qp) comme étant l’ensemble des plongements +τ ′ : L ֒→ K qui donne τ en restriction à Lnr. C’est un ensemble de cardinal e. On fixe +un ordre Στ = {ϕ1, ..., ϕe}. +5.3.1. Définition. Soit F un OS module localement libre muni d’une action de OL tel +que OLnr agit sur F via τ (on rappelle que S est un schéma sur Spec OK). On suppose +qu’il existe un faisceau E localement libre de rang h en tant que OL ⊗OLnr OS-module +tel que F soit un localement un facteur direct de E . +Définition 5.7. Une donnée de Pappas-Rapoport pour (E , F) par rapport à Στ est une +filtration +0 = F (0) ⊂ F (1) ⊂ · · · ⊂ F (e) = F +telle que pour tout 1 ⩽ j ⩽ e +• Les F (j) sont localement des OS-facteurs directs, stables par OL. +• ([̟] − ϕj(̟)) · F (j) ⊂ F (j−1) +• F (j)/F (j−1) est localement libre de rang 1 +La deuxième condition impose que l’action de OL sur le quotient F (j)/F (j−1) se fasse +via le plongement OL +ϕj +−→ OK. +5.3.2. Groupes p-divisibles. Soit S un schéma sur Spec OK. Soit G un groupe p-divisible +sur S de hauteur hd muni d’une action ι : OL → EndS(G). Notons E (G) le cristal asso- +cié. Avec les notations de (2.1), en évaluant ce cristal sur l’épaississement tautologique +(S → S) on obtient la filtration de Hodge : +0 −→ ωG −→ E (G)(S→S) −→ ω∨ +GD −→ 0 +D’après la Proposition 2.1 le faisceau E (G)(S→S) est libre en tant que OL ⊗OLnr OS- +module et l’hypothèse 5.3.1 est donc bien satisfaite. Pour simplifier les notations nous +noterons E = E (G)(S→S). Cette filtration est compatible avec les décompositions in- +duites par 4.1 +E = +� +τ∈Σnr +Eτ, +ωG = +� +τ∈Σnr +ωG,τ +où Σnr = Hom(Lnr, Qp). Autrement dit pour tout τ ∈ Σnr on dispose d’une suite exacte +0 −→ ωG,τ −→ Eτ −→ ω∨ +GD,τ −→ 0 +induite par la filtration de Hodge. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +27 +Définition 5.8. Une donnée de Pappas-Rapoport pour (G, ι) par rapport à (Στ)τ∈Σnr +est la donnée pour tout τ ∈ Σnr d’une filtration de Pappas-Rapoport pour (Eτ, ωG,τ) par +rapport à Στ. Autrement dit c’est la donnée pour tout τ ∈ Σnr d’une filtration +0 = ω(0) +G,τ ⊂ ω(1) +G,τ ⊂ · · · ⊂ ω(e) +G,τ = ωG,τ +telle que pour tout 1 ⩽ j ⩽ e +• Les ω(j) +G,τ sont localement des OS-facteurs directs, stables par OL. +• ([̟] − ϕτ,j(̟)) · ω(j) +G,τ ⊂ ω(j−1) +G,τ +• ω(j) +G,τ/ω(j−1) +G,τ +est localement libre de rang 1 +5.3.3. Cas général. Revenons à notre extension F/Q. Soit donc G un groupe p-divisible +sur S muni d’une action ι : OF → EndS(G). Pour chaque place v|p on note Σnr +v += +Hom(F nr +v , Qp), pour tout τ ∈ Σnr +v +on note Σv,τ = HomF nr +v (Fv, Qp). On a alors une +première décomposition +E = +� +v|p +Ev, +ωG = +� +v +ωG,v +et une deuxième décomposition pour chacune des places v|p : +Ev = +� +τ∈Σnr +v +Ev,τ, +ωG,v = +� +τ∈Σnr +v +ωG,v,τ +(5.3) +Comme précédemment on fixe un ordre sur chacun des Σv,τ. +Définition 5.9. Une donnée de Pappas-Rapoport pour (G, ι) par rapport à (Σv,τ)v,τ est +la donnée pour toute place v|p et tout τ ∈ Σnr +v d’une filtration de Pappas-Rapoport pour +(Ev,τ, ωG,v,τ) par rapport à Σv,τ. Autrement dit c’est la donnée pour toute place v|p et +tout τ ∈ Σnr +v d’une filtration +0 = ω(0) +G,v,τ ⊂ ω(1) +G,v,τ ⊂ · · · ⊂ ω(e) +G,v,τ = ωG,v,τ +telle que pour tout 1 ⩽ j ⩽ e +• Les ω(j) +G,v,τ sont localement des OS-facteurs directs, stables par OF. +• ([̟] − ϕv,τ,j(̟)) · ω(j) +G,v,τ ⊂ ω(j−1) +G,v,τ +• ω(j) +G,v,τ/ω(j−1) +G,v,τ est localement libre de rang 1 +Remarque 5.10. Dans la définition ci dessus nous faisons l’abus d’appeler « donnée +de Pappas Rapoport »ce qui correspond en fait à une « donnée de Pappas Rapoport +dans le cas Hilbert » c’est-à-dire une donnée de Pappas-Rapoport où la dimension des +gradués ω(j) +G,v,τ/ω(j−1) +G,v,τ est égale à dj = 1 pour tout 1 ≤ j ≤ e. On consultera [BH22b] et +[BH16] pour plus de détails sur les filtrations de Pappas-Rapoport dans un cadre PEL +plus général. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +28 +5.3.4. Définition explicite de ShPR. Revenons à notre modèle entier ShPR. Il découle +de la définition (5.2) que le foncteur ShPR associe à un schéma localement noethérien +S → Spec OK, les quintuplés (A, λ, ι, η, ω(·) +G ) à Z× +(p)-isogénie près où +• A → S est un schéma abélien de dimension d. +• λ : A → At est une Z× +(p)-polarisation +• ι : OF → End(A) ⊗Z Z(p) +• η est une structure de niveau rationnelle en dehors de p (voir [Lan13] section +1.4.1) +• ω(·) +G est une donnée de Pappas-Rapoport pour G = A[p∞]. +6. Stratification de Hodge +Désormais tout ce qui suit porte sur la fibre spéciale de nos modèles entiers. Nous +noterons donc désormais pour alléger les notations M +PEL = MPEL et MPR = M +PR. Pour +simplifier nous allons commencer par traiter le cas d’une seule extension totalement +ramifiée L/Qp de degré e (jouant le rôle de Fv/F nr +v . Pour tout 1 ≤ i ≤ e on note +µi = (1, 0) et µ• = (µ1, . . ., µe). On a donc µ := |µ•| = (e, 0). +6.1. Le cas PEL. Dans la section 5.1 nous avons vu que le diagramme de modèle local +nous fournissait dans le cas PEL un morphisme lisse (5.1) +ShPEL −→ +� +MPEL/G +� +D’après la proposition 4.4 on dispose d’un plongement équivariant du modèle local +dans la Grassmannienne affine induisant après passage au quotient un morphisme +� +MPEL/G +� +−→ +� +L+G\LG/L+G +� += Hecke +La description de l’image de ce morphisme 4.4 nous fournit donc finalement un mor- +phisme lisse vers le champ de Hecke borné : +ζ : ShPEL −→ HeckeAdm(µ)K +où Adm(µ)K est définit en 4.5. +6.1.1. Définition. Soit x ∈ |ShPEL|. On appelle polygone de Hodge du point x l’élément +Hodge(x) = ζ(x) +Pour tout λ ∈ Adm(µ)K on définit le sous schéma localement fermé (structure réduite) +ShPEL +λ += ζ−1(λ). Les (ShPEL +λ +)λ∈Adm(µ)K sont appelées les strates de Hodge de ShPEL. +Remarque 6.1. Donnons une définition plus explicite du polygone de Hodge et des +strates de Hodge. Soit x ∈ |ShPEL| un point de corps résiduel k. Pour calculer l’image +ζ(x) il nous faut suivre x dans le diagramme de modèle local : +ShPEL,□ +ShPEL +MPEL +Gr +ϕ +ψ +ι + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +29 +Notons ωx ֒→ Ex la filtration de Hodge associée au point x. On choisit un élément +˜x ∈ ϕ−1(x). L’image ι ◦ ψ(˜x) dépend du choix ˜x, mais son image dans le quotient +Hecke = +� +L+G\Gr +� +ne dépend pas de ce choix. Pour décrire ζ(x) il nous suffit donc de +décrire ι ◦ ψ(˜x). Par définition de ShPEL,□ sur ˜x on dispose d’une trivialisation Ex ≃ Λ. +Autrement dit on dispose d’une base : +Ex ≃ k[[u]] +(ue) ⊕ k[[u]] +(ue) +Quitte à choisir une autre base on peut supposer que la filtration de Hodge soit donnée +par +ωx ≃ uik[[u]] +(ue) ⊕ uj k[[u]] +(ue) +Ensuite d’après (4.3) le plongement du modèle local dans la grassmannienne affine +consiste à prendre l’image inverse de la filtration de Hodge le long de k[[u]] → k[[u]]/(ue). +On obtient finalement +Λωx = uik[[u]] ⊕ ujk[[u]] ⊂ Λ0 +Par conséquent on retrouve la définition usuelle (voir [DP94] Section 4.2, où [BH16] +Définition 1.1.7) du polygone de Hodge d’un groupe p-divisible +Hodge(x) = Hodge(ωx) = (i, j) +Remarque 6.2. En fait le polygone de Hodge correspond plutôt au polygone de pentes +( i +e, j +e) (voir [BH16], Définition 1.1.7). Bien sûr on retrouve une définition à partir de +l’autre et les stratifications induites coïncident. +6.1.2. Propriétés. +La proposition suivante résume les différentes propriétés de la stra- +tification de Hodge : +Proposition 6.3 ([DP94] section 4.2, [NG02] section 4). +(1) Les strates (ShPEL +λ +)λ∈Adm(µ)K forment une bonne stratification de ShPEL. +(2) Pour tout λ ∈ Adm(µ)K la strate ShPEL +λ +est quasi-projective lisse de dimension +⟨2ρ, λ⟩. +(3) La strate ShPEL +(e,0) coïncide avec le lieu lisse de ShPEL. +Démonstration. Pour le point (1), cela découle de la lissité du morphisme ζ : ShPEL → +HeckeAdm(µ)K. Pour le point (3), cela découle du fait que Gr(e,0) coïncide avec le lieu +lisse de Gr≤(e,0) (voir Proposition 3.2) Enfin pour le point (2), il suffit de voir que le +morphisme +ShPEL +λ +→ +� +MPEL +λ +/G +� +est lisse de dimension relative dim G et que le membre de droite était un point de +dimension ⟨2ρ, λ⟩ − dim G. +□ +Remarque 6.4. Pour le point (2) on peut se passer du langage des champs et donner +une preuve à la main : en restriction à une strate λ ∈ Adm(µ)K le diagramme de modèle + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +30 +local devient +ShPEL,□ +λ +ShPEL +λ +MPEL +λ +ϕ +ψ +D’après la proposition 3.2 la strate MPEL +λ +est de dimension ⟨2ρ, λ⟩. Le morphisme ψ est +de dimension relative dim G = dim G et le morphisme ϕ est un G-torseur. On trouve +donc bien +dim ShPEL +λ += ⟨2ρ, λ⟩ + dim G − dim G = ⟨2ρ, λ⟩ +Remarque 6.5. Dans [AG03] la strate ShPEL +(e,0) est appelée lieu de Rapoport car elle +coïncide avec le lieu où le faisceau ω est libre en tant que OL ⊗Zp OS-module. +Remarque 6.6. On peut être plus explicite sur la dimension. Si λ = (i, j) ≤ (e, 0) +alors +⟨2ρ, λ⟩ = ⟨α1 − α2, iλ1 + jλ2⟩ += i − j += e − 2j +où α1 : +� +t1 +0 +0 +t2 +� +�→ t1, α2 : +� +t1 +0 +0 +t2 +� +�→ t2, λ1 : t �→ +� +t +0 +0 +1 +� +et λ2 : t �→ +� +1 +0 +0 +t +� +. +6.2. Le cas Pappas-Rapoport. Nous allons maintenant nous intéresser à la stratifi- +cation du modèle de Pappas-Rapoport ShPR par le polygone Hodge(ω). +6.2.1. Définition. Pour tout λ ∈ Adm(µ)K on définit le sous schéma localement fermé +(structure réduite) ShPR +λ += π−1(ShPEL +λ +) où +π : ShPR → ShPEL +est le morphisme d’oubli. Les (ShPR +λ )λ∈Adm(µ)K sont appelées les strates de Hdoge de +ShPR. Plus explicitement : si x ∈ ShPR alors on peut regarder le polygone de hodge +Hodge(x) := Hodge(ω(e)) où ω(e) = ωx est le dernier cran de la filtration de Pappas- +Rapoport ω(1) ⊂ · · · ⊂ ω(e) associée au point x. +Remarque 6.7. On fera attention à ne pas confondre les deux morphismes : +ShPR +� +L+G\ � +Grµ• +� += Heckeµ• +HeckeAdm(µ)K = +� +L+G\Gr≤|µ•| +� +˜ζ +ζ◦π +Le morphisme de gauche concerne les classes d’isomorphismes de filtrations de Pappas- +Rapoport (ω(1) ⊂ · · · ⊂ ω(e) ⊂ E ), et celui de droite concerne les classes d’isomor- +phismes de filtrations de Hodge (ω ⊂ E ). + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +31 +6.2.2. Propriétés. Le théorème est le suivant : +Théoreme 6.8. +(1) Les strates (ShPR +λ )λ∈Adm(µ)K forment une bonne stratification de ShPR. Autre- +ment dit pour tout λ ∈ Adm(µ)K on a la relation d’adhérence +ShPR +λ += +� +λ′≤λ +ShPR +λ′ +(2) Pour tout λ ∈ Adm(µ)K la strate ShPR +λ +est quasi-projective lisse de dimension +⟨ρ, |µ•| + λ⟩. +Démonstration. Commençons par montrer le point (1). D’après la proposition 3.16 on +dispose d’une bonne stratification du produit de convolution +� +Grµ• = +� +λ∈Adm(µ)K +m−1(Grλ) +Puisque le morphisme de convolution m : � +Grµ• → Gr≤|µ•| est L+G-équivariant, cette +stratification est stable sous l’action de L+G. Autrement dit chacune des strates est +une union de L+G-orbites. Par conséquent cette stratification descend au quotient en +une bonne stratification du champs Heckeµ• = +� +L+G\Grµ• +� +: +Heckeµ• = +� +λ∈Adm(µ)K +� +L+G\m−1(Grλ) +� +On conclut comme pour le cas PEL (voir 6.1.2) en utilisant que le morphisme +˜ζ : ShPR → Heckeµ• +est lisse et donc préserve les relations d’adhérences. +Montrons maintenant le point (2). D’après le théorème 3.7, en restriction à une strate +Grλ associée à λ ∈ Adm(µ)K on dispose d’une fibration localement triviale +m : m−1(Grλ) −→ Grλ +En particulier au dessus d’une telle strate le morphisme est plat et par conséquent on +a la relation +dim m−1(Grλ) = dim Grλ + dim m−1(y) +pour n’importe quel élément y ∈ Grλ. Toujours d’après le théorème 3.7, on a que la +fibre m−1(y) est équidimensionnelle de dimension ⟨ρ, |µ•| − λ⟩ car µ• = (µ1, . . . µe) est +constitué de cocaractères minuscules. On définit au niveau du modèle local MPR +λ += +m−1(Grλ) via l’identification MPR ≃ � +Grµ•. Le modèle local nous donne un morphisme +ShPR +λ +→ +� +MPR +λ /G +� + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +32 +qui est lisse de dimension relative dim G. Le membre de droite est de dimension MPR +λ +− +dim G. Finalement on trouve donc bien +dim ShPR +λ += dim +� +MPR +λ /G +� ++ dim G += dim MPR +λ ++ dim G − dim G += dim m−1(Grλ) += dim Grλ + dim m−1(y) += ⟨2ρ, λ⟩ + ⟨ρ, |µ•| − λ⟩ += ⟨ρ, |µ•| + λ⟩ +□ +Remarque 6.9. Encore une fois on peut être plus explicite concernant la dimension +des strates. Pour λ = (i, j) ≤ (e, 0) on obtient +dim ShPR +(i,j) = ⟨ρ, |µ•| + λ⟩ += 1 +2⟨α1 − α2, (e + i)λ1 + jλ2⟩ += e + i − j += e − j +6.3. Cas général. On revient maintenant au cas d’une extension F/Q de degré d > +1. Pour tout v|p, τ ∈ Σnr +v +et tout 1 ≤ . . . i ≤ ev on note µv,τ,i = (1, 0), µv,τ,• = +(µv,τ,1, . . . , µv,τ,ev), µv,τ = |µv,τ,•| = (ev, 0) et enfin µ := (µv,τ)v,τ La compatibilité du +modèle local au produit décrite en (4.5) suggère la définition suivante : soit x ∈ |ShPR|, +on définit l’invariant : +Hodge(x) = +� +Hodgev,τ(x) +� +v,τ ∈ +� +v,τ +{λ ≤ (ev, 0)} = Adm(µ)K +où Hodgev,τ(x) = Hodge(ωx,v,τ) est l’invariant de Hodge définit en 6.1.1 et ωx,v,τ est +le faisceau associé à x et défini en (5.3). Autrement dit l’invariant Hodge(ω) ci dessus +correspond à la donnée des invariants Hodge(ωv,τ) pour chacun des termes dans la +décomposition +ω = +� +v|p +� +τ∈Σnr +v +ωv,τ +On munit Adm(µ)K de la relation d’ordre suivante : +(λv,τ)v,τ ≤ (λ′ +v,τ)v,τ ⇐⇒ λv,τ ≤ λ′ +v,τ ∀(v, τ) +La généralisation du Théorème 6.8 prend la forme suivante : +Théoreme 6.10. +(1) Les strates (ShPR +λv,τ )λv,τ ∈Adm(µ)K forment une bonne stratification de ShPR. Au- +trement dit pour tout (λv,τ)v,τ ∈ Adm(µ)K on a la relation d’adhérence +ShPR +(λv,τ )v,τ = +� +(λ′v,τ )v,τ ≤(λv,τ )v,τ +ShPR +(λ′v,τ )v,τ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +33 +(2) Pour tout (λv,τ)v,τ ∈ Adm(µ)K la strate ShPR +λv,τ est quasi-projective lisse de di- +mension : +dim ShPR +(λv,τ )v,τ = +� +v,τ +⟨ρ, |µv,τ,•| + λv,τ⟩ +Remarque 6.11. On fera attention au fait que dans le cadre d’une extension L/Qp +d’indice d’inertie f > 1, alors la stratification de Hodge ci-dessus ne coïncide pas avec +la stratification de Hodge définie dans [BH22b]. En fait la stratification définie dans +loc.cit. est moins fine car elle est définie par le polygone de Hodge de [BH16] qui est +un moyenne sur f des invariants de Hodge considérés pour notre stratification. On +consultera [SZ22] Proposition 3.20 pour plus de détails. +6.4. Autres résultats. Lorsque la donnée PEL est ramifiée, l’un des objectifs pour +étudier la géométrie de la fibre spéciale de la variété de Shimura associée est de raffiner +la stratification de Hodge. Autrement dit pour tout λ ∈ Adm(µ)K on aimerait définir +une décomposition : +ShPEL +λ += +� +w∈WΛ +Sw +où (Wλ, ≤) est un certain ensemble partiellement ordonné qui dépend de λ. Donnons +quelques exemples : +(1) Dans [SYZ21] les auteurs définissent pour chaque strate de Hodge un morphisme +lisse : +ζλ : ShPEL +λ +−→ Grdt +0 -ZipJλ +où Grdt +0 +désigne le quotient réductif de G0 = G ⊗Zp Fp et Grdt +0 -Zipλ désigne le +champs des Grdt +0 -Zip de type Jλ (voir [SYZ21] Définition 1.1.5 et Proposition +4.2.6). La stratification est alors définie comme étant celle induite par celle du +champs Grdt +0 -ZipJλ via le morphisme ζλ. Leur construction est beaucoup plus +générale et s’applique aux variétés de Shimura de type abélien sans hypothèse +sur le niveau en p +(2) Dans [AG03] les auteurs calculs explicitement les polygones de Newton sur cha- +cune des strates. Plus précisément si x ∈ ShPEL +(i,j) avec (i, j) ∈ Adm(µ)K alors +existe une (OL ⊗ W(k))- base du module de Dieudonné associé dans laquelle le +Frobenius est donné par la matrice ([AG03], Propositon 4.10) : +F = +� +̟m +c̟i +̟j +0 +� +où m ≥ j et c ∈ (OL ⊗ W(k))×. Ils définissent ensuite les quantités : +n = +� +m +si m ≤ i +i +sinon +λ(n) = min +�n +g , 1 +2 +� +La stratification de ShPEL +(i,j) est alors définie via cet invariant n : +ShPEL +(i,j) = +� +j≤n≤i +ShPEL +(i,j),n + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +34 +Cet invariant leur permet de calculer explicitement le polygone de Newton sur +chacune des strates ShPEL +(i,j),n. Plus précisément si x ∈ ShPEL +(i,j),n alors ([AG03], +Théorème 9.2) : +Newt(x) = {λ(n), . . . , λ(n), 1 − λ(n), . . ., 1 − λ(n)} +(chacune des pentes avec multiplicité e). +(3) Dans [YCO20] les auteurs associent à tout point x ∈ ShPEL un invariant c(x) = +(cτ(x))τ∈Σnr appelé invariant de congruence qui mesure la position relative des +filtrations (voir la [YCO20] Définition 5.2 pour être plus précis) : +F(Mσ−1τ) ⊂ Mτ, +V (Mστ) ⊂ Mτ +où (M, F, V ) désigne le module de Dieudonné associé à x et où les (Mτ)τ∈Σnr +désignent les facteurs directs dans la décomposition +M ≃ +� +τ∈Σnr +Mτ, +F : Mσ−1τ → Mτ, +V : Mστ → Mτ +Cet invariant permet de définir une stratification ([YCO20] Définition 6.1) : +ShPEL +λ += +� +c∈τL +Qc(ShPEL +λ +) +où Qc(ShPEL +λ +) désigne l’ensemble des points d’invariant de congruence c ∈ τL et +τL est un certain ensemble défini dans [YCO20] Définition 3.2.1. +Dans chacun de ces trois articles se pose deux problèmes : +(1) Calculer l’adhérence de Sw dans ShPEL +λ +. +(2) Calculer l’adhérence de Sw dans ShPEL +Dans [YCO20] les auteurs parviennent à calculer le point (1). Dans [AG03] et [SYZ21] +les auteurs parviennent à calculer le point (2). Le théorème suivant montre que le point +(1) est automatique pour ShPR +λ +lorsque l’on regarde le tiré en arrière d’une stratification +de ShPEL +λ +via le morphisme d’oubli π : ShPR → ShPEL. +Théoreme 6.12. Pour tout λ ∈ Adm(µ)K la restriction +π : ShPR +λ +→ ShPEL +λ +est un morphisme plat. +Démonstration. En restriction à une strate d’invariant λ ∈ Adm(µ)K le carré cartésien +(5.2) devient +ShPR +λ +� +MPR +λ /G +� +ShPEL +λ +� +MPEL +λ +/G +� +p +π + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +35 +qui est lui aussi cartésien. Il suffit de montrer que le morphisme p est plat. Ce dernier +s’inscrit dans un carré cartésien +MPR +λ +MPEL +λ +� +MPR +λ /G +� +� +MPEL +λ +/G +� +p +q +p +Or d’après le théorème 3.7 et (4.8) le morphisme p est localement trivial et donc en +particulier plat (le schéma de base Spec k est un corps). Le morphisme q étant fidèlement +plat, on conclut par [Sta18], Lemme 100.25.4. +□ +Corollaire 6.13. Si ShPEL +λ += � +w∈W Sw est une bonne stratification alors +ShPR +λ += +� +w∈W +π−1(Sw) +est également une bonne stratification. +Démonstration. D’après la proposition précédente le morphisme π est plat en restriction +à une strate ShPEL +λ +et est donc en particulier ouvert. +□ +7. Invariants de Hasse partiels +Nous allons maintenant décrire l’interaction entre la stratification de Hodge définie +dans la section précédente, et les invariants de Hasse partiels définis dans [RX14], lorsque +l’indice de ramification satisfait e ≤ 4. Sans perdre de généralité on peut se restreindre +à une place v|p et donc considérer la situation d’un groupe p-divisible G muni d’une +action d’un anneau d’entier OL où L/Qp est une extension de degré d. Pour faciliter la +lecture, nous allons commencer par un rappel des différentes définitions de loc cit. +7.1. Définitions. +7.1.1. Invariant de Hasse. Soit G un groupe de Barsotti-Tate sur un schéma S de +caractéristique p. Le Verschiebung Ver : G(p) → G induit un morphisme : +Ver∗ : ωG −→ ω(p) +G +En prenant le déterminant de ce morphisme on obtient une section +Ha(G) ∈ H0(S, (det ωG)⊗(p−1)) +(7.1) +appelée invariant de Hasse. On a alors la proposition suivante : +Proposition 7.1. Soit G un groupe p-divisible sur un corps k de caractéristique p > 0. +Alors Ha(G) est inversible si et seulement si G est ordinaire. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +36 +7.1.2. Invariants de Hasse partiels. Soit (G, ι) un groupe p-divisible sur S muni d’une +action ι : OL → EndS(G). On suppose que (G, ι) est muni d’une donnée de Pappas- +Rapoport (Eτ, ωG,τ)τ une donnée de Pappas-Rapoport (voir 5.8). Pour tout 2 ≤ i ≤ e on +définit l’application M(i) +τ +comme étant la multiplication par ̟ au niveau des gradués : +M(i) +τ +: ω(i) +G,τ/ω(i−1) +G,τ +−→ ω(i) +G,τ/ω(i−2) +G,τ +Ce morphisme induit une section appelée invariant de Hasse primitif : +m(i) +τ (G) ∈ H0(S, det (ω[i−1] +G,τ /ω[i−2] +G,τ ) ⊗ det (ω[i] +G,τ/ω[i−1] +G,τ )−1) +Dans le cas Hilbert par exemple, c’est-à-dire le cas où dim(G) = dg = d, on a la +proposition suivante caractérisant le lieu de Rapoport : +Proposition 7.2. On suppose que S = Spec k. Les conditions suivantes sont équiva- +lentes : +(1) Hodgeτ(G, ι) = (e, 0) +(2) m(i) +τ ̸= 0 pour tout 2 ≤ i ≤ e +(3) ωG,τ est un OL ⊗Fp OS-module libre de rang 1. +Démonstration. (1) implique (2) et (3) implique (1) sont évidents. Pour (2) implique +(3), il suffit de prendre un vecteur v ∈ ω(e) +τ \ω(e−1) +τ +, et de voir qu ⟨̟e−1v, . . ., ̟v, v⟩ est +une base de ω(e) +τ . +□ +Remarque 7.3. La même preuve fonctionne également pour le cas Hilbert-Siegel. +Définition 7.4. Soit (G, ι, ω(·) +G,τ) un groupe p-divisible sur un corps k avec donnée de +Pappas-Rapoport. Pour tout 1 ≤ i ≤ e on définit : +Hodge(ω(i) +G,τ) := Hodge(Λω(i) +G,τ) = Inv(Λω(i) +G,τ, Λ0) ∈ X∗(T)+ +via la construction (4.3) (voir la section 6.1 pour la description explicite de Hodge(ω(e) +G,τ)). +Remarque 7.5. Dans la définition ci-dessus on adopte une convention différente que +celle utilisée dans [Bij22]. Plus précisément, si (G, ι, ω(·) +G,τ) un groupe p-divisible sur un +corps k avec donnée de Pappas-Rapoport, alors le faisceau ωi +G,τ est un OL ⊗ k-module +satisfaisant ̟i · ω(i) +G,τ = 0. Via l’identification ̟ �→ u on peut donc le voir comme un +k[u]/(ue)-module ou comme un k[u]/(ui)-module. Dans la construction (4.3) on le voit +comme un k[u]/(ue)-module, alors que dans [Bij22] il est considéré comme un k[u]/(ui)- +module. +Par la suite nous aurons besoin du lemme suivant : +Lemme 7.6. Soit τ ∈ Σnr, 2 ≤ i ≤ e et x ∈ ShPR. Notons (G, ι, ω(·) +G,τ) le groupe +p-divisible avec donnée de Pappas-Rapoport associé. Si m(i) +τ (x) = 0 alors on a l’égalité +Hodge(ω(i) +G,τ) = Hodge(ω(i−2) +G,τ ) − (1, 1) +Démonstration. Par définition si m(i) +τ += 0 on a ω(i) +τ +⊂ ̟−1(ω(i−2) +τ +) (ici on utilise que les +gradués sont de dimension 1). Un simple calcul des dimensions respectives montre que +cette inclusion est une égalité. Le résultat en découle. +□ + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +37 +Remarque 7.7. On fera attention au fait que la réciproque est bien sûr fausse. +Remarque 7.8. On fera également attention au fait que le lemme précédent n’est pas +vrai dans le cas Hilbert-Siegel pour g ≥ 2. Cela vient du fait que le determinant peut +être nul sans que l’application M(i) +τ +soit nulle pour autant. +7.1.3. +Le Verschiebung V : G(p) → G induit une application +Vτ : Eτ → ω(p) +G,σ−1τ = (ω(e) +G,σ−1τ)(p) +(7.2) +(voir 2.1.4). On considère la composée : +Eτ[̟] ≃ Eτ/Eτ[̟e−1] +Vτ +−→ (ω(e) +G,σ−1τ/ω(e−1) +G,σ−1τ)(p) +Autrement dit si x ∈ Eτ[̟] alors x = ̟e−1 · y pour un certains y ∈ Eτ, on lui associe +alors Vτ(y). En restreignant ce morphisme à ω(1) +G,τ ⊂ Eτ[̟] on obtient finalement un +morphisme : +Haτ : ω(1) +G,τ → (ω(e) +G,σ−1τ/ω(e−1) +G,σ−1τ)(p) +(7.3) +et donc une section +haτ(G) ∈ H0(S, det (ω(e) +G,σ−1τ/ω(e−1) +G,σ−1τ)⊗p ⊗ det (ω(1) +G,τ)−1) +également appelé invariant de Hasse partiel. +Remarque 7.9. Pour uniformiser les notations on définit m(1) +τ +:= haτ. +7.2. Stratification. Dans [DK22] les auteurs se sont intéressés à la stratification de la +fibre spéciale induite par ces invariants de Hasse partiels. Plus précisément si on note +ShPR +T += {x ∈ ShPR| mi +τ(x) = 0 ssi (τ, i) ∈ T}, +T ⊂ Σnr × {1, . . ., e} +les sous schémas localement fermés définis comme les lieux d’annulations des invariants +de Hasse partiels d’indice contenu dans T, alors on a le théorème suivant +Théoreme 7.10 ([DK22], Proposition 5.8). Les (ShPR +T )T pour T ⊂ Σnr × {1, . . . , e} +définissent une bonne stratification de ShPR. Plus précisément on a pour tout T ⊂ +Σnr × {1, . . ., e} : +ShPR +T += +� +T ′⊂T +ShPR +T ′ +De plus chacune des strates ShPR +T +est non vide, quasi affine, et équidimensionnelle de +dimension d − |T|. +7.3. Interaction avec la stratification de Hodge : le cas e = 4. On dispose de +deux stratifications de la fibre spéciale du modèle de Pappas Rapoport : +ShPR = +� +λ∈Adm(µ)K +ShPR +λ , +ShPR = +� +T∈T +ShPR +T +où T = P(Σnr×{1, . . . , e}) désigne les sous ensembles de Σnr×{1, . . ., e}. Il est naturel +de se demander dans quelle mesure la décomposition +ShPR = +� +(λ,T)∈Adm(µ)K×T +ShPR +(λ,T), +ShPR +(λ,T) := ShPR +λ +∩ ShPR +T +fourni une bonne stratification. Deux problèmes se posent alors : + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +38 +(1) Quelles sont les strates ShPR +(λ,T) qui sont non vides ? +(2) Si cette stratification est une bonne stratification, quelle est la relation d’ordre +sur les couples (λ, T) qui décrit les relations d’adhérences des strates ? +Pour le point (1), la Proposition 7.2 nous dit que pour λ = (e, 0) alors ShPR +(λ,T) = ∅ si +et seulement si il existe un indice (τ, i) ∈ T ∈ T avec i ≥ 2. De manière générale, +il semble difficile de prédire quelles sont les strates non-vides. Nous allons répondre +à cette question et décrire l’interaction entre le stratification de Hodge et celle par +les invariants (m(i))i dans le cadre d’une extension L/Qp totalement ramifiée de degré +e = 4. L’extension étant totalement ramifiée, nous pouvons omettre l’indice τ et on a +donc T = P({1, . . . , e}). +Définition 7.11. On définit les ensembles : +Tλ = {T ∈ P({2, . . ., e}) | ShPR +(λ,T) ̸= ∅} +∀ λ ∈ Adm(µ)K +(7.4) +Tµ• = {(λ, T) ∈ Adm(µ)K × P({2, . . ., e}) | ShPR +(λ,T) ̸= ∅} +Par définition de ces ensembles on a une décomposition de l’ensemble Tµ• : +Tµ• = +� +λ∈Adm(µ)K +Tλ +Remarque 7.12. La stratification définie par l’ensemble Tµ• est purement « linéaire » +dans le sens où elle est définie par les invariants mi pour i ≥ 2 et par l’invariant de +Hodge, qui sont des invariants « linéaires » (en opposition à l’invariant m1 = ha qui est +un invariant σ-linéaire). +Remarque 7.13. En fait la stratification induite par l’ensemble Tµ• correspond à une +stratification du produit de convolution � +Grµ•. La stratification définie ci-dessus est donc +« linéaire » dans le sens où elle ne dépend que du modèle local MPR. +Définition 7.14. On définit les ensembles suivants : +T (m1=0) +µ• += {(λ, T) ∈ Adm(µ)K × T | ShPR +(λ,T) ̸= ∅, m1 ∈ T} +T (m1̸=0) +µ• += {(λ, T) ∈ Adm(µ)K × T | ShPR +(λ,T) ̸= ∅, m1 /∈ T} +Aµ• = {(λ, T) ∈ Adm(µ)K × T | ShPR +(λ,T) ̸= ∅} +On a par définition de ces ensembles : +Aµ• = T (m1=0) +µ• +� +T (m1̸=0) +µ• +Dans un premier temps nous allons décrire l’interaction entre les strates de Hodge et +les strates définies par les m(i) pour i = 2, 3, 4. Nous intégrerons l’invariant m1 = ha à +notre raisonnement par la suite. Notre problème devient alors simplement un problème +d’algèbre linéaire sur la filtration de Pappas-Rapoport ω(·). Pour alléger les notations +nous noterons X(mi)i∈T +λ += ShPR +(λ,T). + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +39 +7.3.1. Invariant (mi)i≥2. Le schéma ci-dessous décrit les strates non vides et les rela- +tions d’adhérences entre elles avec la convention A → B si A ⊂ B. +X(4,0) +X(m2) +(3,1) +X(m3) +(3,1) +X(m4) +(3,1) +X(m2,m3) +(3,1) +X(m3) +(2,2) +X(m3,m4) +(3,1) +X(m2,m4) +(2,2) +X(m2,m3,m4) +(2,2) +(7.5) +Le problème étant pour le moment purement un problème d’algèbre linéaire, nous +allons utiliser les notations utilisées dans le cadre des grassmanniennes affines (via la +construction (4.3)). +Tout d’abord commençons par prouver l’assertion sur les strates vides. Nous allons +donner tous les détails pour le cas (1). Les autres cas étant très similaire, nous donnerons +seulement les arguments importants. +(1) X(m2,m4) +(3,1) += ∅. A priori on a une décomposition en union disjoint : +X(m2,m4) = X(m2,m4) +(3,1) +∪ X(m2,m4) +(2,2) +(on rappelle que la strate X(m2,m4) +(4,0) +est vide d’après le Lemme 7.2). Soit x ∈ +X(m2,m4). Pour simplifier les notations on note Λ1 ⊂ · · · ⊂ Λ4 la filtration +Λω(1) +x +⊂ · · · ⊂ Λω(4) +x +associée à ce point (voir (4.3)). D’après le Lemme 7.6 +puisque m2(x) = m4(x) = 0 on a +Hodge(Λ4) = Hodge(Λ2) − (1, 1) += Hodge(Λ0) − (1, 1) − (1, 1) += (2, 2) +Par conséquent X(m2,m4) = X(m2,m4) +(2,2) +et X(m2,m4) +(3,1) += ∅. +(2) X(m4) +(2,2) = ∅. Puisque m4 = 0 on a d’après 7.6 : +Hodge(Λ4) = Hodge(Λ2) − (1, 1) += (4, 2) − (1, 1) += (3, 1) +où l’on a utilisé m2 ̸= 0 pour obtenir Hodge(Λ2) = (4, 2). + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +40 +(3) X(m3,m4) +(2,2) += ∅. Même raison que ci-dessus. +(4) X(m2) +(2,2) = ∅. Soit v ∈ ω(4)\ω(3). On a u · v ∈ ω(3)\ω(2) car m4 ̸= 0. Ensuite +puisque m2 = 0 on a ω(2) = E [u] et par conséquent u2 · v ̸= 0. En particulier +Hodge(ω(4)) > (2, 2) +(5) X(m2,m3) +(2,2) += ∅. Même raison que ci-dessus. +Il assez simple de trouver des filtrations adéquates pour prouver que les strates du +schéma ci-dessus sont non vides. En d’autres termes on a calculé les ensembles Tλ pour +tout λ ∈ Adm(µ)K (avec les notations de (7.4)) : +T(4,0) = {∅} +(7.6) +T(3,1) = {(m2), (m3), (m4), (m2, m3), (m3, m4)} +(7.7) +T(2,2) = {(m3), (m2, m4), (m4), (m2, m3, m4)} +(7.8) +Montrons maintenant l’assertion sur les relations d’adhérences. +D’après le Théorème 7.10 les invariants (mi)i forment une bonne stratification de +ShPR. En particulier d’après le Lemme 2.10 cela veut dire que l’on peut « inverser »un +invariant sans toucher aux autres. Plus précisément : +Proposition 7.15. Si x ∈ X(mi)i∈T +λ +avec T ̸= ∅ alors pour tout i0 ∈ T il existe λ′ ∈ +Adm(µ)K et y ∈ X +(mi)i∈T ′ +λ′ +tel que x ∈ {y} où T ′ = T\{i0}. +Démonstration. C’est la combinaison de 7.10 et 2.10. +□ +Le problème est que dans la proposition ci dessus on ne contrôle pas le polygone +de Hodge lors de la déformation. Cependant, nous avons prouver que certaines strates +X(mi)i∈T +λ +étaient vides, et nous pouvons l’utiliser pour décrire le polygone de Hodge lors +des déformations. +Exemple 7.16. Soit x ∈ X(m2,m3,m4) +(2,2) +et soit y ∈ X(m2,m4) +λ +un point tel que x ∈ {y} +(fourni par la proposition précédente). D’après 7.3.1 la strate X(m2,m4) +(3,1) +est vide ce qui +impose l’égalité λ = (2, 2). Cela montre que pour tout point x ∈ X(m2,m3,m4) +(2,2) +il existe un +y ∈ X(m2,m4) +(2,2) +. Autrement dit cela prouve la relation d’adhérence X(m2,m3,m4) +(2,2) +⊂ X(m2,m4) +(2,2) +. +Nous pouvons également utiliser le fait que l’invariant de Hodge ne peut qu’augmenter +par générisation (autrement dit le polygone de Hodge s’abaisse par générisation). Plus +précisément si x ∈ Xλ et y ∈ Xλ′ tel que x ∈ {y} alors λ ≤ λ′, ce qui peut être déduit +du Théorème 6.8 qui est cependant beaucoup plus fort. +Exemple 7.17. Soit x ∈ X(m2,m3) +(3,1) +et soit y ∈ X(m3) +λ +un point tel que x ∈ {y} (fourni +par la proposition précédente). L’invariant de Hodge ne pouvant qu’augmenter par +générisation on a nécessairement Hodge(y) ≥ (3, 1). Le cas (4, 0) étant impossible +d’après 7.3.1 on a nécessairement Hodge(y) = (3, 1). Cela montre que pour tout point +x ∈ X(m2,m3) +(3,1) +il existe un y ∈ X(m3) +(3,1) . Autrement dit cela prouve la relation d’adhérence +X(m2,m3) +(3,1) +⊂ X(m3) +(3,1) . + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +41 +Les deux exemples ci dessus fonctionnent pour toutes les relations d’adhérences à +l’exception de : +X(m2,m3,m4) +(2,2) +⊂ X(m3) +(2,2) , +X(m3) +(2,2) ⊂ X(m3) +(3,1) +Dans le premier cas ce qui pose problème c’est que la strate X(m3) +(3,1) est non vide et par +conséquent le raisonnement ci dessus ne fonctionne pas. Dans le second cas il s’agit de +déformer le polygone de Hodge au sein de la strate définie par l’équation m3 = 0 et +m2, m4 ̸= 0. +(1) X(m2,m3,m4) +(2,2) +⊂ X(m3) +(2,2) . Soit x ∈ X(m2,m3,m4) +(2,2) +et 0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3) ⊂ ω(4) la +filtration associée. Par définition on a les égalités : +ω(2) = E [u] = ω(1) ⊕ ⟨v2⟩, +ω(3) = u−1(ω(1)), +ω(4) = u−1(ω(2)) = E [u2] +pour un certain v2 ∈ ω(2) que l’on fixe. On définit la déformation sur k[[t]] comme +suit : +�E = E ⊗k k[[t]], +�ω(1) = ω(1) ⊗k k[[t]] +�ω(2) = �ω(1) ⊕ ⟨v2 + tv⟩, +�ω(3) = u−1(�ω(1)), +�ω(4) = �E [u2] +pour un certain v ∈ u−1(�ω(1))\ �E [u] que l’on choisit. Par construction en fibre +générique cette filtration satisfait bien les équations m2 ̸= 0, m4 ̸= 0 , m3 = 0 +et Hodge(�ω(4)) = (2, 2). +(2) X(m3) +(2,2) ⊂ X(m3) +(3,1) . Soit x ∈ X(m3) +(2,2) et 0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3) ⊂ ω(4) la filtration +associée. Par définition on a les égalités : +ω(3) = u−1(ω(1)), +ω(4) = E [u2] +Soit v(4) ∈ ω(4) tel que ω(4) = ω(3)⊕⟨v(4)⟩. On choisit un élément v ∈ u−1(ω(3))\E [u2]. +On définit la déformation sur k[[t]] comme suit : +�E = E ⊗k k[[t]], +�ω(i) = ω(i) ⊗k k[[t]], +i = 1, 2, 3 +et +�ω(4) = �ω(3) ⊕ ⟨v(4) + tv⟩ +En fibre générique cette filtration satisfait bien les équations m2m4 ̸= 0 , m3 = 0 +et Hodge(�ω(4)) = (3, 1) car par construction u2(v4 + tv) ̸= 0. +7.3.2. Invariant m1 = ha. Nous allons maintenant nous intéresser à l’invariant m1 = +ha. On aimerait pouvoir « inverser » ha sans modifier les autres invariants de Hasse +partiels et sans modifier l’invariant de Hodge. On a la proposition suivante valable pour +une extension L/Qp totalement ramifiée de degré quelconque : +Proposition 7.18. Soit x ∈ ShPR +(λ,T). On suppose que 1 ∈ T c’est-à-dire que m1(x) = 0. +Alors il existe y ∈ ShPR +(λ,T ′) tel que y ⇝ x où T ′ = T\{1}. +Démonstration. Soit x ∈ ShPR +(λ,T) défini sur un corps k. On note E le cristal associé. Nous +allons construire une déformation sur k[[t]] de ce cristal telle qu’en fibre générique on ait +ha ̸= 0, et sans que les autres invariants ne soient modifiés. Soit F (1) := F((ω(e+1))(p)) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +42 +où ω(e+1) := ̟−1(ω(e−1)). D’après [Bij16] Proposition 3.12, il existe un isomorphisme +E [̟]/F (1) ≃ +� +ω(e)/ω(e−1)�(p) faisant commuter le diagramme suivant : +ω(1) +E [̟]/F (1) +� +ω(e)/ω(e−1)�(p) +≃ +ha +Puisque les gradués sont de dimension 1, la condition ha = 0 est équivalente à l’égalité +ω(1) = F (1) dans E [̟]. Dans tout ce qui suit on notera E(n), F (1) +(n), ω(k) +(n) etc les objets +définis sur Rn = k[t]/(tn). +D’après le lemme 2.7 on dispose d’un relèvement F (1) +(2) sur R2 qui ne dépend pas +du choix des relèvements ω(e) +(2) et ω(e−1) +(2) +sur R2. Soit Λ(1) +(1) ⊂ · · · ⊂ Λ(e) +(1) ∈ � +Grµ•(R1) la +filtration associée à 0 ⊂ ω(1) +(1) ⊂ · · · ⊂ ω(e) +(1) via 4.4. Soit (g1, . . ., ge) ∈ GL2(R1((u))) tel +que : +Λ(i) +(1) = gi · Λ(i−1) +(1) +, +Λ(0) +(1) = ue · Λ0 +Soit ΛF (1) +(2) le R2[[u]]-réseau associé à F (1) +(2) via 4.4. On définit ensuite : +Λ(i+1) +(2) += gi+1 · Λ(i) +(1), +Λ(1) +(2) ̸= ΛF (1) +(2) +où chacun des gi+1 ∈ GL2(R2((u))) est vu ici via la section R1 → R2 et Λ(1) +(2) désigne +n’importe quel R2[[u]]-réseau relevant Λ(1) +(1) et satisfaisant la condition Λ(1) +(2) ̸= ΛF (1) +(2) . Si +on note gF ∈ GL2(R2((u))) le lacet définissant ΛF (1) +(2) alors il suffit par exemple de +choisir Λ(1) +(2) = (g · gF) · ueΛ0 avec g ∈ GL2(R2[[u]]) relevant Id ∈ GL2(R2[[u]]) et g ̸= Id. +Toujours grâce au procédé 4.4 on obtient une filtration 0 ⊂ ω(1) +(2) ⊂ · · · ⊂ ω(e) +(2) satisfaisant +ω(1) +(2) ̸= F (1) +(2) . On continue le processus par récurrence sur n ≥ 2. Supposons donnés +E(n), ω(k) +(n) etc sur Rn. On définit alors la déformation sur Rn+1 comme étant (avec les +notations similaires au cas n = 2) : +Λ(i+1) +(n+1) = gi+1 · Λ(i) +(n+1), +Λ(1) +(n+1) ⊗Rn+1 Rn = Λ(1) +(n) +où Λ(1) +(n+1) désigne n’importe quel relèvement de Λ(1) +(n) sur Rn+1. Cette propriété de relè- +vement nous assure que la condition Λ(1) +(n+1) ̸= ΛF (1) +(n+1) soit satisfaite (car elle l’est après +réduction à Rn). Toujours via le procédé 4.4 on obtient une filtration 0 ⊂ ω(1) +(n+1) ⊂ +· · · ⊂ ω(e) +(n+1) sur Rn+1 satisfaisant ω(1) +(n+1) ̸= F (1) +(n+1). On obtient par passage à la limite +un groupe p-divisible G ∈ BTPR(R) muni d’une action ι : OL → End(G) et d’une filtra- +tion de Pappas-Rapoport. Par Gronthendieck-Messing et Serre-Tate cela nous fournit +un morphisme : +Spec k[[t]] → ShPR + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +43 +En fibre générique la filtration satisfait les propriétés souhaitées : par construction les +positions relatives de la filtration n’ont pas changé et par conséquent les invariants +(mi)i et l’invariant de Hodge n’ont pas changé. Enfin lim F (1) +(n) ̸= lim ω(1) +(n), toujours par +construction, ce qui nous assure qu’en fibre générique on ait bien ha ̸= 0. +□ +Remarque 7.19. Ce qui est assez surprenant dans la preuve ci-dessus est que l’on +change une donnée σ-linéaire (l’invariant m1) via une déformation d’une donnée linéaire +(filtration de Pappas-Rapoport). C’est le Lemme 2.7 qui rend ce processus possible. Si +nous avions voulu déformer directement sur k[[t]], il aurait été difficile de calculer le +faisceau F (1). +Remarque 7.20. La proposition ci-dessus nous dit que l’on peut inverser l’invariant +m1 sans toucher aux autres invariants. C’est une déformation au sein d’une strate fixée +par les invariants (mi)i≥2 et l’invariant de Hodge. +Nous devons maintenant traiter la réciproque : l’équation m1 = 0 est-elle satisfaite +sur la fibre générique des déformations définies en 7.5 ? Autrement dit, peut on modifier +l’invariant de Hodge et les (mi)i tout en satisfaisant ha = 0 ? Pour les mêmes raisons +que dans les exemples 7.16 et 7.17 il suffit de traiter les cas : +X(m1,m2,m3,m4) +(2,2) +⊂ X(m1,m3) +(2,2) +, +X(m1,m3) +(2,2) +⊂ X(m1,m3) +(3,1) +Le problème est que les déformations de 7.3.1 ne sont pas assez précises car on ne +contrôle pas le Frobenius lors de la déformation. Nous allons donc adapter ces déforma- +tions à l’idée de la preuve de la Proposition 7.18 qui est de déformer étape par étape le +long des Rn+1 → Rn et d’utiliser le Lemme 2.7. +(1) X(m1,m2,m3,m4) +(2,2) +⊂ X(m1,m3) +(2,2) +. Soit x ∈ X(m1m2,m3,m4) +(2,2) +et 0 ⊂ ω(1) +(1) ⊂ ω(2) +(1) ⊂ ω(3) +(1) ⊂ +ω(4) +(1) la filtration associée. Par définition on a les égalités : +ω(1) +(1) = F (1) +(1) , +ω(2) +(1) = E(1)[u] +ω(3) +(1) = u−1(ω(1) +(1)), +ω(4) +(1) = u−1(ω(2) +(1)) = E(1)[u2] +On fixe des bases : +⟨v(1) +(1)⟩ = ω(1) +(1), +⟨v(1) +(1), v(2) +(1)⟩ = ω(2) +(1) +Comme dans la preuve de la Proposition 7.18, d’après le Lemme 2.7, on dispose +d’un relèvement F (1) +(2) de F (1) +(1) sur R2. On fixe un relèvement v(1) +(2) sur R2 de v(1) +(1) tel +que F (1) +(2) = ⟨v(1) +(2)⟩ et un élément α(2) ∈ u−1(F (1) +(2) )\E(2)[u]. On choisit également +un élément v(2) +(2) ∈ E(2)[u] qui relève v(2) +(1). On définit alors la déformation sur R2 +comme suit : +E(2) = E ⊗R1 R2 +ω(1) +(2) = ⟨v(1) +(2)⟩ = F (1) +(2) +ω(2) +(2) = ω(1) +(2) ⊕ ⟨v(2) +(2) + tα(2)⟩, +ω(3) +(2) = u−1(ω(1) +(2)), +ω(4) +(2) = E(2)[u2] +Par construction on a bien u · (v(2) +(2) + tα(2)) ̸= 0. On continue le processus +par récurrence sur n ≥ 2 . Supposons donnés E(n), ω(k) +(n) etc sur Rn. D’après le + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +44 +lemme 2.7 on dispose d’un relèvement canonique F (1) +(n+1) sur Rn+1. On définit la +déformation sur Rn+1 comme suit : +E(n+1) = E ⊗R1 Rn+1 +ω(1) +(n+1) = F (1) +(n+1) +ω(3) +(n+1) = u−1(ω(1) +(n+1)), +ω(4) +(n+1) = E(n+1)[u2] +et on prend n’importe quel relèvement ω(2) +(n+1) ⊂ u−1(F (1) +(n+1)) de ω(2) +(n), la condition +ω(2) +(n+1) ̸= E(n+1)[u] étant assurée par cette propriété de relèvement. On obtient +par passage à la limite un groupe p-divisible G ∈ BTPR(R) qui satisfait les +propriétés souhaitées. +(2) X(m1,m3) +(2,2) +⊂ X(m1,m3) +(3,1) +. Soit x ∈ X(m3) +(2,2) et 0 ⊂ ω(1) +(1) ⊂ ω(2) +(1) ⊂ ω(3) +(1) ⊂ ω(4) +(1) la filtration +associée. Par définition on a les égalités : +ω(1) +(1) = F (1) +(1) , +ω(3) +(1) = u−1(ω(1) +(1)), +ω(4) +(1) = E(1)[u2] +Soit v(4) +(1) ∈ ω(4) +(1) tel que ω(4) +(1) = ω(3) +(1) ⊕ ⟨v(4) +(1)⟩. Comme dans la preuve de la Propo- +sition 7.18, d’après le Lemme 2.7, on dispose d’un relèvement F (1) +(2) de F (1) +(1) sur +R2. On définit la déformation sur R2 comme suit : +E(2) = E ⊗R1 R2 +ω(1) +(2) = F (1) +(2) , +ω(3) +(2) = u−1(ω(1) +(2)) +Ensuite on choisit un élément α(2) ∈ u−1(ω(3) +(2))\E(2)[u2] et un élément v(4) +(2) ∈ +u−1(ω(3) +(2) ∩ E(2)[u] qui relève v(4) +(1). On définit alors : +ω(4) +(2) = ω(3) +(2) ⊕ ⟨v(4) +(1) + tα(2)⟩ +On continue le processus par récurrence sur n ≥ 2 . Supposons donnés E(n), ω(k) +(n) +etc sur Rn. D’après le lemme 2.7 on dispose d’un relèvement canonique F (1) +(n+1) +sur Rn+1. On définit la déformation sur Rn+1 comme suit : +E(n+1) = E ⊗R1 Rn+1, +ω(1) +(n+1) = F (1) +(n+1), +ω(3) +(n+1) = u−1(ω(1) +(n+1)) +et on prend n’importe quel relèvement ω(2) +(n+1), ω(4) +(n+1). Les propriétés ω(2) +(n+1) ̸= +E(n+1)[u] et ω(2) +(n+1) ̸= E(n+1)[u2] sont assurées par cette propriété de relèvement. +On obtient par passage à la limite un groupe p-divisible G ∈ BTPR(R) qui +satisfait les propriétés souhaitées. +Avant de conclure, il reste à calculer l’ensemble Aµ• des strates non vides. C’est +l’objet du lemme suivant : +Lemme 7.21. Soit (λ, T) ∈ Tµ•. On a : +Sh(m1=0) +(λ,T) +̸= ∅, +Sh(m1̸=0) +(λ,T) +̸= ∅ +En d’autres termes l’ensemble des strates non vides est donné par : +Aµ• = Tµ• ∪ (Tµ• × {m1}) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +45 +Remarque 7.22. Avec des mots plus concrets, le lemme ci dessus, combiné aux pro- +position précédentes, dit que l’invariant σ-linéaire m1 interagit naïvement avec la stra- +tification définie par l’ensemble Tµ• (linéaire). +Démonstration. D’après la proposition 7.18 et les calculs de déformations précédents +il suffit de montrer que la strate X(m1,m2,m3,m4) +(2,2) +est non vide. Nous allons utiliser les +travaux de Goren et Andreatta [AG03]. Soit x ∈ ShPEL +(2,2) un point défini sur un corps k +algébriquement clos et G le groupe p-divisible associé. D’après la Proposition 4.10 de +loc cit il existe une base (e1, e2) de E (G) telle que la filtration de Hodge soit donnée +par : +ωG = ⟨u2e1, u2e2⟩ +et que le Frobenius soit donné dans cette base par : +F = +� +um +cu2 +u2 +0 +� +où m ≥ 2 et c ∈ (OL ⊗ k)×. On définit la filtration de Pappas Rapoport comme suit : +ω(1) = ⟨u3e2⟩, +ω(2) = ⟨u3e1, u3e2⟩, +ω(3) = ⟨u3e1u2e2⟩, +ω(4) = ⟨u2e1, u2e2⟩ +Cela définit un point ˜x ∈ ShPR. Un simple calcul montre que pour cette filtration on +a : +F (1) := F(u−1(ω(3))(p)) = ⟨cu3e2⟩ +En particulier ω(1) = F (1) et par conséquent m1(˜x) = 0. Il est également simple de voir +que m2(˜x) = m3(˜x) = m4(˜x) = 0. Par conséquent ˜x ∈ X(m1,m2,m3,m4) +(2,2) +et cette dernière +strate est donc non vide. +□ +Théoreme 7.23. Pour L/Qp totalement ramifiée de degré e = 4 la stratification : +ShPR = +� +(λ,T)∈Aµ• +ShPR +(λ,T) +est une bonne stratification où les relations d’adhérences sont données par la relation +d’ordre naïve sur Aµ• +8. Conjecture +8.1. Conjecture. Une fois la stratification par le polygone de Hodge de ShPEL réin- +terprétée via le morphisme lisse +ShPEL → Hecke = +� +L+G\Gr +� +il est naturel de stratifier le modèle de Pappas-Rapoport ShPR via le morphisme lisse +ShPR → Heckeµ• = +� +L+G\ � +Grµ• +� +Le problème est alors de décrire les L+G-orbites dans � +Grµ•. Dans cet article nous nous +sommes intéressé au morphisme de convolution +m : � +Grµ• → Gr≤|µ•| + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +46 +Ce morphisme étant L+G-équivariant les strates (m−1(Grλ))λ≤|µ•| sont des unions de +L+G-orbites dans � +Grµ•. Si la preuve de la proposition 3.16 semble être « à la main »et +si elle ne se généralise pas à d’autres situations (voir [BH22b] Proposition 3.9) c’est +parce qu’elle n’est pas naturelle : la stratification la plus naturelle est celle induite par +la décomposition en orbites. +Conjecture 8.1. Il existe un ensemble fini partiellement ordonné (X, <) décrivant les +L+G-orbites dans � +Grµ•. En d’autres termes on dispose d’un homéomorphisme +|Heckeµ•| ≃ X +où X est muni de la topologie induite par <. +On consultera [PWZ12] (§2.1, §2.2) pour plus de détails concernant la stratification +d’un espace en orbites sous l’action d’un groupe. +8.2. Le cas de petites dimensions. Dans [Bij22], S.Bijakowski a répondu positive- +ment à la conjecture ci-dessus dans le cas Unitaire et dans le Hilbert-Siegel lorsque e ≤ 3 +. Énonçons le résultat dans le cas Hilbert totalement ramifié d’indice de ramification +e = 3 +Théoreme 8.2 ([Bij22]). La classe d’isomorphisme d’une filtration de Pappas-Rapoport +(0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3)) est entièrement déterminée par les invariants Hodge(ω(3)), +Hodge(ω(2)) et Hodge(ω(3)/ω(1)). De plus ces invariants définissent une bonne stratifi- +cation de ShPR où la relation d’ordre est la relation naïve : (λ1, λ2, λ3) ≤ (λ′ +1, λ′ +2, λ′ +3) si +λi ≤ λ′ +i pour i = 1, 2, 3. +Faisons quelques commentaires : +(1) Dans le théorème ci-dessous, le fait que les classes d’isomorphismes induisent une +bonne stratification est automatique : les classes d’isomorphismes de filtration +de Pappas-Rapoport correspondent aux L+G-orbites dans � +Grµ•. Le résultat non +trivial du théorème ci-dessus concernant la stratification réside dans le calcul +explicite des relations d’adhérences entres orbites. +(2) Dans le cas Hilbert, c’est-à-dire dans le cas G = GL2 et µ• = (1, 0)e les gradués +Λ(i)/Λ(i−1) sont de dimension 1 et on obtient donc les équivalences 1 : +Hodge(Λ(2)) = (3, 3) ⇔ Hodge(Λ(2)/Λ(0)) = (2, 0) ⇔ m2 ̸= 0 +Hodge(Λ(3)/Λ(1)) = (2, 0) ⇔ m3 ̸= 0 +En d’autres termes dans le cas e = 3 la donnée (Hodge(ω), m3, m2) détermine +entièrement la classe d’isomorphisme. En fait on peut être plus précis : (m3, m2) +détermine Hodge(ω) car dans le cas e = 3 il y a que deux possibilités (2, 1) et +(3, 0). +(3) Pour le cas e = 4 nous avons vu que les strates X(m3) +(2,2) et X(m3) +(3,1) étaient non vides. +Dans cette situation les invariants (mi)i ne déterminent donc pas l’invariant de +Hodge, et donc en particulier ils ne détectent pas la classe d’isomorphisme non +plus. +1. On utilise la notation abusive Hodge(Λ(i)/Λ(i−2)) = Inv(Λ(i), Λ(i−2)) + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +47 +(4) Dans le cas Hilbert-Siegel, c’est-à-dire pour G = GSp2g, les invariants ci-dessous +ne sont pas assez fins pour détecter la classe d’isomorphisme. En fait dans le cas +général on a l’équivalence suivante : +Hodge(Λ(i)/Λ(i−2)) = (2, 0)g ⇔ mi ̸= 0 +En d’autres termes le lieu de non annulation de l’invariant mi coïncide avec +la strate maximale de la stratification par l’invariant Hodge(Λ(i)/Λ(i−2)). Ce +dernier est donc le bon invariant à considérer en dimension supérieure. + +STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION +48 +Références +[AG03] Fabrizio Andreatta and Eyal Z. Goren. Geometry of Hilbert modular varieties over totally +ramified primes. International Mathematics Research Notices, 2003 :1785–1835, 2003. +[BBM82] Pierre Berthelot, Lawrence Breen, and William Messing. Théorie de Dieudonné Cristalline +II. 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Ekedahl–Oort and Newton strata for Shimura varieties +of PEL type. Mathematische Annalen, 356(4) :1493–1550, jan 2013. +[YCO20] Chia-Fu Yu, Ching-Li Chai, and Frans Oort. Stratifying Lie Strata of Hilbert Modular Va- +rieties. Taiwanese Journal of Mathematics, 24(6) :1307 – 1352, 2020. +[Zhu16] X. Zhu. An introduction to affine Grassmannians and the geometric Satake equivalence. 2016. +https://arxiv.org/abs/1603.05593. +[Zin02] Thomas Zink. The display of a formal p-divisible group. Astérisque, pages 127–248, 2002. + diff --git a/5tE4T4oBgHgl3EQfbwzK/content/tmp_files/load_file.txt b/5tE4T4oBgHgl3EQfbwzK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7e0c1d4e7934a8e6eba55f4923b598def0c3319 --- /dev/null +++ b/5tE4T4oBgHgl3EQfbwzK/content/tmp_files/load_file.txt @@ -0,0 +1,1530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf,len=1529 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='05078v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='AG] 12 Jan 2023 STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION DIEGO BERGER Résumé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans ce papier on étudie la géométrie de la fibre spéciale des modèles de Pappas-Rapoport des variétés de Shimura dans le cas Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément on prouve que la stratification induite par le polygone de Hodge est une bonne stratifi- cation, ce qui est faux dans le cas Hilbert-Siegel pour g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ensuite on utilise des résultats connus sur le produit de convolution de grassmanniennes affines pour décrire la dimension des strates et obtenir que le morphisme d’oubli vers le modèle PEL de Kottwitz est plat en restriction aux strates de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Table des matières 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Préliminaires 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Grassmanniennes Affines 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèles locaux 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèles entiers 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Stratification de Hodge 28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariants de Hasse partiels 35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Conjecture 45 Références 48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’étude de la géométrie de la réduction modulo p des variétés de Shimura est un vaste sujet qui a eu de nombreuses conséquences arithmétiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans cet article nous nous concentrons sur les variétés modulaires de Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce sont des variétés sur Qp qui peuvent être vues comme des espaces de modules de variétés abéliennes munies de certaines données.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose de plusieurs modèles sur Zp de ces variétés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le modèle de Kottwitz, noté ShPEL, est défini comme un prolongement sur Zp du problème de module initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément ShPEL est défini comme l’espace de module des quadruplés (A, λ, ι, κ) où A est une variété abélienne de dimension d > 0, λ : A → At est une Z× (p)-polarisation, ι : OF → End(A) une action d’un anneau d’entiers d’un corps de nombre totalement réel F/Q sur A tel que [F : Q] = d, et κ une structure de niveau en dehors de p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Lorsque l’extension F/Q est ramifiée, le modèle ShPEL n’est plus lisse sur Spec Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pappas et Rapoport ont défini dans [PR02] un modèle lisse ShPR sur Spec OK où K est une extension contenant les clôtures galoisiennes de toutes 1 STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 2 les extensions Fv où v|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce modèle est défini via une résolution des singularités du modèle local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est un espace de module classifiant les 5-uplés (A, λ, ι, κ, Fil(ω)) où (A, λ, ι, κ) est un quadruplé de ShPEL et Fil(ω) est une filtration du faisceau conormal satisfaisant certaines propriétés (voir 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8 pour plus de détails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’un des objectifs dans l’étude de la géométrie de la réduction modulo p des variétés de Shimura est de définir des stratifications, dont les strates possèdent de bonnes propriétés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On peut trouver dans la littérature de nombreux travaux sur les stratifications de la réduction modulo p du modèle ShPEL (voir par exemple [VW13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Concernant le modèle ShPR plusieurs stratifications ont été définies dans [DK22], [RX14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans cet article nous allons nous intéresser à la stratification de Hodge de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Celle-ci est définie via le polygone de Hodge du faisceau conormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Principaux résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour ne pas alourdir les notations nous allons énoncer les principaux résultats de cet article dans le cadre d’une extension L/Qp totalement ramifiée (l’extension L/Qp jouant le rôle de Fv/Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Mis à part le Théorème 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3, les résultats qui suivent sont valables et sont démontrés dans le cadre générale d’une ex- tension totalement réelle F/Q sans condition sur la ramification en p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit L/Qp une extension totalement ramifiée de degré e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note ShPEL la fibre spé- ciale du modèle de Kottwitz et ShPR celle du modèle de Pappas-Rapoport des variétés modulaires de Hilbert (voir les Définitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’oubli de la filtration induit un morphisme : π : ShPR −→ ShPEL On dispose d’une stratification appelée stratification de Kottwitz-Rapoport (KR) du modèle PEL : ShPEL = � λ∈Adm(µ)K ShPEL λ où Adm(µ)K désigne l’ensemble µ-admissible (voir la Remarque 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans le cas Hilbert la stratification (KR) de ShPEL coïncide avec la stratification par le polygone de Hodge (voir Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il est naturel de se demander si cette stratification induit une (bonne) stratification (voir Définition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8) de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On prouve dans cet article que la réponse est oui : Théorème 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Les strates (ShPR λ )λ∈Adm(µ)K forment une bonne stratification de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autre- ment dit pour tout λ ∈ Adm(µ)K on a la relation d’adhérence ShPR λ = � λ′≤λ ShPR λ′ (2) Pour tout λ ∈ Adm(µ)K la strate ShPR λ est quasi-projective lisse de dimension ⟨ρ, |µ•| + λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce théorème est surprenant car il est faux dans des cas plus généraux que le cas Hilbert (voir [BH22b]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La preuve du point (2) repose sur les travaux de Haines ([Hai06]) sur la géométrie du morphisme de convolution dans le cadre des Grassmanniennes STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 3 affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ses travaux nous permettent également de prouver le théorème suivant qui concerne la géométrie du morphisme π : Théoreme 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout λ ∈ Adm(µ)K la restriction π : ShPR λ → ShPEL λ est un morphisme plat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les travaux de Haines n’ayant pas de restriction sur le groupe réductif, le théorème ci-dessus est en fait valable dans le cas Hilbert-Siegel plus général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans [RX14] les auteurs ont défini des invariants de Hasse partiels (mi)1≤i≤e et dans [DK22] il est prouvé que les sous schémas localement fermés définis comme les lieux d’annulation de ces sections définissent une bonne stratification de ShPR : ShPR = � T⊂{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=',e} ShPR T où ShPR T est le sous schéma localement fermé défini par l’annulation des (mi)i∈T et l’inversibilité des (mi)i/∈T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il est alors naturel d’étudier l’interaction entre la stratification par le polygone de Hodge, et celle définit par ces invariants de Hasse partiels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout λ ∈ Adm(µ) et tout T ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e} on note ShPR (λ,T) l’intersection de la strate ShPR λ et de la strate ShPR T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On montre le résultat suivant dans le cas e = 4 : Théoreme 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour L/Qp totalement ramifiée de degré e = 4 la stratification : ShPR = � (λ,T)∈Aµ• ShPR (λ,T) est une bonne stratification où les relations d’adhérences sont données par la relation d’ordre naïve sur Aµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans le théorème ci-dessus l’ensemble Aµ• désigne le sous ensemble des couples (λ, T) tels que la strate ShPR (λ,T) soit non vide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il semble difficile de calculer ce sous ensemble en général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remerciements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Je tiens à remercier Stéphane Bijakowski de m’avoir encouragé à écrire cet article et de m’avoir expliqué les résultats de déformations de la première section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Je tiens également à remercier Benoit Stroh, Thibault Alexandre et Arnaud Eteve pour toutes les discussions qui m’ont aidé à écrire cet article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Enfin je tiens à remercier Xu Shen pour ses commentaires et remarques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Préliminaires Dans tout ce qui suit on fixe un nombre premier p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette première partie est composée essentiellement de rappels sur la théorie de Dieu- donné cristalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous présentons tous les outils nécessaires pour pouvoir déformer un groupe p-divisible le long d’un morphisme k[[t]] → k où k est un corps de caractéristique p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous aurons besoin de la théorie de Dieudonné cristalline [BBM82], de la théorie des display de Zink et Lau [Zin02], [Lau09], [Lau14] et de Grothendieck-Messing [Mes72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait, nous pourrions nous contenter de la théorie des display car elle englobe les anneaux locaux complet de corps résiduels parfait comme kperf[[t]] (où kperf désigne la perfection de k), ce qui nous suffit amplement pour nos problèmes de déformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Théorie de Dieudonné cristalline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit k un corps parfait de caractéristique p et S un schéma sur k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit A → S un schéma abélien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note G = A[p∞] le groupe p-divisible associé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Σ = Spec (W(k)), et G le faisceau sur le site cristallin Cris(S/Σ) induit par G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En suivant les notations de [BBM82] on note : E (G) := E xt1 S/Σ(G, OS/Σ) le cristal de Dieudonné contravariant de G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est un OS/Σ-cristal localement libre de rang h où h désigne la hauteur de G ([BBM82], Corollaire 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez qu’avec cette convention, les applications : F : E (G)(p) → E (G), V : E (G) → E (G)(p) (où ( · )(p) désigne le twist par le Frobenius) sont induites respectivement par : F : G → G(p), V : G(p) → G Si l’on évalue ce cristal sur l’épaississement (S id −→ S) on obtient une filtration de OS- modules localement libres ([BBM82], Corollaire 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5) 0 −→ ωG −→ E (G) (S id−→S) −→ ω∨ GD −→ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) appelée filtration de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [BBM82] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 on dispose d’un iso- morphisme E xt1 S/Σ(A, OS/Σ) ≃ E xt1 S/Σ(G, OS/Σ) reliant le cristal de A/S et celui de son groupe p-divisible, compatible aux filtrations de Hodge respectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En combinant maintenant avec l’isomorphisme ([BBM82], Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8) : E xt1 S/Σ(A, OS/Σ)(S→S) ≃ H1 dR(A/S) on retrouve la filtration de Hodge induit par la suite spectrale de Hodge bien connue : 0 −→ ωA −→ H1 dR(A/S) −→ ω∨ At −→ 0 STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plaçons nous maintenant dans le cas où S = Spec(k) est le spectre d’un corps k parfait de caractéristique p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note : D(G) := HomS(G, CW) le module de Dieudonné contravariant au sens de Fontaine ([Fon77] ou [BBM82] Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En évaluant notre cristal le long de l’épaississement (W(k) ։ k) on obtient un isomorphisme de W(k) module compatible avec F et V des deux cotés : E (G)(W (k)։k) ≃ D(G)(p) (voir [BBM82] Théorème 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La réduction modulo p de cet isomorphisme permet l’identification de la p-torsion : E xt1 S/Σ(G[p], OS/Σ) ≃ D(G[p])(p) ≃ (D(G)/pD(G))(p) où E xt1 S/Σ(G[p], OS/Σ) est le cristal associé au groupe fini plat de p-torsion G[p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition E (G) étant un cristal, il commute aux changement de bases et par conséquent la réduction modulo p ci dessus redonne l’identification bien connue entre le module de Dieudonné de la p-torsion et le premier groupe de cohomologie de De Rham : E (G)(W (k)։k) ⊗W (k) k ≃ E (G)(k→k) ≃ H1 dR(A/k) Toujours d’après [BBM82] (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10) on dispose d’un isomorphisme : ω(p) G ≃ E (G)(S→S)/F(E (G)(p) (S→S)) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On s’intéresse au cas où A/S est muni d’une action de OL où L/Qp est une extension de degré fini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [Far06] on dispose du résultat suivant : Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 ([Far06], Lemme B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit E un F-cristal en OS/Σ-modules loca- lement libre de rang fini sur Cris(S/Σ) muni d’une action de OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors E est un OS/Σ ⊗Zp OL-module localement libre sur Cris(S/Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [BBM82] on dispose d’un isomorphisme : ωG ≃ E xt1(G, JS/Σ)(S→S) où JS/Σ est le faisceau d’idéal à puissances divisées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Or E xt1(G, JS/Σ) n’est à priori pas un cristal et donc la proposition ci-dessus ne s’applique pas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les morphismes F et V de G induisent un diagramme commutatif aux lignes horizontales exactes : 0 ωG E (G)(S→S) ω∨ GD 0 0 ω(p) G E (G)(p) (S→S) (ω∨ GD)(p) 0 0 ωG E (G)(S→S) ω∨ GD 0 V V V F F F STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 6 Les composées verticales sont nulles car égales à p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque F est nul sur ω(p) G le mor- phisme F se factorise en : F : (ω∨ GD)(p) −→ E (G)(S→S) De même V est nul sur ω∨ GD (utiliser que VG = F D GD) donc on obtient une factorisation : V : E (G)(S→S) → ω(p) G Nous aurons besoin de cette factorisation pour définir l’invariant de Hasse primitif dans la section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Déformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Grothendieck-Messing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant rappeler quelques résultats sur les déformations des groupes de Barsotti-Tate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit S0 ֒→ S une immersion fermée nilpotente munie d’une structure de puissances divisées (PD-structure), avec p localement nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose d’un morphisme sur le site Cris(S0/Σ) : (S0 → S0, γ0) −→ (S0 ֒→ S, γ) Si F est un cristal en OS0/Σ-module alors par définition on dispose d’un isomorphisme de OS0-modules canonique : F(S0֒→S) ⊗OS OS0 ≃ F(S0→S0) On note BT(S) la catégorie des groupes de Barsotti-Tate sur S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G ∈ BT(S) un tel groupe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note E (G) le cristal associé et G|S0 = G ×S S0 le changement de base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors un isomorphisme canonique de OS-module : E (G)(S→S) ≃ E (G|S0)(S0֒→S) En effet d’un coté on dispose d’un isomorphisme : E (G|S0)(S0֒→S) ≃ E (G)(S0֒→S) provenant essentiellement de la définition du cristal E (G) ([BM79], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' De l’autre coté puisque E (G) est un cristal, le morphisme (S0 → S) → (S → S) dans Cris(S/Σ) induit un isomorphisme : E (G)(S0֒→S) ⊗OS OS = E (G)(S→S) et le résultat en découle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cet isomorphisme est compatible aux filtrations de Hodge : ωG E (G)(S→S) E (G|S0)(S0֒→S) ωG|S0 E (G|S0)(S0→S0) E (G|S0)(S0֒→S) ⊗OS OS0 ≃ ≃ dans le sens où le diagramme est commutatif et la ligne du bas correspond à la réduction de la ligne du haut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 7 Définition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G0 ∈ BT(S0) un groupe de Barsotti-Tate sur S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une filtration Fil1 ⊂ E (G0)(S0֒→S) est dite admissible si c’est un OS-module localement facteur direct de E (G0)(S0֒→S) qui relève ωG0 ⊂ E (G0)(S0→S0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après ce qui précède si G ∈ BT(S) relève G0 ∈ BT(S0) alors ωG ⊂ E (G)(S→S) ≃ E (G|S0)(S0֒→S) est une filtration admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note DefBT(S0 ֒→ S) la catégorie dont les objets sont les couples (G0, Fil1) où G0 ∈ BT(S0), Fil1 ⊂ E (G0)(S0֒→S) est une filtration admissible et où les morphismes (G0, Fil1) → (G′ 0, Fil1′) sont les morphismes G0 → G′ 0 compatibles avec les filtrations respectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On peut maintenant énoncer le théorème de Grothendieck-Messing : Théoreme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4 ([Mes72] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le foncteur BT(S) −→ DefBT(S0 ֒→ S) G �−→ � G|S0, ωG ⊂ E (G|S0)(S0֒→S) � est une équivalence de catégorie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans cette section nous allons voir comment appliquer le théorème de déformation de Grothendieck-Messing le long de l’immersion fermée k[[t]] → k, qui n’est pas munie de PD-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela nous sera utile lorsque nous voudrons calculer des relations d’adhérences entres strates de nos variétés de Shimura (voir Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit R un anneaux de caractéristique p et I un idéal tel que I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On peut munir I d’une PD-structure faisant du morphisme Spec R/I → Spec R une immersion fermée nilpotente avec PD-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En effet il suffit de poser γ1(x) = x et γn(x) = 0 pour tout n ≥ 2 et tout x ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons appliquer cette remarque à la suite d’immersions fermées : · · → k[t]/(tn+1) → k[t]/(tn) → k[t]/(tn−1) → · · · → k[t]/(t2) → k Pour tout n ≥ 1 on note Rn = k[t]/(tn), In = Ker � k[t]/(tn) → k[t]/(tn−1) � et Sn = Spec Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Chaque immersion fermée Sn−1 ֒→ Sn est définie par un idéal In = (tn−1)/(tn) satisfaisant I2 n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après ce qui précède on peut donc munir chacune de ces immer- sions fermées d’une PD-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le résultat suivant fonctionne pour tout anneau R local complet de corps résiduel parfait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5 ([Lau09], Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G1 ∈ BT(k) et Fil1 ⊂ E (G1)(k→k) ⊗k k[[t]] un relèvement de ωG1 ⊂ E (G1)(k→k) qui est localement un facteur direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors il existe G ∈ BT(k[[t]]) tel que : (1) G est un relèvement de G1 le long de Spec k → Spec k[[t]] (2) E (G)(k[[t]]→k[[t]]) ≃ E (G)(k→k) ⊗k k[[t]] (3) ωG ≃ Fil1 (via l’identification ci dessus) Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons utiliser la théorie des display de Zink et Lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Posons R = k[[t]] et S = Spec R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez que se donner un groupe p-divisible G ∈ BT(S) c’est se donner un système compatible de groupes p-divisibles (Gn)n≥1 où Gn ∈ BT(Sn) ([Lau14] STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 8 Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons donc construire pour tout n ≥ 2 un groupe p-divisible Gn ∈ BT(Sn) satisfaisant les propriétés souhaitées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque pour tout n ≥ 2 l’anneau Rn est un anneau local artinien sur lequel p est nilpotent et de corps résiduel parfait, on dispose donc d’une équivalence de catégorie BT(Sn) ∼= Disp(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note P1 = (P1, Q1, F1, V −1 1 ) le display associé à G1 par Lau ([Lau14], Théorème A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Toujours d’après loc cit, Théorème A, on dispose d’un isomorphisme canonique entre le cristal de Dieudonné de G1 et le cristal associé à P1 par Zink ([Zin02] ou [Lau14]) : E (G1) ≃ D(P1) Le cristal D(P1) étant défini sur le site Crisadm(R) ⊂ Cris(R) des épaississements avec pd-structure (B → A, δ) avec A admissible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' si le nilradical NR est « bounded nil- potent » et que Rred = R/NR est un anneau parfait de caractéristique p), l’isomorphisme ci-dessus est un isomorphisme de cristaux sur le site Crisadm(R) et on fait ici l’abus de notation d’également noter E (G1) la restriction du cristal E (G1) au site Crisadm(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier en évaluant sur l’épaississement tautologique (S1 → S1) on obtient un isomorphisme canonique de R1-modules : E (G1)(S1→S1) ≃ P1 ⊗W(R1) R1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) D’après la remarque précédente, R2 → R1 est muni d’une PD-structure nilpotente, et par conséquent se donner un relèvement P2 ∈ Disp(R2) de P1 c’est se donner un relèvement Fil1 (2) ⊂ D(P1)(R2։R1) (voir [Lau14], Corollaire 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10, ou [Lau09] Lemme 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On pose P2 = P1 ⊗W(R1) W(R2) (notez qu’on dispose pour tout n ≥ 2 d’un morphisme R1 → Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [Lau14], Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6, on a par construction du cristal D(P1) un isomorphisme : D(P1)(R2։R1) ≃ P2 ⊗W(R2) R2 = P1 ⊗W(R1) R2 Via l’isomorphisme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) on obtient : E (G1)(S1→S1) ⊗R1 R2 ≃ D(P1)(R2։R1) On définit le relèvement de la filtration de Hodge de P1 comme étant : Fil1 (2) := Fil1 ⊗R R2 ⊂ E (G1)(S1→S1) ⊗R1 R2 ≃ D(P1)(R2։R1) Ce relèvement définit une display P2 ∈ Disp(R2) et donc un groupe p-divisible G2 ∈ BT(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose de nouveau d’une identification entre cristaux E (G2) ≃ D(P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque G2 est un relèvement de G1 le long de S1 ֒→ S2 on a : E (G2)(S2→S2) ≃ E (G1)(S1֒→S2) ≃ D(P1)(R2։R1) ≃ E (G1)(S1→S1) ⊗R1 R2 ≃ E ⊗R R2 où l’on a posé E := E (G1)(S1→S1) ⊗R1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On continue le processus par induction en posant pour tout n ≥ 3 : Pn := P1 ⊗W(R1) W(Rn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Fil1 (n) := Fil1 ⊗R Rn ⊂ D(Pn−1)(Rn։Rn−1) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 9 ce qui nous fournit à chaque étape un display Pn ∈ Disp(Rn) et un groupe p-divisible Gn ∈ BT(Sn) satisfaisant les équations : ωGn ≃ Fil1 (n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' E (Gn)(Sn→Sn) ≃ E ⊗R Rn Par passage à la limite on obtient un groupe p-divisible G ∈ BT(S) satisfaisant par construction : E (G)(S→S) ≃ lim ←− n E ⊗R Rn ≃ E ωG ≃ lim ←− n Fil1 (n) ≃ lim ←− n Fil1 ⊗R Rn ≃ Fil1 ce qui démontre les points (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) En fait la proposition ci-dessus est une reformulation en termes de cristaux de Dieudonné des lemmes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='15 et 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16 de [Lau14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette reformulation est possible grâce à l’identification entre le cristal de Dieudonné E (G) et le cristal associé à un display D(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous aurions très bien pu nous passer de la théorie de Dieudonné cristalline et simplement utiliser la théorie des displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) L’un des apports de la théorie des displays dans la preuve ci-dessus est qu’elle rend explicite le calcul du faisceau E (Gn)(Rn+1։Rn) : il suffit de prendre n’importe quel Pn+1 qui relève le display Pn (voir [Zin02] Théorème 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela découle du fait que le morphisme de « frame »(DRn+1/Rn → DRn) est cristallin (voir [Lau14] Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la preuve ci-dessus à chaque étape nous avons défini la display Pn+1 comme étant le changement de base de P1 le long de la section R1 → Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La proposition ci-dessus n’est donc qu’un passage à la limite d’un fait bien connu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Énonçons pour finir un lemme dont nous aurons besoin dans la section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 : Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit S0 = Spec R/I ֒→ S = Spec R une immersion fermée telle que I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G0 ∈ BT(S0) un groupe p-divisible sur S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit Fil1 (1), Fil1 (2) ⊂ E (G0)(S0֒→S) deux filtrations admissibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans E (G0)(p) (S0֒→S) on dispose de l’égalité : (Fil1 (1))(p) = (Fil1 (2))(p) En particulier le faisceau ω(p) G ne dépend pas du relèvement G ∈ BT(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est un simple résultat d’algèbre commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On pose M = E (G0)(S0֒→S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est un R-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque E est un cristal on dispose de l’identification E (G0)(S0→S0) = M/IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit donc N1, N2 ⊂ M tels que N1/IM = N2/IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On veut montrer que N(p) 1 = N(p) 2 ⊂ M ⊗R,σ R où σ : R → R désigne le Frobenius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Mais c’est clair puisque 0 = IM(p) ⊂ M ⊗R,σ R (voir que im ⊗ 1 = m ⊗ σ(i) et que σ(i) = 0 car par hypothèse I2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Relation d’adhérence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit S un espace topologique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une stratification de S par rapport à un ensemble partiellement ordonné (I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ≤) est une décomposition : S = � i∈I Si telle que pour tout i ∈ I on ait la relation d’adhérence : Si ⊂ � j≤i Sj Une stratification est appelée bonne stratification si de plus elle satisfait pour tout i ∈ I : Si = � j≤i Sj Dans cette partie nous allons énoncer un résultat dont nous aurons besoin par la suite lorsque nous nous intéresserons aux relations d’adhérences entre strates de nos variétés de Shimura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce résultat est certainement bien connu et est notamment utiliser dans [BH22a] (Preuve du théorème 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11) mais à défaut d’avoir trouvé une preuve, nous allons en proposer une ci-dessous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit X un un schéma noethérien de caractéristique p > 0 et Y ⊂ X un sous schéma localement fermé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les conditions suivantes sont équivalentes : (1) x ∈ Y (2) Il existe y ∈ Y tel que y ⇝ x (3) Il existe une extension de corps k/κ(x) et un morphisme de schémas Spec k[[t]] → X qui envoi le point fermé sur x et le point générique sur y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3) ⇒ (1) découle de la continuité du morphisme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) ⇒ (2) découle du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 ci dessous et du fait que l’espace topologique sous-jacent à un schéma soit sobre et qu’un espace topologique noethérien sobre soit spectral (dans un espace topologique noethérien tout sous-ensemble est quasi-compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Montrons que (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11 ci dessous, il existe un anneau de valuation discrète R satisfaisant les propriétés du point (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le théorème de structure de Cohen nous assure que �R ≃ k[[t]] où �R désigne la complétion le long de l’idéal maximal m ⊂ R et k = R/m désigne le corps résiduel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le morphisme Spec �R → X satisfait les propriétés souhaitées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit X un espace topologique spectral, et Y ⊂ X un sous ensemble constructible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors Y = � y∈Y {y} Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Voir [Sta18] Lemme 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit X un schéma noethérien et y ⇝ x une spécialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors il existe un anneau de valuation discrète R et un morphisme Spec R → X qui envoie le point fermé sur x et le point générique sur y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Voir [Sta18] Lemme 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Grassmanniennes Affines 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Grassmanienne affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le début de cette section est constitué essentiellement de rappels sur la géométrie des grassmanniennes affines que l’on peut retrouver dans [Zhu16] par exemple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G un groupe réductif connexe déployé lisse sur un corps k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe un tore déployé T ⊂ G sur k et un borel B le contenant On notera ⟨, ⟩ : X∗(T) × X∗(T) → Z le produit scalaire, X∗(T)+ l’ensemble des cocaractères B-dominants et 2ρ ∈ X∗(T) la somme des racines positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Algk la catégorie des k-algèbres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit le groupes de lacets LG et le groupe d’arcs L+G comme les foncteurs Algk → Sets qui à une k algèbre R associent : LG(R) = G(R((u))), L+G(R) = G(R[[u]]) On définit la grassmannienne affine pour le groupe G comme le quotient (pour la topo- logie étale ou fppf puisque G est lisse) GrG = LG/L+G On montre que ce quotient est ind-représentable par un schéma propre sur Spec k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En utilisant le fait que G soit supposé lisse on peut montrer que l’on a en fait une description explicite de ce quotient : GrG(R) = � (E, β) ����� E est un G torseur sur DR, β : E|D∗ R ≃ E0|D∗ R est une trivialisation � où DR = Spec R[[u]] désigne le disque unité, D∗ R = Spec R((u)) le disque unité épointé et E0 le G-torseur trivial sur DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la définition ci dessus E est un G-torseur pour la topologie fppf ou étale (encore une fois puisque G est supposé lisse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par la suite nous noterons E ��� E0 la trivialisation β : E|D∗ R ≃ E0|D∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par la suite nous aurons besoin du lemme facile suivant : Lemme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G1, G2 deux groupes réductifs connexes déployés lisses sur k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose d’un isomorphisme canonique GrG1×G2 ≃ GrG1 × GrG2 Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il suffit de voir que le résultat est vrai au niveau des foncteurs : L(G1 × G2)(R) = (G1 × G2)(R((u))) = G1(R((u))) × G2(R((u))) □ On dispose d’une action à gauche : L+G × GrG −→ GrG (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' β)) �−→ (E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' g · β) dont le quotient est appelé champs de Hecke HeckeG = [L+G\\LG/L+G] STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 12 Par construction il associe à une k algèbre R HeckeG(R) = � (E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' E′β) ����� E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' E′ sont des G torseurs sur DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' β : E|D∗ R ≃ E′|D∗ R est un isomorphisme � Désormais nous ferons l’abus de notation de retirer l’indice G et de noter Hecke et Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On rappelle que par la décomposition de Cartan, on peut associer à une modification β : E ��� E′, sa position relative Inv(β) ∈ X∗(T)+ via la bijection G(k[[u]])\\G(k((u)))/G(k[[u]]) −→ X∗(T)+ [g] �−→ Inv(g) [uλ] ←−� λ où uλ = λ(u) ∈ T(k((u))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout élément λ ∈ X∗(T)+ on définit Grλ := {(E, β) ∈ Gr | Inv(β) = λ } , Gr≤λ := {(E, β) ∈ Gr | Inv(β) ≤ λ } De la même manière on définit Heckeλ et Hecke≤λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La proposition suivante résume la plupart des propriétés de la décomposition de Gr en L+G-orbites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Pour tout λ ∈ X∗(T)+ on a Grλ = L+G · uλ est une L+G- orbite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Grλ est une variété quasi-projective lisse de dimension ⟨2ρ, λ⟩ (3) On dispose d’une décomposition de Gr en L+G-orbites Gr = � λ∈X∗(T)+ Grλ (4) Pour tout λ ∈ X∗(T)+ on a la relation d’adhérence Grλ = � λ′≤λ Grλ′ (5) L’ouvert dense Grλ ⊂ Gr≤λ coïncide avec le lieu lisse de Gr≤λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les points (1), (2), (3), (4) sont démontrés dans [Zhu16] (Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour (5) on pourra trouver une preuve dans [MOV03] (Corollary B) □ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La proposition précédente nous fournit une description de l’espace topologique sous-jacent au champs de Hecke borné : |Hecke≤λ| ≃ {λ′ ∈ X∗(T)+ | λ′ ≤ λ} L’identification ci-dessus est un homéomorphisme où la topologie du membre de droite est celle induite par la relation d’ordre sur X∗(T)+ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si µ ∈ X∗(T)+ est minuscule alors on dispose d’une identification Grµ ≃ G/Pµ où Pµ désigne le sous groupe parabolique associé à µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En effet on dispose de deux morphismes dont on montre qu’ils sont réciproques l’un de l’autre : Grµ −→ G/Pµ g · uµ �−→ [ev(g)] , G/Pµ −→ Grµ [g] �−→ g · uµ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 13 où ev : L+G → G, g �→ g (mod u) et G ֒→ L+G désigne le groupe des lacets « constants ».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Produit de convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit le produit de n-convolution Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr comme étant le champ paramétrant les modifications (βi : Ei ��� Ei−1)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='n : En ��� En−1 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ��� E1 ��� E0 On dispose d’un morphisme : Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr −→ Gr (En ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ��� E0) �−→ (En ��� E0) appelé morphisme de convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce procédé fournit également pour tout 1 ≤ i ≤ n un morphisme : mi : Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr −→ Gr (En ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ��� E0) �−→ (Ei ��� E0) Ces morphismes mis ensemble nous donnent un isomorphisme : n � i=1 mi : Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr ≃ Gr × · · · × Gr En particulier le produit de convolution Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr est ind-représentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' De la même manière que pour Gr, on dispose d’une uniformisation du produit de convolution : Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr ≃ LG ×L+G · · · ×L+G LG ×L+G Gr Via cet isomorphisme le morphisme de convolution devient le morphisme de multipli- cation : LG ×L+G · · · ×L+G LG ×L+G Gr −→ LG/L+G (gn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , [g1]) �−→ [gn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' g1] Par la suite pour alléger les notations et lorsque le nombre n est explicite nous note- rons le produit de n-convolution � Gr = Gr˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout n-uplet de cocaractères λ• = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λn) on définit les sous schémas localement fermés du produit de convolu- tion : � Grλ• := � (Ei, βi) ∈ � Gr | Inv(βi) = λi � , � Gr≤λ• := � (Ei, βi) ∈ � Gr | Inv(βi) ≤ λi � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) En particulier � Grλ• est représentable par un schéma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il n’est pas difficile de montrer qu’on dispose alors d’une bonne stratification pour tout µ• ∈ (X∗(T)+)n : � Gr≤µ• = � λ•≤µ• � Grλ• où λ• ≤ µ• ⇔ λi ≤ µi ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez que si chacun des µi est minuscule cette stratification est constituée d’une seule strate à savoir � Gr≤µ• = � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par la suite si λ• = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λn) est un n-uplet de cocaractères on notera |λ•| := λ1 + · · · + λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' � Grµ• est lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier si µi est minuscule pour tout i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='n, alors � Gr≤µ• est lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 14 Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On commence par regarder le morphisme de projection � Gr(µ1,µ2) −→ Grµ1 (E2 ��� E1 ��� E0) �−→ (E1 ��� E0) En termes de groupes de lacets, ce morphisme correspond à la projection sur la première coordonnée : LGµ1 ×L+G Grµ2 → Grµ1 où LGµ1 = p−1(Grµ1) ⊂ LG avec p : LG → Gr la projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition ce mor- phisme est une fibration avec pour fibre Grµ2 (identification non canonique) et est donc lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Enfin puisque Grµ1 est lisse, il s’en suit que � Gr(µ1,µ2) l’est également.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le résultat s’en déduit par récurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier si µ• = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' µn) avec chacun des µi minuscule, alors le morphisme de convolution mµ• : � Gr≤µ• → Gr≤|µ•| est une résolution des singularités.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Elle est parfois appelé résolution de Demazure en référence à [Dem74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans le cas G = GL2 et µi = (1, 0), la preuve précédente nous dit que le produit de convolution est obtenu par des fibrations successives en Grµi = G/Pµi ≃ P1 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le théorème suivant décrit la géométrie du morphisme de convolution : Théoreme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='Haines, [Hai06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit µ• = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' µn) un n-uplet de cocaractères quelconques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors (1) Le morphisme mµ• : � Gr≤µ• → Gr≤|µ•| est localement trivial en restriction à Grλ ⊂ Gr≤|µ•| pour tout λ ≤ |µ•|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Si chacun des µi est minuscule alors pour tout y ∈ Grλ la fibre m−1 µ• (y) est équidimensionnelle de dimension ⟨ρ, |µ•| − λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le point (1) correspond au Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 de [Hai06] et le point (2) cor- respond au Théorème 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 de loc cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le point (1) découle directement d’un fait plus général : si p : X → Y est un morphisme G-équivariant tel que G agit transitivement sur Y , alors en choisissant un point de base y0 ∈ Y on obtient un isomorphisme G-équivariant : G ×H p−1(y0) ≃ X où H = StabG(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans notre situation le morphisme m : m−1(Grλ) → Grλ est bien L+G équivariant et Grλ est une L+G-orbite par définition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Exemples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En utilisant le fait qu’un GLn-torseur E sur DR (où R est une k- algèbre) correspond à un R[[u]]-module Λ localement libre de rang n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' on obtient une description de la grassmannienne affine pour GLn en termes de réseaux : Gr(R) = � Λ ⊂ R((u))n ����� Λ R[[u]]-module localement libre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Λ ⊗R[[u]] R((u)) ≃ R((u))n � STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 15 En prenant T = Gn m le tore des matrices diagonales la décomposition de Cartan prend la forme : (Zn)+ −→ GLn(k[[u]])\\GLn(k((u)))/GLn(k[[u]]) λ = (λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λn) �−→ uλ = diag(uλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' uλn) En termes de réseaux cela se traduit comme suit : pour tout réseau Λ ⊂ k((u))n il existe une base (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , en) de Λ0 = k[[u]]n et λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λn) ∈ (Zn)+ tels que (uλ1e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', uλnen) soit une base de Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ici on a utilisé la notation (Zn)+ = {(λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λn) | λ1 ≥ · · · ≥ λn } Désormais nous noterons Λ0 := k[[u]]n le réseau associé au GLn-torseur trivial E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Exemple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans tout ce qui suit R désigne une k-algèbre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Voici quelques exemples : (1) Pour µ = (1d, 0n−d) on obtient en termes de réseaux : Gr(1d,0n−d)(R) = {Λ ⊂ Λ0 := R[[u]]n | uΛ0 ⊂ Λ ⊂ Λ0, dimkΛ0/Λ = d } On retrouve bien la variété GLn/Pµ = Grass(n − d, d) via : Λ �→ (Rn = Λ0/uΛ0 → Λ0/Λ) (2) Pour GL2 et µ = (e, 0) on trouve : Gr≤(e,0)(R) = {Λ | ueΛ0 ⊂ Λ ⊂ Λ0, dimkΛ0/Λ = e } Soit Λ ∈ Gr≤(e,0)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il existe un plus petit entier i ≤ e tel que uiΛ ⊂ ueΛ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notons N(Λ) ce nombre (notation non standard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On obtient alors la description suivante des différentes strates Gr(i,e−i)(R) = {Λ | N(Λ) = i, dimkΛ0/Λ = e } (3) On s’intéresse au produit de e-convolution pour le groupe GL2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour des coca- ractères µi = (1, 0) on a la description suivante Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0)(R) = {Λe ⊂ · · · ⊂ Λ1 ⊂ Λ0 | uΛi ⊂ Λi−1, dimkΛi/Λi−1 = 1 } Le morphisme de convolution prend la forme m : Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0) −→ Gr≤(e,0) (Λe ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Λ1 ⊂ Λ0) �−→ Λe (4) On peut donner une autre interprétation du produit de convolution précédent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit M(R) = � Λ1 ⊂ · · · ⊂ Λe ⊂ Λ0 ����� ∀ i < e : uΛi ⊂ Λi−1, dimkΛi/Λi−1 = 1, Λe ∈ Gr≤(e,0) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) On dispose d’un isomorphisme Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0) −→ M (Λe ⊂ · · · ⊂ Λ1 ⊂ Λ0) �−→ (ue−1Λ1 ⊂ · · · ⊂ uΛe−1 ⊂ Λe) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 16 Cet isomorphisme s’insère dans le diagramme commutatif suivant Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0) M Gr≤(e,0) π m (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4) où π : (Λ1 ⊂ · · · ⊂ Λe ⊂ Λ0) �→ Λe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans les exemples ci-dessus nous avons fait quelques abus de no- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si R ∈ Algk est une k-algèbre et (E, β) ∈ Gr(R), alors Inv(β) n’est pas bien défini : la position relative n’est définie qu’en un point x ∈ Spec R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier nous aurions dû adopter la notation plus rigoureuse : Gr≤µ(R) = {(E, β) | Invx(β) ≤ µ pour tout x ∈ Spec R} Nous n’avons donc pas nécessairement Invx(β) = Invx′(β) pour x ̸= x′ ∈ Spec R (prendre par exemple Spec R non connexe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (Non standard) Soit Λ ⊂ k((u))n un réseau défini par un point x ∈ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit : Hodge(x) = Hodge(Λ) := Inv(Λ, Λ0) ∈ X•(T)+ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Concrètement pour GLn si Λ ⊂ k((u))n est un réseau alors pour N assez grand uNΛ0 ⊂ Λ et l’invariant Hodge(Λ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , an) est caractérisé par : Λ/uNΛ0 ≃ n � i=1 k[u]/(uN−ai) Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si (Λ1 ⊂ · · · ⊂ Λe) est un point de � Grµ• alors on notera Hodge(Λk) = Inv(Λk, Λ0) ∈ X•(T)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Exemple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les invariants de Hodge de la filtration : Λ1 = ⟨u2e1, u3e2⟩ ⊂ Λ2 = ⟨u2e1, u2e2⟩ ⊂ Λ3 = ⟨u2e1, ue2⟩ sont : Hodge(Λ1) = (3, 2), Hodge(Λ2) = (2, 2), Hodge(Λ3) = (2, 1) Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette notation prendra sens lorsque nous aurons relié la stratification de Hodge de notre variété de Shimura à celle de la Grassmannienne affine (voir 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On en vient maintenant au principal résultat de cet article : Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si µi = (1, 0) pour tout i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e, alors {m−1(Grλ)}λ≤|µ| défi- nit une bonne stratification de Grµ1 ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Grµe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En d’autres termes si on note Xλ := m−1(Grλ) alors pour tout λ ≤ |µ•| Xλ = � λ′≤λ Xλ′ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 17 Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons démontrer l’assertion pour l’espace de module M de l’exemple (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9) (équation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2)) car c’est cet espace de module que nous allons considérer par la suite dans le cadre des modèles entiers des variétés de Shimura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Bien sûr pour obtenir le résultat pour Grµ1 ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Grµe il suffit de réécrire la preuve ci dessous en appliquant l’isomorphisme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) et d’utiliser la commutativité du diagramme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit λ = (i, j) < (e, 0) = |µ•| (pour λ = (e, 0) il n’y a rien à démontrer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ Xλ un point de corps résiduel k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notons Λ1 ⊂ · · · ⊂ Λe la filtration associée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (vk)k⩾1 des éléments tels que Λk = Λk−1 ⊕(k · vk) (somme directe en tant que k-espaces vectoriels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit des entiers (sk)k pour tout k ≥ 1 sk = min{s | usΛk ⊂ ueΛ0} = N(Λk) où N(Λk) est l’entier définit en (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9) et Λ0 = k[[u]]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a Hodge(Λk) = (e − k + sk, e − sk) On note k0 = max{k|sk = sk−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez que k0 ̸= 0 car on a supposé (i, j) < (e, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' No- tons (a, b) = Hodge(Λk0) avec a ⩾ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors par hypothèse (a+1, b) = Hodge(Λk0−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (e1, e2) une base de Λ0 telle que : Λk0−1 = ⟨ua+1e1, ube2⟩ Dans cette base on peut écrire vk0 = xuae1 + yub−1e2 On a nécessairement y = 0 car sinon on aurait usk0 ·vk0 /∈ ueΛ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par conséquent on peut écrire vk0 = uae1, Λk0 = Λk0−1 ⊕ (k · uae1) = ⟨uae1, ube2⟩ Ensuite par définition pour tout n ≥ 1 on a : u · vk0+n ∈ Λk0+n−1 = Λk0−1 n−1 � ℓ=0 (k · vk0+ℓ) On peut donc fixer une décomposition par récurrence : vk0+n = wn u + n−1 � ℓ=0 xn,ℓ vk0+ℓ u , wn ∈ Λk0−1, xn,ℓ ∈ k Nous allons maintenant définir une déformation sur R = k�t� de notre filtration initiale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit J := { n | xn,n−1 = 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On travaille dans (k�t�⊗k�u�)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On commence par déformer trivialement la filtration pour tout ℓ ≤ k0 − 1 : ˜Λℓ := Λℓ ⊗ k�t� ∀ℓ ≤ k0 − 1 Pour ℓ = k0 on définit ˜vk0 = uae1 + tub−1e2 = vk0 + tub−1e2 et on pose ˜Λk0 = ˜Λk0−1 ⊕ (k · ˜vk0) Ensuite on déforme par récurrence sur n en fonction de si n ∈ J ou n /∈ J : STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 18 (1) (n ∈ J) Dans ce cas on définit ˜vk0+n = vk0+n + t˜vk0+(n−1) u (2) (n /∈ J) Dans ce cas on définit ˜vk0+n = wn u + n−1 � ℓ=0 xn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='ℓ ˜vk0+ℓ u Dans les deux situations on définit la déformation de Λk0+n comme étant : ˜Λk0+n = ˜Λk0+(n−1) ⊕ (k · ˜vk0+n) Il faut vérifier que l’équation u · ˜vk0+n ∈ ˜Λk0+(n−1) est bien satisfaite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour la situation (2) c’est évident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour la situation (1) il faut voir que par hypothèse on a u · ˜vk0+n = wn + n−2 � ℓ=0 xn,ℓ˜vk0+ℓ + t˜vk0+(n−1) ∈ ˜Λk0+(n−2) ⊕ (k · ˜vk0+(n−1)) = ˜Λk0+(n−1) Au point fermé t = 0 on a ˜vk0 = vk0 et par suite ˜vk0+n = vk0+n pour tout n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par conséquent cette filtration correspond bien à une déformation de notre filtration initiale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour calculer Hodge(˜Λe ⊗ k((t))) il faut voir que par construction on a ˜sk0+n = ˜sk0+n + 1 ∀n ≥ 1 et que par conséquent puisque ˜sk0 = sk0 + 1 on trouve ˜se = ˜sk0+e−k0 = ˜sk0 + e − k0 = sk0 + 1 + e − k0 = se + 1 On a donc bien en fibre générique : Hodge(˜Λe ⊗ k((t))) = (i + 1, j − 1) □ Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’idée de la preuve est la suivante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Si Λe = ⟨uie1, uje2⟩ alors on aimerait déformer sur k�t� en prenant l’élément ˜ve = uie1 + tuj−1e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le problème est que cet élément ne satisfait pas nécessairement u · ˜ve ∈ Λe−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il faut donc déformer Λe−1 également.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le problème est que cette déformation doit de nouveau satisfaire l’équation u · ˜Λe−1 ⊂ Λe−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Il existe un rang k0 tel que là déformation ˜Λk0 existe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit on peut trouver un élément ˜vk0 de la « bonne valuation », c’est-à-dire celle de vk0 moins 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3) Ensuite on déforme par récurrence les vk0+n en à « divisant par u »à chaque étape de sorte à faire apparaître du uj−1 dans la décomposition de ˜ve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Donnons un exemple explicite de déformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On considère la filtration Λ1 = ⟨u3e1, u2e2⟩, Λ2 = ⟨u2e2, u2e1⟩, Λ3 = ⟨ue2, u2e1⟩ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 19 On représente cette filtration par une matrice \uf8ee \uf8ef\uf8f0 u2e1 ue1 e1 u2e2 ue2 e2 v1 ∗ v2 ∗ v3 ∗ \uf8f9 \uf8fa\uf8fb La multiplication par u consiste à décaler les colonnes vers la gauche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La déformation construit dans la preuve précédente est donnée par la matrice : \uf8ee \uf8ef\uf8f0 u2e1 ue1 e1 u2e2 ue2 e2 v1 ∗ v2 ∗ t2 v3 t3 ∗ t3t2 \uf8f9 \uf8fa\uf8fb Remarque 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le théorème ci-dessus n’est pas vrai dans le cas général (voir [BH22b] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9 pour un contre exemple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait dans la preuve ci dessus on utilise un fait spécifique au cas G = GL2 : si Λ ∈ Gr≤(e,0) alors avec les notations de 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9 : Λ ∈ Gr(i,j) ⇔ N(Λ) = i L’invariant N(Λ) est égal à l’indice de nilpotence de u ∈ End(Λ/ueΛ0) ce qui rend le calcul de Hodge(Λ) = Inv(Λ, Λ0) beaucoup plus simple à calculer en pratique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La preuve du Théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16 consiste simplement à déformer une filtration Λ1 ⊂ · · · ⊂ Λe de sorte à faire apparaître le bon indice de nilpotence en fibre générique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèles locaux 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit F un corps totalement réel de degré d > 1 sur Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note OF son anneau d’entiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout v|p on note ev l’indice de ramification et fv le degré résiduel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Fv la complétion de F en v et Ov son anneau d’entiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note F nr v la sous extension maximale non ramifiée et Onr v son anneau d’entiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit K/Qp une extension qui contient tous les plongement Fv → Qp pour tout v|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note OK son anneau d’entiers, k son corps résiduel, et on fixe une uniformisante ̟ ∈ OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose d’une décomposition OF ⊗Z OK ∼= � v|p � τ∈Σnr v Ov ⊗Onr v ,τ OK (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) où Σnr v = Hom(F nr v , Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si on fixe une uniformisante ̟v de Ov alors on peut identifier Ov ⊗Onr v ,τ k ≃ k[u]/(uev) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) On obtient donc une décomposition non canonique : OF ⊗Z k ≃ � v|p � τ∈Σnr v k[u]/(uev) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèle local PEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le groupe réductif qui nous intéresse est : G = ResF/QGL2 Lorsqu’on le change de base à K on obtient une décomposition G ⊗Q K = � v|p � τ∈Σnr v (ResFv/F nr v GL2) ⊗Qp K Pour simplifier les notations nous allons dans un premier temps décrire le modèle local PEL pour le groupe G = ResL/LnrGL2 avec L/Qp une extension finie d’indice de ramification e (jouant le rôle de Fv/Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe un plongement τ : Lnr ֒→ Qp et on note Στ = HomLnr(L, Qp) l’ensemble des plongements qui prolongent τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe une numérotation Στ ≃ {ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', ϕe}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit V un L-espace vectoriel de dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe une base (e1, e2) de V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Λ le OL-module libre de base (e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le modèle local PEL pour le groupe G noté MPEL, est le schéma sur Spec Onr L représentant le foncteur qui à (S → Spec Onr L ) associe l’ensemble MPEL(S) des OL ⊗Onr L OS-sous modules F ⊂ ΛS := Λ ⊗Onr L OS tels que F est localement sur S (pour la topologie Zariski) un OS-facteur direct de ΛS de rang e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout a ∈ OL on a l’égalité polynomiale suivante det(a | F) = � ϕi∈Στ ϕi(a) On note G = AutOL(Λ) le schéma en groupe sur Spec Onr L des automorphismes de Λ compatibles avec l’action de OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors la proposition suivante : Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' G est lisse sur Spec Onr L Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Voir [RZ96] (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ En fait la proposition ci dessus est démontrée pour G = AutOL((Λ)i∈I) où (Λ)i∈I est une chaîne périodique de réseaux (voir [RZ96]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette situation apparaît lorsque l’on autorise du niveau en p, ce qui n’est pas notre cas ici.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans notre situation ce groupe est en fait explicite : G = ResOL/Onr L GL2 et est bien sûr lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèle local de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On considère maintenant le foncteur MPR qui à un schéma (S → Spec OK) associe l’ensemble MPR(S) des filtrations (F (i))i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=',e de OL ⊗OLnr OS-sous modules de ΛS : 0 = F (0) ⊂ F (1) ⊂ · · · ⊂ F (e) ⊂ ΛS telles que : Les F (i) sont Zariski-localement des OS-facteurs directs de ΛS de rang i Pour tout a ∈ L et pour tout i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e (a ⊗ 1 − 1 ⊗ ϕi(a)) · F (i) ⊂ F (i−1) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 21 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce foncteur est représentable par uns schéma projectif sur Spec OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il suffit de voir que l’on peut le plonger dans un produit de Grassman- niennes convenables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément, on peut utiliser 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 et le fait que le produit de convolution soit représentable (voir 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ On dispose d’un morphisme d’oubli π : MPR −→ MPEL ⊗Onr L OK (F (i)) �−→ F (e) Désormais pour alléger les notations nous noterons de la même manière MPEL = MPEL ⊗Onr L OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plongement dans les grassmanniennes affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant dé- crire les plongements des fibres spéciales des modèles MPR et MPEL dans certaines grass- manniennes affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On suit presque à la lettre [PR02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note M PEL = MPEL ⊗OK k et M PR = MPR ⊗OK k les fibres spéciales de ces deux modèles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Comme en (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) le choix d’une uniformisante ̟ ∈ OL nous fournit une identification : OL ⊗Zp k ≃ k[[u]]/(ue), ̟ ⊗ 1 �→ u Cela induit un isomorphisme de OL ⊗Zp k-modules : Λ ⊗Zp k ≃ Λ0 ⊗k[[u]] k[[u]]/(ue) (on rappelle que Λ ⊂ V est un OL-réseau fixé (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) et que Λ0 = k[[u]]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note p la projection : p : Λ0 → Λ0 ⊗k[[u]] k[[u]]/(ue) Soit (S → Spec k) un schéma et F ∈ M PEL(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’identification précédente permet de voir F comme un sous module de Λ0 ⊗k[[u]] OS[[u]]/(ue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit le OS[[u]]-module ΛF comme étant ΛF := p−1 � F ⊂ Λ0 ⊗k[[u]] OS[[u]]/(ue) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) On obtient donc finalement par ce procédé une immersion fermée M PEL ֒↛ Gr Remarque 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par construction on dispose des inclusions de réseaux ueΛ0,S ⊂ ΛF ⊂ Λ0,S dont les gradués sont des OS-modules localement libres de rang e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez que l’image de cette immersion est entièrement caractérisée par les inclusions ci-dessus et le rang des gradués.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’immersion fermée ι est équivariante pour l’action de G⊗k à gauche et L+G à droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Elle induit un isomorphisme M PEL ≃ � λ⩽(e,0) Grλ Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Découle de la remarque précédente et de l’exemple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 22 Remarque 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En d’autres termes l’ensemble des copoids µ-admissibles (voir [Goe01], section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) est ici tout à fait explicite Adm(µ)K = {λ ⩽ (e, 0)} (Ici on est dans le cas parahorique maximal K = G(Zp)) De la même manière on dispose d’une immersion fermée pour le modèle de Pappas- Rapoport dans un produit de grassmanniennes affines : ιPR : M PR −→ Gr × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' × Gr (F (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', F (e)) �−→ (ΛF (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , ΛF (e)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4) Remarque 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par construction on dispose des inclusions ueΛ0 ⊂ ΛF (1) ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ⊂ ΛF (e) ⊂ Λ0 dont les gradués sont localement libres de rang 1 pour j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e − 1 et de rang e pour j = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après l’exemple 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9 cette immersion fermée induit un isomorphisme M PR ≃ M où M est l’espace de module décrit en (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’immersion fermée ιPR induit un isomorphisme équivariant pour l’action de G à gauche et L+G à droite : M PR ≃ Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0) où le produit de convolution est pris e fois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Découle de la remarque précédente en composant avec l’isomorphisme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le carré suivant est cartésien M PR Gr(1,0) ˜× .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ˜×Gr(1,0) M PEL Gr≤(e,0) π m Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il s’agit essentiellement de montrer que le diagramme suivant est com- mutatif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela découle du fait que le diagramme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4) est commutatif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cas général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Revenons maintenant au cas général où l’on considère une exten- sion F/Q de degré d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note G = ResF/QGL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout v|p et tout τ ∈ Σnr v , on note MPEL v,τ le modèle local pour le groupe ResFv/F nr v GL2 définit dans la section précédente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèles locaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit V un F-espace vectoriel de dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe une base (e1, e2) de V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Λ le OF-module libre de base (e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe (à conjugaison près) un cocaractère µ : Gm,C → GC, et on suppose que ce dernier induit une décomposition en espaces propres : V ⊗ C = V0 ⊕ V1 STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 23 Enfin on suppose que le corps de définition de µ est Q (ces hypothèses seront vérifiées dans le cadre des variétés de Shimura de type Hilbert).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit le modèle local MPEL comme étant le foncteur qui à (S → Spec Zp) associe l’ensemble MPEL(S) des OF ⊗Zp OS-sous modules F ⊂ ΛS := Λ ⊗Zp OS tels que F est Zariski-localement sur S un OS-facteur direct de ΛS de rang [F : Q] = d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout a ∈ OF on a l’égalité polynomiale suivante (condition de Kottwitz) : det(a | F) = det(a | V0) On note G = AutOL(Λ) le schéma en groupe sur Spec Zp des automorphismes de Λ compatibles avec l’action de OF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors la proposition suivante : Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Après changement de base à OK, on dispose d’un isomorphisme canonique MPEL ⊗Zp OK ≃ � v|p � τ∈Σnr v MPEL v,τ Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Découle de (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Comme précédemment on note G = AutOF (Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors pour les mêmes raisons une décomposition G ⊗ Zp ≃ � v|p ResOv/ZpGL2 En particulier G ⊗ Zp est bien lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette décomposition suggère la définition suivante pour le modèle de Pappas-Rapoport dans le cas général d’une extension F/Q Définition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le modèle local de Pappas-Rapoport pour le groupe G = ResF/QGL2 est : MPR = � v|p � τ∈Σnr v MPR v,τ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plongements dans les grassmanniennes affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La « compatibilité »des grass- mannienne affines avec le produit vu au Lemme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 combinée aux décompositions des modèles locaux de la section précédente nous donne sans trop d’efforts les isomorphismes non canoniques (dépend des différents choix d’uniformisantes) : M PEL ≃ � v|p � τ∈Σnr v Gr≤(ev,0), M PR ≃ � v|p � τ∈Σnr v � Gr(ev,0) où � Gr(ev,0) désigne le produit de convolution de ev copie de Gr(1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Finalement le carré cartésien de la proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8 devient M PR � v|p � τ∈Σnr v � Gr(ev,0) M PEL � v|p � τ∈Σnr v Gr≤(ev,0) π (mv,τ )v,τ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 24 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèles entiers 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèle entier PEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On garde les notations de 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 et 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On considère l’espace de module ShPEL sur Spec Zp qui à un schéma localement noethérien (S → Spec Zp) associe l’ensemble des quadruplés (A, λ, ι, η) à Z× (p)-isogénie où : (1) A → S est un schéma abélien de dimension g = d = [F : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) λ : A → A∨ est une Z× (p)-polarisation (3) η est une structure de niveau en dehors de p (voir [Lan13], section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) (4) ι : OF ֒→ End(A) ⊗Z Z(p) morphisme respectant les involutions des deux cotés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (5) (A, λ, ι, η) satisfait la condition de déterminant de Kottwitz : det(a | ωA/S) = det(a | V0) ∀ a ∈ OF (voir [Lan13] Définition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 pour plus de détails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors le résultat suivant dû à Mumford puis Kottwitz : Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 ([MFK94], [Kot92] section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ShPEL est représentable par un schéma quasi-projectif sur Spec Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’espace de module définit ci-dessus est celui définit par Kottwitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On consultera l’article de I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='Vollaard [Vol03] pour plus de détails concernant l’équivalence des différents espaces de modules considérés dans le cas Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant expliquer le lien entre le modèle entier ShPEL est le modèle local MPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On considère le foncteur ShPEL,□ qui à un schéma localement noethérien (S → Spec Zp) associe les 5-uplés (A, λ, ι, η, γ) où γ : H1 dR(A/S) ≃ Λ ⊗Zp OS est une trivialisation du premier groupe de cohomologie de deRham (en tant que OF ⊗Zp OS-module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notez que ShPEL,□ est muni d’une action de G = AutOF (Λ) où ce dernier agit sur la trivialisation γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’oubli de la trivialisation γ fournit un morphisme ϕ : ShPEL,□ → ShPEL L’un des résultats majeurs de [RZ96] est le suivant Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3 ([RZ96] Théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ϕ : ShPEL,□ → ShPEL est un G-torseur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La trivialisation γ nous fournit un morphisme vers le modèle local PEL ψ : ShPEL,□ −→ MPEL (A, λ, ι, γ) �−→ γ(ωA/S) ⊂ ΛS Le fait que ce morphisme soit bien défini découle essentiellement de la définition de l’espace de module ShPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors la proposition suivante qui résulte de [dJ91] et [RZ96] : Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ψ : ShPEL,□ −→ MPEL est un morphisme lisse G-équivariant de dimension relative dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 25 Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le fait que le morphisme soit G-équivariant est évident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La lissité dé- coule de Grothendieck-Messing, voir [dJ91] (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5 du document .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='dvi du même nom sur sa page web).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ En d’autres termes on dispose d’un diagramme de modèle local au sens de [RZ96] : ShPEL,□ ShPEL MPEL ϕ ψ Ce qui correspond à un morphisme de champs algébriques ShPEL → � MPEL/G � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) lisse de dimension relative dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Modèle entier de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit le modèle entier de Pappas- Rapoport ShPEL comme le produit cartésien : ShPR � MPR/GOK � ShPEL OK � MPEL OK /GOK � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) où l’indice (·)OK désigne le changement de base ⊗ZpOK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Avant de rendre la définition de ce modèle explicite, mentionnons un corollaire direct de la Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 : Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ShPR est représentable par un schéma quasi projectif lisse sur Spec OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque le carré (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) est cartésien il suffit de montrer que MPR est représentable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela découle de la proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le même raisonnement et le fait que le morphisme de convolution soit propre montrent que ShPR est quasi projectif puisque ShPEL l’est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour la lissité, il suffit de voir que le modèle local MPR est lisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [PR02] Théorème 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3 le modèle local est plat par conséquent il suffit de montrer que la fibre spéciale est lisse (la fibre générique étant toujours lisse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La lissité de MPR découle de la Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait la preuve du théorème 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3 de [PR02] repose sur un diagramme de torseurs reliant le modèle local MPR à un produit de modèles locaux non ramifiés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans le cas PEL sans niveau en p, ces derniers sont lisses (et pas seulement plats) ce qui donne directement la lissité sur Spec OK, sans passer par la fibre spéciale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Donnée de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant donner une définition plus explicite du modèle entier de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour ce faire nous allons devoir définir la notion de « donnée de Pappas-Rapoport »pour un groupe p-divisible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour simplifier les notations, comme pour le §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 nous allons ici travailler avec une seule place v|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soient L/Qp une extension finie de degré d > 1 (jouant le rôle de Fv/Qp), K/Qp une extension contenant la clôture Galoisienne de L, et S un schéma sur Spec OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note Lnr l’extension maximale non ramifiée dans L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note e l’indice de ramification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit ̟ une uniformisante de L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe un plongement τ : Lnr ֒→ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit l’ensemble Στ = HomLnr(L, Qp) comme étant l’ensemble des plongements τ ′ : L ֒→ K qui donne τ en restriction à Lnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est un ensemble de cardinal e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe un ordre Στ = {ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', ϕe}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit F un OS module localement libre muni d’une action de OL tel que OLnr agit sur F via τ (on rappelle que S est un schéma sur Spec OK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On suppose qu’il existe un faisceau E localement libre de rang h en tant que OL ⊗OLnr OS-module tel que F soit un localement un facteur direct de E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une donnée de Pappas-Rapoport pour (E , F) par rapport à Στ est une filtration 0 = F (0) ⊂ F (1) ⊂ · · · ⊂ F (e) = F telle que pour tout 1 ⩽ j ⩽ e Les F (j) sont localement des OS-facteurs directs, stables par OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ([̟] − ϕj(̟)) · F (j) ⊂ F (j−1) F (j)/F (j−1) est localement libre de rang 1 La deuxième condition impose que l’action de OL sur le quotient F (j)/F (j−1) se fasse via le plongement OL ϕj −→ OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Groupes p-divisibles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit S un schéma sur Spec OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G un groupe p-divisible sur S de hauteur hd muni d’une action ι : OL → EndS(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notons E (G) le cristal asso- cié.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Avec les notations de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1), en évaluant ce cristal sur l’épaississement tautologique (S → S) on obtient la filtration de Hodge : 0 −→ ωG −→ E (G)(S→S) −→ ω∨ GD −→ 0 D’après la Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 le faisceau E (G)(S→S) est libre en tant que OL ⊗OLnr OS- module et l’hypothèse 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 est donc bien satisfaite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour simplifier les notations nous noterons E = E (G)(S→S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette filtration est compatible avec les décompositions in- duites par 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 E = � τ∈Σnr Eτ, ωG = � τ∈Σnr ωG,τ où Σnr = Hom(Lnr, Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit pour tout τ ∈ Σnr on dispose d’une suite exacte 0 −→ ωG,τ −→ Eτ −→ ω∨ GD,τ −→ 0 induite par la filtration de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 27 Définition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une donnée de Pappas-Rapoport pour (G, ι) par rapport à (Στ)τ∈Σnr est la donnée pour tout τ ∈ Σnr d’une filtration de Pappas-Rapoport pour (Eτ, ωG,τ) par rapport à Στ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit c’est la donnée pour tout τ ∈ Σnr d’une filtration 0 = ω(0) G,τ ⊂ ω(1) G,τ ⊂ · · · ⊂ ω(e) G,τ = ωG,τ telle que pour tout 1 ⩽ j ⩽ e Les ω(j) G,τ sont localement des OS-facteurs directs, stables par OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ([̟] − ϕτ,j(̟)) · ω(j) G,τ ⊂ ω(j−1) G,τ ω(j) G,τ/ω(j−1) G,τ est localement libre de rang 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cas général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Revenons à notre extension F/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit donc G un groupe p-divisible sur S muni d’une action ι : OF → EndS(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour chaque place v|p on note Σnr v = Hom(F nr v , Qp), pour tout τ ∈ Σnr v on note Σv,τ = HomF nr v (Fv, Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors une première décomposition E = � v|p Ev, ωG = � v ωG,v et une deuxième décomposition pour chacune des places v|p : Ev = � τ∈Σnr v Ev,τ, ωG,v = � τ∈Σnr v ωG,v,τ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) Comme précédemment on fixe un ordre sur chacun des Σv,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une donnée de Pappas-Rapoport pour (G, ι) par rapport à (Σv,τ)v,τ est la donnée pour toute place v|p et tout τ ∈ Σnr v d’une filtration de Pappas-Rapoport pour (Ev,τ, ωG,v,τ) par rapport à Σv,τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit c’est la donnée pour toute place v|p et tout τ ∈ Σnr v d’une filtration 0 = ω(0) G,v,τ ⊂ ω(1) G,v,τ ⊂ · · · ⊂ ω(e) G,v,τ = ωG,v,τ telle que pour tout 1 ⩽ j ⩽ e Les ω(j) G,v,τ sont localement des OS-facteurs directs, stables par OF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' ([̟] − ϕv,τ,j(̟)) · ω(j) G,v,τ ⊂ ω(j−1) G,v,τ ω(j) G,v,τ/ω(j−1) G,v,τ est localement libre de rang 1 Remarque 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la définition ci dessus nous faisons l’abus d’appeler « donnée de Pappas Rapoport »ce qui correspond en fait à une « donnée de Pappas Rapoport dans le cas Hilbert » c’est-à-dire une donnée de Pappas-Rapoport où la dimension des gradués ω(j) G,v,τ/ω(j−1) G,v,τ est égale à dj = 1 pour tout 1 ≤ j ≤ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On consultera [BH22b] et [BH16] pour plus de détails sur les filtrations de Pappas-Rapoport dans un cadre PEL plus général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 28 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition explicite de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Revenons à notre modèle entier ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il découle de la définition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) que le foncteur ShPR associe à un schéma localement noethérien S → Spec OK, les quintuplés (A, λ, ι, η, ω(·) G ) à Z× (p)-isogénie près où A → S est un schéma abélien de dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' λ : A → At est une Z× (p)-polarisation ι : OF → End(A) ⊗Z Z(p) η est une structure de niveau rationnelle en dehors de p (voir [Lan13] section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) ω(·) G est une donnée de Pappas-Rapoport pour G = A[p∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Stratification de Hodge Désormais tout ce qui suit porte sur la fibre spéciale de nos modèles entiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous noterons donc désormais pour alléger les notations M PEL = MPEL et MPR = M PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour simplifier nous allons commencer par traiter le cas d’une seule extension totalement ramifiée L/Qp de degré e (jouant le rôle de Fv/F nr v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout 1 ≤ i ≤ e on note µi = (1, 0) et µ• = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', µe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a donc µ := |µ•| = (e, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le cas PEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 nous avons vu que le diagramme de modèle local nous fournissait dans le cas PEL un morphisme lisse (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) ShPEL −→ � MPEL/G � D’après la proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4 on dispose d’un plongement équivariant du modèle local dans la Grassmannienne affine induisant après passage au quotient un morphisme � MPEL/G � −→ � L+G\\LG/L+G � = Hecke La description de l’image de ce morphisme 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4 nous fournit donc finalement un mor- phisme lisse vers le champ de Hecke borné : ζ : ShPEL −→ HeckeAdm(µ)K où Adm(µ)K est définit en 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ |ShPEL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On appelle polygone de Hodge du point x l’élément Hodge(x) = ζ(x) Pour tout λ ∈ Adm(µ)K on définit le sous schéma localement fermé (structure réduite) ShPEL λ = ζ−1(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les (ShPEL λ )λ∈Adm(µ)K sont appelées les strates de Hodge de ShPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Donnons une définition plus explicite du polygone de Hodge et des strates de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ |ShPEL| un point de corps résiduel k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour calculer l’image ζ(x) il nous faut suivre x dans le diagramme de modèle local : ShPEL,□ ShPEL MPEL Gr ϕ ψ ι STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 29 Notons ωx ֒→ Ex la filtration de Hodge associée au point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On choisit un élément ˜x ∈ ϕ−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’image ι ◦ ψ(˜x) dépend du choix ˜x, mais son image dans le quotient Hecke = � L+G\\Gr � ne dépend pas de ce choix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour décrire ζ(x) il nous suffit donc de décrire ι ◦ ψ(˜x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition de ShPEL,□ sur ˜x on dispose d’une trivialisation Ex ≃ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit on dispose d’une base : Ex ≃ k[[u]] (ue) ⊕ k[[u]] (ue) Quitte à choisir une autre base on peut supposer que la filtration de Hodge soit donnée par ωx ≃ uik[[u]] (ue) ⊕ uj k[[u]] (ue) Ensuite d’après (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) le plongement du modèle local dans la grassmannienne affine consiste à prendre l’image inverse de la filtration de Hodge le long de k[[u]] → k[[u]]/(ue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On obtient finalement Λωx = uik[[u]] ⊕ ujk[[u]] ⊂ Λ0 Par conséquent on retrouve la définition usuelle (voir [DP94] Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2, où [BH16] Définition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7) du polygone de Hodge d’un groupe p-divisible Hodge(x) = Hodge(ωx) = (i, j) Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait le polygone de Hodge correspond plutôt au polygone de pentes ( i e, j e) (voir [BH16], Définition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Bien sûr on retrouve une définition à partir de l’autre et les stratifications induites coïncident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Propriétés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La proposition suivante résume les différentes propriétés de la stra- tification de Hodge : Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3 ([DP94] section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2, [NG02] section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Les strates (ShPEL λ )λ∈Adm(µ)K forment une bonne stratification de ShPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Pour tout λ ∈ Adm(µ)K la strate ShPEL λ est quasi-projective lisse de dimension ⟨2ρ, λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3) La strate ShPEL (e,0) coïncide avec le lieu lisse de ShPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour le point (1), cela découle de la lissité du morphisme ζ : ShPEL → HeckeAdm(µ)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour le point (3), cela découle du fait que Gr(e,0) coïncide avec le lieu lisse de Gr≤(e,0) (voir Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) Enfin pour le point (2), il suffit de voir que le morphisme ShPEL λ → � MPEL λ /G � est lisse de dimension relative dim G et que le membre de droite était un point de dimension ⟨2ρ, λ⟩ − dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour le point (2) on peut se passer du langage des champs et donner une preuve à la main : en restriction à une strate λ ∈ Adm(µ)K le diagramme de modèle STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 30 local devient ShPEL,□ λ ShPEL λ MPEL λ ϕ ψ D’après la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 la strate MPEL λ est de dimension ⟨2ρ, λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le morphisme ψ est de dimension relative dim G = dim G et le morphisme ϕ est un G-torseur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On trouve donc bien dim ShPEL λ = ⟨2ρ, λ⟩ + dim G − dim G = ⟨2ρ, λ⟩ Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans [AG03] la strate ShPEL (e,0) est appelée lieu de Rapoport car elle coïncide avec le lieu où le faisceau ω est libre en tant que OL ⊗Zp OS-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On peut être plus explicite sur la dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si λ = (i, j) ≤ (e, 0) alors ⟨2ρ, λ⟩ = ⟨α1 − α2, iλ1 + jλ2⟩ = i − j = e − 2j où α1 : � t1 0 0 t2 � �→ t1, α2 : � t1 0 0 t2 � �→ t2, λ1 : t �→ � t 0 0 1 � et λ2 : t �→ � 1 0 0 t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le cas Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant nous intéresser à la stratifi- cation du modèle de Pappas-Rapoport ShPR par le polygone Hodge(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout λ ∈ Adm(µ)K on définit le sous schéma localement fermé (structure réduite) ShPR λ = π−1(ShPEL λ ) où π : ShPR → ShPEL est le morphisme d’oubli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les (ShPR λ )λ∈Adm(µ)K sont appelées les strates de Hdoge de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus explicitement : si x ∈ ShPR alors on peut regarder le polygone de hodge Hodge(x) := Hodge(ω(e)) où ω(e) = ωx est le dernier cran de la filtration de Pappas- Rapoport ω(1) ⊂ · · · ⊂ ω(e) associée au point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fera attention à ne pas confondre les deux morphismes : ShPR � L+G\\ � Grµ• � = Heckeµ• HeckeAdm(µ)K = � L+G\\Gr≤|µ•| � ˜ζ ζ◦π Le morphisme de gauche concerne les classes d’isomorphismes de filtrations de Pappas- Rapoport (ω(1) ⊂ · · · ⊂ ω(e) ⊂ E ), et celui de droite concerne les classes d’isomor- phismes de filtrations de Hodge (ω ⊂ E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Propriétés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le théorème est le suivant : Théoreme 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Les strates (ShPR λ )λ∈Adm(µ)K forment une bonne stratification de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autre- ment dit pour tout λ ∈ Adm(µ)K on a la relation d’adhérence ShPR λ = � λ′≤λ ShPR λ′ (2) Pour tout λ ∈ Adm(µ)K la strate ShPR λ est quasi-projective lisse de dimension ⟨ρ, |µ•| + λ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Commençons par montrer le point (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16 on dispose d’une bonne stratification du produit de convolution � Grµ• = � λ∈Adm(µ)K m−1(Grλ) Puisque le morphisme de convolution m : � Grµ• → Gr≤|µ•| est L+G-équivariant, cette stratification est stable sous l’action de L+G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit chacune des strates est une union de L+G-orbites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par conséquent cette stratification descend au quotient en une bonne stratification du champs Heckeµ• = � L+G\\Grµ• � : Heckeµ• = � λ∈Adm(µ)K � L+G\\m−1(Grλ) � On conclut comme pour le cas PEL (voir 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) en utilisant que le morphisme ˜ζ : ShPR → Heckeµ• est lisse et donc préserve les relations d’adhérences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Montrons maintenant le point (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7, en restriction à une strate Grλ associée à λ ∈ Adm(µ)K on dispose d’une fibration localement triviale m : m−1(Grλ) −→ Grλ En particulier au dessus d’une telle strate le morphisme est plat et par conséquent on a la relation dim m−1(Grλ) = dim Grλ + dim m−1(y) pour n’importe quel élément y ∈ Grλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Toujours d’après le théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7, on a que la fibre m−1(y) est équidimensionnelle de dimension ⟨ρ, |µ•| − λ⟩ car µ• = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' µe) est constitué de cocaractères minuscules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit au niveau du modèle local MPR λ = m−1(Grλ) via l’identification MPR ≃ � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le modèle local nous donne un morphisme ShPR λ → � MPR λ /G � STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 32 qui est lisse de dimension relative dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le membre de droite est de dimension MPR λ − dim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Finalement on trouve donc bien dim ShPR λ = dim � MPR λ /G � + dim G = dim MPR λ + dim G − dim G = dim m−1(Grλ) = dim Grλ + dim m−1(y) = ⟨2ρ, λ⟩ + ⟨ρ, |µ•| − λ⟩ = ⟨ρ, |µ•| + λ⟩ □ Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Encore une fois on peut être plus explicite concernant la dimension des strates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour λ = (i, j) ≤ (e, 0) on obtient dim ShPR (i,j) = ⟨ρ, |µ•| + λ⟩ = 1 2⟨α1 − α2, (e + i)λ1 + jλ2⟩ = e + i − j = e − j 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cas général.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On revient maintenant au cas d’une extension F/Q de degré d > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout v|p, τ ∈ Σnr v et tout 1 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' i ≤ ev on note µv,τ,i = (1, 0), µv,τ,• = (µv,τ,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , µv,τ,ev), µv,τ = |µv,τ,•| = (ev, 0) et enfin µ := (µv,τ)v,τ La compatibilité du modèle local au produit décrite en (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5) suggère la définition suivante : soit x ∈ |ShPR|, on définit l’invariant : Hodge(x) = � Hodgev,τ(x) � v,τ ∈ � v,τ {λ ≤ (ev, 0)} = Adm(µ)K où Hodgev,τ(x) = Hodge(ωx,v,τ) est l’invariant de Hodge définit en 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 et ωx,v,τ est le faisceau associé à x et défini en (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit l’invariant Hodge(ω) ci dessus correspond à la donnée des invariants Hodge(ωv,τ) pour chacun des termes dans la décomposition ω = � v|p � τ∈Σnr v ωv,τ On munit Adm(µ)K de la relation d’ordre suivante : (λv,τ)v,τ ≤ (λ′ v,τ)v,τ ⇐⇒ λv,τ ≤ λ′ v,τ ∀(v, τ) La généralisation du Théorème 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8 prend la forme suivante : Théoreme 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) Les strates (ShPR λv,τ )λv,τ ∈Adm(µ)K forment une bonne stratification de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Au- trement dit pour tout (λv,τ)v,τ ∈ Adm(µ)K on a la relation d’adhérence ShPR (λv,τ )v,τ = � (λ′v,τ )v,τ ≤(λv,τ )v,τ ShPR (λ′v,τ )v,τ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 33 (2) Pour tout (λv,τ)v,τ ∈ Adm(µ)K la strate ShPR λv,τ est quasi-projective lisse de di- mension : dim ShPR (λv,τ )v,τ = � v,τ ⟨ρ, |µv,τ,•| + λv,τ⟩ Remarque 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fera attention au fait que dans le cadre d’une extension L/Qp d’indice d’inertie f > 1, alors la stratification de Hodge ci-dessus ne coïncide pas avec la stratification de Hodge définie dans [BH22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait la stratification définie dans loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' est moins fine car elle est définie par le polygone de Hodge de [BH16] qui est un moyenne sur f des invariants de Hodge considérés pour notre stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On consultera [SZ22] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='20 pour plus de détails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autres résultats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Lorsque la donnée PEL est ramifiée, l’un des objectifs pour étudier la géométrie de la fibre spéciale de la variété de Shimura associée est de raffiner la stratification de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit pour tout λ ∈ Adm(µ)K on aimerait définir une décomposition : ShPEL λ = � w∈WΛ Sw où (Wλ, ≤) est un certain ensemble partiellement ordonné qui dépend de λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Donnons quelques exemples : (1) Dans [SYZ21] les auteurs définissent pour chaque strate de Hodge un morphisme lisse : ζλ : ShPEL λ −→ Grdt 0 -ZipJλ où Grdt 0 désigne le quotient réductif de G0 = G ⊗Zp Fp et Grdt 0 -Zipλ désigne le champs des Grdt 0 -Zip de type Jλ (voir [SYZ21] Définition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5 et Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La stratification est alors définie comme étant celle induite par celle du champs Grdt 0 -ZipJλ via le morphisme ζλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Leur construction est beaucoup plus générale et s’applique aux variétés de Shimura de type abélien sans hypothèse sur le niveau en p (2) Dans [AG03] les auteurs calculs explicitement les polygones de Newton sur cha- cune des strates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément si x ∈ ShPEL (i,j) avec (i, j) ∈ Adm(µ)K alors existe une (OL ⊗ W(k))- base du module de Dieudonné associé dans laquelle le Frobenius est donné par la matrice ([AG03], Propositon 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10) : F = � ̟m c̟i ̟j 0 � où m ≥ j et c ∈ (OL ⊗ W(k))×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ils définissent ensuite les quantités : n = � m si m ≤ i i sinon λ(n) = min �n g , 1 2 � La stratification de ShPEL (i,j) est alors définie via cet invariant n : ShPEL (i,j) = � j≤n≤i ShPEL (i,j),n STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 34 Cet invariant leur permet de calculer explicitement le polygone de Newton sur chacune des strates ShPEL (i,j),n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément si x ∈ ShPEL (i,j),n alors ([AG03], Théorème 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) : Newt(x) = {λ(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , λ(n), 1 − λ(n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', 1 − λ(n)} (chacune des pentes avec multiplicité e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3) Dans [YCO20] les auteurs associent à tout point x ∈ ShPEL un invariant c(x) = (cτ(x))τ∈Σnr appelé invariant de congruence qui mesure la position relative des filtrations (voir la [YCO20] Définition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 pour être plus précis) : F(Mσ−1τ) ⊂ Mτ, V (Mστ) ⊂ Mτ où (M, F, V ) désigne le module de Dieudonné associé à x et où les (Mτ)τ∈Σnr désignent les facteurs directs dans la décomposition M ≃ � τ∈Σnr Mτ, F : Mσ−1τ → Mτ, V : Mστ → Mτ Cet invariant permet de définir une stratification ([YCO20] Définition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) : ShPEL λ = � c∈τL Qc(ShPEL λ ) où Qc(ShPEL λ ) désigne l’ensemble des points d’invariant de congruence c ∈ τL et τL est un certain ensemble défini dans [YCO20] Définition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans chacun de ces trois articles se pose deux problèmes : (1) Calculer l’adhérence de Sw dans ShPEL λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Calculer l’adhérence de Sw dans ShPEL Dans [YCO20] les auteurs parviennent à calculer le point (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans [AG03] et [SYZ21] les auteurs parviennent à calculer le point (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le théorème suivant montre que le point (1) est automatique pour ShPR λ lorsque l’on regarde le tiré en arrière d’une stratification de ShPEL λ via le morphisme d’oubli π : ShPR → ShPEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Théoreme 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout λ ∈ Adm(µ)K la restriction π : ShPR λ → ShPEL λ est un morphisme plat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En restriction à une strate d’invariant λ ∈ Adm(µ)K le carré cartésien (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) devient ShPR λ � MPR λ /G � ShPEL λ � MPEL λ /G � p π STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 35 qui est lui aussi cartésien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il suffit de montrer que le morphisme p est plat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce dernier s’inscrit dans un carré cartésien MPR λ MPEL λ � MPR λ /G � � MPEL λ /G � p q p Or d’après le théorème 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 et (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8) le morphisme p est localement trivial et donc en particulier plat (le schéma de base Spec k est un corps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le morphisme q étant fidèlement plat, on conclut par [Sta18], Lemme 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Corollaire 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si ShPEL λ = � w∈W Sw est une bonne stratification alors ShPR λ = � w∈W π−1(Sw) est également une bonne stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après la proposition précédente le morphisme π est plat en restriction à une strate ShPEL λ et est donc en particulier ouvert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariants de Hasse partiels Nous allons maintenant décrire l’interaction entre la stratification de Hodge définie dans la section précédente, et les invariants de Hasse partiels définis dans [RX14], lorsque l’indice de ramification satisfait e ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Sans perdre de généralité on peut se restreindre à une place v|p et donc considérer la situation d’un groupe p-divisible G muni d’une action d’un anneau d’entier OL où L/Qp est une extension de degré d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour faciliter la lecture, nous allons commencer par un rappel des différentes définitions de loc cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariant de Hasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G un groupe de Barsotti-Tate sur un schéma S de caractéristique p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le Verschiebung Ver : G(p) → G induit un morphisme : Ver∗ : ωG −→ ω(p) G En prenant le déterminant de ce morphisme on obtient une section Ha(G) ∈ H0(S, (det ωG)⊗(p−1)) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1) appelée invariant de Hasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a alors la proposition suivante : Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit G un groupe p-divisible sur un corps k de caractéristique p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors Ha(G) est inversible si et seulement si G est ordinaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariants de Hasse partiels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (G, ι) un groupe p-divisible sur S muni d’une action ι : OL → EndS(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On suppose que (G, ι) est muni d’une donnée de Pappas- Rapoport (Eτ, ωG,τ)τ une donnée de Pappas-Rapoport (voir 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout 2 ≤ i ≤ e on définit l’application M(i) τ comme étant la multiplication par ̟ au niveau des gradués : M(i) τ : ω(i) G,τ/ω(i−1) G,τ −→ ω(i) G,τ/ω(i−2) G,τ Ce morphisme induit une section appelée invariant de Hasse primitif : m(i) τ (G) ∈ H0(S, det (ω[i−1] G,τ /ω[i−2] G,τ ) ⊗ det (ω[i] G,τ/ω[i−1] G,τ )−1) Dans le cas Hilbert par exemple, c’est-à-dire le cas où dim(G) = dg = d, on a la proposition suivante caractérisant le lieu de Rapoport : Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On suppose que S = Spec k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les conditions suivantes sont équiva- lentes : (1) Hodgeτ(G, ι) = (e, 0) (2) m(i) τ ̸= 0 pour tout 2 ≤ i ≤ e (3) ωG,τ est un OL ⊗Fp OS-module libre de rang 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) implique (2) et (3) implique (1) sont évidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour (2) implique (3), il suffit de prendre un vecteur v ∈ ω(e) τ \\ω(e−1) τ , et de voir qu ⟨̟e−1v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', ̟v, v⟩ est une base de ω(e) τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La même preuve fonctionne également pour le cas Hilbert-Siegel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (G, ι, ω(·) G,τ) un groupe p-divisible sur un corps k avec donnée de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour tout 1 ≤ i ≤ e on définit : Hodge(ω(i) G,τ) := Hodge(Λω(i) G,τ) = Inv(Λω(i) G,τ, Λ0) ∈ X∗(T)+ via la construction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) (voir la section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 pour la description explicite de Hodge(ω(e) G,τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la définition ci-dessus on adopte une convention différente que celle utilisée dans [Bij22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément, si (G, ι, ω(·) G,τ) un groupe p-divisible sur un corps k avec donnée de Pappas-Rapoport, alors le faisceau ωi G,τ est un OL ⊗ k-module satisfaisant ̟i · ω(i) G,τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Via l’identification ̟ �→ u on peut donc le voir comme un k[u]/(ue)-module ou comme un k[u]/(ui)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans la construction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) on le voit comme un k[u]/(ue)-module, alors que dans [Bij22] il est considéré comme un k[u]/(ui)- module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par la suite nous aurons besoin du lemme suivant : Lemme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit τ ∈ Σnr, 2 ≤ i ≤ e et x ∈ ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notons (G, ι, ω(·) G,τ) le groupe p-divisible avec donnée de Pappas-Rapoport associé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si m(i) τ (x) = 0 alors on a l’égalité Hodge(ω(i) G,τ) = Hodge(ω(i−2) G,τ ) − (1, 1) Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition si m(i) τ = 0 on a ω(i) τ ⊂ ̟−1(ω(i−2) τ ) (ici on utilise que les gradués sont de dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Un simple calcul des dimensions respectives montre que cette inclusion est une égalité.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le résultat en découle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 37 Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fera attention au fait que la réciproque est bien sûr fausse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fera également attention au fait que le lemme précédent n’est pas vrai dans le cas Hilbert-Siegel pour g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela vient du fait que le determinant peut être nul sans que l’application M(i) τ soit nulle pour autant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le Verschiebung V : G(p) → G induit une application Vτ : Eτ → ω(p) G,σ−1τ = (ω(e) G,σ−1τ)(p) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) (voir 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On considère la composée : Eτ[̟] ≃ Eτ/Eτ[̟e−1] Vτ −→ (ω(e) G,σ−1τ/ω(e−1) G,σ−1τ)(p) Autrement dit si x ∈ Eτ[̟] alors x = ̟e−1 · y pour un certains y ∈ Eτ, on lui associe alors Vτ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En restreignant ce morphisme à ω(1) G,τ ⊂ Eτ[̟] on obtient finalement un morphisme : Haτ : ω(1) G,τ → (ω(e) G,σ−1τ/ω(e−1) G,σ−1τ)(p) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3) et donc une section haτ(G) ∈ H0(S, det (ω(e) G,σ−1τ/ω(e−1) G,σ−1τ)⊗p ⊗ det (ω(1) G,τ)−1) également appelé invariant de Hasse partiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour uniformiser les notations on définit m(1) τ := haτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans [DK22] les auteurs se sont intéressés à la stratification de la fibre spéciale induite par ces invariants de Hasse partiels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément si on note ShPR T = {x ∈ ShPR| mi τ(x) = 0 ssi (τ, i) ∈ T}, T ⊂ Σnr × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e} les sous schémas localement fermés définis comme les lieux d’annulations des invariants de Hasse partiels d’indice contenu dans T, alors on a le théorème suivant Théoreme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 ([DK22], Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les (ShPR T )T pour T ⊂ Σnr × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , e} définissent une bonne stratification de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément on a pour tout T ⊂ Σnr × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e} : ShPR T = � T ′⊂T ShPR T ′ De plus chacune des strates ShPR T est non vide, quasi affine, et équidimensionnelle de dimension d − |T|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Interaction avec la stratification de Hodge : le cas e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On dispose de deux stratifications de la fibre spéciale du modèle de Pappas Rapoport : ShPR = � λ∈Adm(µ)K ShPR λ , ShPR = � T∈T ShPR T où T = P(Σnr×{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , e}) désigne les sous ensembles de Σnr×{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il est naturel de se demander dans quelle mesure la décomposition ShPR = � (λ,T)∈Adm(µ)K×T ShPR (λ,T), ShPR (λ,T) := ShPR λ ∩ ShPR T fourni une bonne stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Deux problèmes se posent alors : STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 38 (1) Quelles sont les strates ShPR (λ,T) qui sont non vides ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Si cette stratification est une bonne stratification, quelle est la relation d’ordre sur les couples (λ, T) qui décrit les relations d’adhérences des strates ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour le point (1), la Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 nous dit que pour λ = (e, 0) alors ShPR (λ,T) = ∅ si et seulement si il existe un indice (τ, i) ∈ T ∈ T avec i ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' De manière générale, il semble difficile de prédire quelles sont les strates non-vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons répondre à cette question et décrire l’interaction entre le stratification de Hodge et celle par les invariants (m(i))i dans le cadre d’une extension L/Qp totalement ramifiée de degré e = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’extension étant totalement ramifiée, nous pouvons omettre l’indice τ et on a donc T = P({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' , e}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit les ensembles : Tλ = {T ∈ P({2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e}) | ShPR (λ,T) ̸= ∅} ∀ λ ∈ Adm(µ)K (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4) Tµ• = {(λ, T) ∈ Adm(µ)K × P({2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', e}) | ShPR (λ,T) ̸= ∅} Par définition de ces ensembles on a une décomposition de l’ensemble Tµ• : Tµ• = � λ∈Adm(µ)K Tλ Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La stratification définie par l’ensemble Tµ• est purement « linéaire » dans le sens où elle est définie par les invariants mi pour i ≥ 2 et par l’invariant de Hodge, qui sont des invariants « linéaires » (en opposition à l’invariant m1 = ha qui est un invariant σ-linéaire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait la stratification induite par l’ensemble Tµ• correspond à une stratification du produit de convolution � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La stratification définie ci-dessus est donc « linéaire » dans le sens où elle ne dépend que du modèle local MPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Définition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit les ensembles suivants : T (m1=0) µ• = {(λ, T) ∈ Adm(µ)K × T | ShPR (λ,T) ̸= ∅, m1 ∈ T} T (m1̸=0) µ• = {(λ, T) ∈ Adm(µ)K × T | ShPR (λ,T) ̸= ∅, m1 /∈ T} Aµ• = {(λ, T) ∈ Adm(µ)K × T | ShPR (λ,T) ̸= ∅} On a par définition de ces ensembles : Aµ• = T (m1=0) µ• � T (m1̸=0) µ• Dans un premier temps nous allons décrire l’interaction entre les strates de Hodge et les strates définies par les m(i) pour i = 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous intégrerons l’invariant m1 = ha à notre raisonnement par la suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Notre problème devient alors simplement un problème d’algèbre linéaire sur la filtration de Pappas-Rapoport ω(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour alléger les notations nous noterons X(mi)i∈T λ = ShPR (λ,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariant (mi)i≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le schéma ci-dessous décrit les strates non vides et les rela- tions d’adhérences entre elles avec la convention A → B si A ⊂ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' X(4,0) X(m2) (3,1) X(m3) (3,1) X(m4) (3,1) X(m2,m3) (3,1) X(m3) (2,2) X(m3,m4) (3,1) X(m2,m4) (2,2) X(m2,m3,m4) (2,2) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5) Le problème étant pour le moment purement un problème d’algèbre linéaire, nous allons utiliser les notations utilisées dans le cadre des grassmanniennes affines (via la construction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Tout d’abord commençons par prouver l’assertion sur les strates vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons donner tous les détails pour le cas (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les autres cas étant très similaire, nous donnerons seulement les arguments importants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) X(m2,m4) (3,1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' A priori on a une décomposition en union disjoint : X(m2,m4) = X(m2,m4) (3,1) ∪ X(m2,m4) (2,2) (on rappelle que la strate X(m2,m4) (4,0) est vide d’après le Lemme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m2,m4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour simplifier les notations on note Λ1 ⊂ · · · ⊂ Λ4 la filtration Λω(1) x ⊂ · · · ⊂ Λω(4) x associée à ce point (voir (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le Lemme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6 puisque m2(x) = m4(x) = 0 on a Hodge(Λ4) = Hodge(Λ2) − (1, 1) = Hodge(Λ0) − (1, 1) − (1, 1) = (2, 2) Par conséquent X(m2,m4) = X(m2,m4) (2,2) et X(m2,m4) (3,1) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) X(m4) (2,2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Puisque m4 = 0 on a d’après 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6 : Hodge(Λ4) = Hodge(Λ2) − (1, 1) = (4, 2) − (1, 1) = (3, 1) où l’on a utilisé m2 ̸= 0 pour obtenir Hodge(Λ2) = (4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 40 (3) X(m3,m4) (2,2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Même raison que ci-dessus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (4) X(m2) (2,2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit v ∈ ω(4)\\ω(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a u · v ∈ ω(3)\\ω(2) car m4 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ensuite puisque m2 = 0 on a ω(2) = E [u] et par conséquent u2 · v ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier Hodge(ω(4)) > (2, 2) (5) X(m2,m3) (2,2) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Même raison que ci-dessus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il assez simple de trouver des filtrations adéquates pour prouver que les strates du schéma ci-dessus sont non vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En d’autres termes on a calculé les ensembles Tλ pour tout λ ∈ Adm(µ)K (avec les notations de (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4)) : T(4,0) = {∅} (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='6) T(3,1) = {(m2), (m3), (m4), (m2, m3), (m3, m4)} (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7) T(2,2) = {(m3), (m2, m4), (m4), (m2, m3, m4)} (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8) Montrons maintenant l’assertion sur les relations d’adhérences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le Théorème 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 les invariants (mi)i forment une bonne stratification de ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En particulier d’après le Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 cela veut dire que l’on peut « inverser »un invariant sans toucher aux autres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément : Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si x ∈ X(mi)i∈T λ avec T ̸= ∅ alors pour tout i0 ∈ T il existe λ′ ∈ Adm(µ)K et y ∈ X (mi)i∈T ′ λ′ tel que x ∈ {y} où T ′ = T\\{i0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est la combinaison de 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 et 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Le problème est que dans la proposition ci dessus on ne contrôle pas le polygone de Hodge lors de la déformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cependant, nous avons prouver que certaines strates X(mi)i∈T λ étaient vides, et nous pouvons l’utiliser pour décrire le polygone de Hodge lors des déformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Exemple 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m2,m3,m4) (2,2) et soit y ∈ X(m2,m4) λ un point tel que x ∈ {y} (fourni par la proposition précédente).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 la strate X(m2,m4) (3,1) est vide ce qui impose l’égalité λ = (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela montre que pour tout point x ∈ X(m2,m3,m4) (2,2) il existe un y ∈ X(m2,m4) (2,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit cela prouve la relation d’adhérence X(m2,m3,m4) (2,2) ⊂ X(m2,m4) (2,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous pouvons également utiliser le fait que l’invariant de Hodge ne peut qu’augmenter par générisation (autrement dit le polygone de Hodge s’abaisse par générisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Plus précisément si x ∈ Xλ et y ∈ Xλ′ tel que x ∈ {y} alors λ ≤ λ′, ce qui peut être déduit du Théorème 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='8 qui est cependant beaucoup plus fort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Exemple 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m2,m3) (3,1) et soit y ∈ X(m3) λ un point tel que x ∈ {y} (fourni par la proposition précédente).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' L’invariant de Hodge ne pouvant qu’augmenter par générisation on a nécessairement Hodge(y) ≥ (3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le cas (4, 0) étant impossible d’après 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 on a nécessairement Hodge(y) = (3, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cela montre que pour tout point x ∈ X(m2,m3) (3,1) il existe un y ∈ X(m3) (3,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit cela prouve la relation d’adhérence X(m2,m3) (3,1) ⊂ X(m3) (3,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 41 Les deux exemples ci dessus fonctionnent pour toutes les relations d’adhérences à l’exception de : X(m2,m3,m4) (2,2) ⊂ X(m3) (2,2) , X(m3) (2,2) ⊂ X(m3) (3,1) Dans le premier cas ce qui pose problème c’est que la strate X(m3) (3,1) est non vide et par conséquent le raisonnement ci dessus ne fonctionne pas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans le second cas il s’agit de déformer le polygone de Hodge au sein de la strate définie par l’équation m3 = 0 et m2, m4 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) X(m2,m3,m4) (2,2) ⊂ X(m3) (2,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m2,m3,m4) (2,2) et 0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3) ⊂ ω(4) la filtration associée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition on a les égalités : ω(2) = E [u] = ω(1) ⊕ ⟨v2⟩, ω(3) = u−1(ω(1)), ω(4) = u−1(ω(2)) = E [u2] pour un certain v2 ∈ ω(2) que l’on fixe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la déformation sur k[[t]] comme suit : �E = E ⊗k k[[t]], �ω(1) = ω(1) ⊗k k[[t]] �ω(2) = �ω(1) ⊕ ⟨v2 + tv⟩, �ω(3) = u−1(�ω(1)), �ω(4) = �E [u2] pour un certain v ∈ u−1(�ω(1))\\ �E [u] que l’on choisit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par construction en fibre générique cette filtration satisfait bien les équations m2 ̸= 0, m4 ̸= 0 , m3 = 0 et Hodge(�ω(4)) = (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) X(m3) (2,2) ⊂ X(m3) (3,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m3) (2,2) et 0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3) ⊂ ω(4) la filtration associée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition on a les égalités : ω(3) = u−1(ω(1)), ω(4) = E [u2] Soit v(4) ∈ ω(4) tel que ω(4) = ω(3)⊕⟨v(4)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On choisit un élément v ∈ u−1(ω(3))\\E [u2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la déformation sur k[[t]] comme suit : �E = E ⊗k k[[t]], �ω(i) = ω(i) ⊗k k[[t]], i = 1, 2, 3 et �ω(4) = �ω(3) ⊕ ⟨v(4) + tv⟩ En fibre générique cette filtration satisfait bien les équations m2m4 ̸= 0 , m3 = 0 et Hodge(�ω(4)) = (3, 1) car par construction u2(v4 + tv) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Invariant m1 = ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons maintenant nous intéresser à l’invariant m1 = ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On aimerait pouvoir « inverser » ha sans modifier les autres invariants de Hasse partiels et sans modifier l’invariant de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a la proposition suivante valable pour une extension L/Qp totalement ramifiée de degré quelconque : Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ ShPR (λ,T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On suppose que 1 ∈ T c’est-à-dire que m1(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Alors il existe y ∈ ShPR (λ,T ′) tel que y ⇝ x où T ′ = T\\{1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ ShPR (λ,T) défini sur un corps k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On note E le cristal associé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons construire une déformation sur k[[t]] de ce cristal telle qu’en fibre générique on ait ha ̸= 0, et sans que les autres invariants ne soient modifiés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit F (1) := F((ω(e+1))(p)) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 42 où ω(e+1) := ̟−1(ω(e−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après [Bij16] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='12, il existe un isomorphisme E [̟]/F (1) ≃ � ω(e)/ω(e−1)�(p) faisant commuter le diagramme suivant : ω(1) E [̟]/F (1) � ω(e)/ω(e−1)�(p) ≃ ha Puisque les gradués sont de dimension 1, la condition ha = 0 est équivalente à l’égalité ω(1) = F (1) dans E [̟].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans tout ce qui suit on notera E(n), F (1) (n), ω(k) (n) etc les objets définis sur Rn = k[t]/(tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 on dispose d’un relèvement F (1) (2) sur R2 qui ne dépend pas du choix des relèvements ω(e) (2) et ω(e−1) (2) sur R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit Λ(1) (1) ⊂ · · · ⊂ Λ(e) (1) ∈ � Grµ•(R1) la filtration associée à 0 ⊂ ω(1) (1) ⊂ · · · ⊂ ω(e) (1) via 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=', ge) ∈ GL2(R1((u))) tel que : Λ(i) (1) = gi · Λ(i−1) (1) , Λ(0) (1) = ue · Λ0 Soit ΛF (1) (2) le R2[[u]]-réseau associé à F (1) (2) via 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit ensuite : Λ(i+1) (2) = gi+1 · Λ(i) (1), Λ(1) (2) ̸= ΛF (1) (2) où chacun des gi+1 ∈ GL2(R2((u))) est vu ici via la section R1 → R2 et Λ(1) (2) désigne n’importe quel R2[[u]]-réseau relevant Λ(1) (1) et satisfaisant la condition Λ(1) (2) ̸= ΛF (1) (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si on note gF ∈ GL2(R2((u))) le lacet définissant ΛF (1) (2) alors il suffit par exemple de choisir Λ(1) (2) = (g · gF) · ueΛ0 avec g ∈ GL2(R2[[u]]) relevant Id ∈ GL2(R2[[u]]) et g ̸= Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Toujours grâce au procédé 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4 on obtient une filtration 0 ⊂ ω(1) (2) ⊂ · · · ⊂ ω(e) (2) satisfaisant ω(1) (2) ̸= F (1) (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On continue le processus par récurrence sur n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Supposons donnés E(n), ω(k) (n) etc sur Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit alors la déformation sur Rn+1 comme étant (avec les notations similaires au cas n = 2) : Λ(i+1) (n+1) = gi+1 · Λ(i) (n+1), Λ(1) (n+1) ⊗Rn+1 Rn = Λ(1) (n) où Λ(1) (n+1) désigne n’importe quel relèvement de Λ(1) (n) sur Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Cette propriété de relè- vement nous assure que la condition Λ(1) (n+1) ̸= ΛF (1) (n+1) soit satisfaite (car elle l’est après réduction à Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Toujours via le procédé 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='4 on obtient une filtration 0 ⊂ ω(1) (n+1) ⊂ · · ⊂ ω(e) (n+1) sur Rn+1 satisfaisant ω(1) (n+1) ̸= F (1) (n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On obtient par passage à la limite un groupe p-divisible G ∈ BTPR(R) muni d’une action ι : OL → End(G) et d’une filtra- tion de Pappas-Rapoport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par Gronthendieck-Messing et Serre-Tate cela nous fournit un morphisme : Spec k[[t]] → ShPR STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 43 En fibre générique la filtration satisfait les propriétés souhaitées : par construction les positions relatives de la filtration n’ont pas changé et par conséquent les invariants (mi)i et l’invariant de Hodge n’ont pas changé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Enfin lim F (1) (n) ̸= lim ω(1) (n), toujours par construction, ce qui nous assure qu’en fibre générique on ait bien ha ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce qui est assez surprenant dans la preuve ci-dessus est que l’on change une donnée σ-linéaire (l’invariant m1) via une déformation d’une donnée linéaire (filtration de Pappas-Rapoport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est le Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 qui rend ce processus possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si nous avions voulu déformer directement sur k[[t]], il aurait été difficile de calculer le faisceau F (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La proposition ci-dessus nous dit que l’on peut inverser l’invariant m1 sans toucher aux autres invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est une déformation au sein d’une strate fixée par les invariants (mi)i≥2 et l’invariant de Hodge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous devons maintenant traiter la réciproque : l’équation m1 = 0 est-elle satisfaite sur la fibre générique des déformations définies en 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='5 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Autrement dit, peut on modifier l’invariant de Hodge et les (mi)i tout en satisfaisant ha = 0 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour les mêmes raisons que dans les exemples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16 et 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='17 il suffit de traiter les cas : X(m1,m2,m3,m4) (2,2) ⊂ X(m1,m3) (2,2) , X(m1,m3) (2,2) ⊂ X(m1,m3) (3,1) Le problème est que les déformations de 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1 ne sont pas assez précises car on ne contrôle pas le Frobenius lors de la déformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons donc adapter ces déforma- tions à l’idée de la preuve de la Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18 qui est de déformer étape par étape le long des Rn+1 → Rn et d’utiliser le Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (1) X(m1,m2,m3,m4) (2,2) ⊂ X(m1,m3) (2,2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m1m2,m3,m4) (2,2) et 0 ⊂ ω(1) (1) ⊂ ω(2) (1) ⊂ ω(3) (1) ⊂ ω(4) (1) la filtration associée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition on a les égalités : ω(1) (1) = F (1) (1) , ω(2) (1) = E(1)[u] ω(3) (1) = u−1(ω(1) (1)), ω(4) (1) = u−1(ω(2) (1)) = E(1)[u2] On fixe des bases : ⟨v(1) (1)⟩ = ω(1) (1), ⟨v(1) (1), v(2) (1)⟩ = ω(2) (1) Comme dans la preuve de la Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18, d’après le Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7, on dispose d’un relèvement F (1) (2) de F (1) (1) sur R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On fixe un relèvement v(1) (2) sur R2 de v(1) (1) tel que F (1) (2) = ⟨v(1) (2)⟩ et un élément α(2) ∈ u−1(F (1) (2) )\\E(2)[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On choisit également un élément v(2) (2) ∈ E(2)[u] qui relève v(2) (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit alors la déformation sur R2 comme suit : E(2) = E ⊗R1 R2 ω(1) (2) = ⟨v(1) (2)⟩ = F (1) (2) ω(2) (2) = ω(1) (2) ⊕ ⟨v(2) (2) + tα(2)⟩, ω(3) (2) = u−1(ω(1) (2)), ω(4) (2) = E(2)[u2] Par construction on a bien u · (v(2) (2) + tα(2)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On continue le processus par récurrence sur n ≥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Supposons donnés E(n), ω(k) (n) etc sur Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 44 lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 on dispose d’un relèvement canonique F (1) (n+1) sur Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la déformation sur Rn+1 comme suit : E(n+1) = E ⊗R1 Rn+1 ω(1) (n+1) = F (1) (n+1) ω(3) (n+1) = u−1(ω(1) (n+1)), ω(4) (n+1) = E(n+1)[u2] et on prend n’importe quel relèvement ω(2) (n+1) ⊂ u−1(F (1) (n+1)) de ω(2) (n), la condition ω(2) (n+1) ̸= E(n+1)[u] étant assurée par cette propriété de relèvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On obtient par passage à la limite un groupe p-divisible G ∈ BTPR(R) qui satisfait les propriétés souhaitées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) X(m1,m3) (2,2) ⊂ X(m1,m3) (3,1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ X(m3) (2,2) et 0 ⊂ ω(1) (1) ⊂ ω(2) (1) ⊂ ω(3) (1) ⊂ ω(4) (1) la filtration associée.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par définition on a les égalités : ω(1) (1) = F (1) (1) , ω(3) (1) = u−1(ω(1) (1)), ω(4) (1) = E(1)[u2] Soit v(4) (1) ∈ ω(4) (1) tel que ω(4) (1) = ω(3) (1) ⊕ ⟨v(4) (1)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Comme dans la preuve de la Propo- sition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18, d’après le Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7, on dispose d’un relèvement F (1) (2) de F (1) (1) sur R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la déformation sur R2 comme suit : E(2) = E ⊗R1 R2 ω(1) (2) = F (1) (2) , ω(3) (2) = u−1(ω(1) (2)) Ensuite on choisit un élément α(2) ∈ u−1(ω(3) (2))\\E(2)[u2] et un élément v(4) (2) ∈ u−1(ω(3) (2) ∩ E(2)[u] qui relève v(4) (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit alors : ω(4) (2) = ω(3) (2) ⊕ ⟨v(4) (1) + tα(2)⟩ On continue le processus par récurrence sur n ≥ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Supposons donnés E(n), ω(k) (n) etc sur Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='7 on dispose d’un relèvement canonique F (1) (n+1) sur Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la déformation sur Rn+1 comme suit : E(n+1) = E ⊗R1 Rn+1, ω(1) (n+1) = F (1) (n+1), ω(3) (n+1) = u−1(ω(1) (n+1)) et on prend n’importe quel relèvement ω(2) (n+1), ω(4) (n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Les propriétés ω(2) (n+1) ̸= E(n+1)[u] et ω(2) (n+1) ̸= E(n+1)[u2] sont assurées par cette propriété de relèvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On obtient par passage à la limite un groupe p-divisible G ∈ BTPR(R) qui satisfait les propriétés souhaitées.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Avant de conclure, il reste à calculer l’ensemble Aµ• des strates non vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' C’est l’objet du lemme suivant : Lemme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit (λ, T) ∈ Tµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On a : Sh(m1=0) (λ,T) ̸= ∅, Sh(m1̸=0) (λ,T) ̸= ∅ En d’autres termes l’ensemble des strates non vides est donné par : Aµ• = Tµ• ∪ (Tµ• × {m1}) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 45 Remarque 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Avec des mots plus concrets, le lemme ci dessus, combiné aux pro- position précédentes, dit que l’invariant σ-linéaire m1 interagit naïvement avec la stra- tification définie par l’ensemble Tµ• (linéaire).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Démonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après la proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='18 et les calculs de déformations précédents il suffit de montrer que la strate X(m1,m2,m3,m4) (2,2) est non vide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Nous allons utiliser les travaux de Goren et Andreatta [AG03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Soit x ∈ ShPEL (2,2) un point défini sur un corps k algébriquement clos et G le groupe p-divisible associé.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' D’après la Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='10 de loc cit il existe une base (e1, e2) de E (G) telle que la filtration de Hodge soit donnée par : ωG = ⟨u2e1, u2e2⟩ et que le Frobenius soit donné dans cette base par : F = � um cu2 u2 0 � où m ≥ 2 et c ∈ (OL ⊗ k)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On définit la filtration de Pappas Rapoport comme suit : ω(1) = ⟨u3e2⟩, ω(2) = ⟨u3e1, u3e2⟩, ω(3) = ⟨u3e1u2e2⟩, ω(4) = ⟨u2e1, u2e2⟩ Cela définit un point ˜x ∈ ShPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Un simple calcul montre que pour cette filtration on a : F (1) := F(u−1(ω(3))(p)) = ⟨cu3e2⟩ En particulier ω(1) = F (1) et par conséquent m1(˜x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il est également simple de voir que m2(˜x) = m3(˜x) = m4(˜x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Par conséquent ˜x ∈ X(m1,m2,m3,m4) (2,2) et cette dernière strate est donc non vide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' □ Théoreme 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Pour L/Qp totalement ramifiée de degré e = 4 la stratification : ShPR = � (λ,T)∈Aµ• ShPR (λ,T) est une bonne stratification où les relations d’adhérences sont données par la relation d’ordre naïve sur Aµ• 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Une fois la stratification par le polygone de Hodge de ShPEL réin- terprétée via le morphisme lisse ShPEL → Hecke = � L+G\\Gr � il est naturel de stratifier le modèle de Pappas-Rapoport ShPR via le morphisme lisse ShPR → Heckeµ• = � L+G\\ � Grµ• � Le problème est alors de décrire les L+G-orbites dans � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans cet article nous nous sommes intéressé au morphisme de convolution m : � Grµ• → Gr≤|µ•| STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 46 Ce morphisme étant L+G-équivariant les strates (m−1(Grλ))λ≤|µ•| sont des unions de L+G-orbites dans � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Si la preuve de la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='16 semble être « à la main »et si elle ne se généralise pas à d’autres situations (voir [BH22b] Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='9) c’est parce qu’elle n’est pas naturelle : la stratification la plus naturelle est celle induite par la décomposition en orbites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Conjecture 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Il existe un ensemble fini partiellement ordonné (X, <) décrivant les L+G-orbites dans � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En d’autres termes on dispose d’un homéomorphisme |Heckeµ•| ≃ X où X est muni de la topologie induite par <.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On consultera [PWZ12] (§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='1, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2) pour plus de détails concernant la stratification d’un espace en orbites sous l’action d’un groupe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le cas de petites dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans [Bij22], S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='Bijakowski a répondu positive- ment à la conjecture ci-dessus dans le cas Unitaire et dans le Hilbert-Siegel lorsque e ≤ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Énonçons le résultat dans le cas Hilbert totalement ramifié d’indice de ramification e = 3 Théoreme 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='2 ([Bij22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' La classe d’isomorphisme d’une filtration de Pappas-Rapoport (0 ⊂ ω(1) ⊂ ω(2) ⊂ ω(3)) est entièrement déterminée par les invariants Hodge(ω(3)), Hodge(ω(2)) et Hodge(ω(3)/ω(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' De plus ces invariants définissent une bonne stratifi- cation de ShPR où la relation d’ordre est la relation naïve : (λ1, λ2, λ3) ≤ (λ′ 1, λ′ 2, λ′ 3) si λi ≤ λ′ i pour i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Faisons quelques commentaires : (1) Dans le théorème ci-dessous, le fait que les classes d’isomorphismes induisent une bonne stratification est automatique : les classes d’isomorphismes de filtration de Pappas-Rapoport correspondent aux L+G-orbites dans � Grµ•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Le résultat non trivial du théorème ci-dessus concernant la stratification réside dans le calcul explicite des relations d’adhérences entres orbites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (2) Dans le cas Hilbert, c’est-à-dire dans le cas G = GL2 et µ• = (1, 0)e les gradués Λ(i)/Λ(i−1) sont de dimension 1 et on obtient donc les équivalences 1 : Hodge(Λ(2)) = (3, 3) ⇔ Hodge(Λ(2)/Λ(0)) = (2, 0) ⇔ m2 ̸= 0 Hodge(Λ(3)/Λ(1)) = (2, 0) ⇔ m3 ̸= 0 En d’autres termes dans le cas e = 3 la donnée (Hodge(ω), m3, m2) détermine entièrement la classe d’isomorphisme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait on peut être plus précis : (m3, m2) détermine Hodge(ω) car dans le cas e = 3 il y a que deux possibilités (2, 1) et (3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' (3) Pour le cas e = 4 nous avons vu que les strates X(m3) (2,2) et X(m3) (3,1) étaient non vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Dans cette situation les invariants (mi)i ne déterminent donc pas l’invariant de Hodge, et donc en particulier ils ne détectent pas la classe d’isomorphisme non plus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On utilise la notation abusive Hodge(Λ(i)/Λ(i−2)) = Inv(Λ(i), Λ(i−2)) STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 47 (4) Dans le cas Hilbert-Siegel, c’est-à-dire pour G = GSp2g, les invariants ci-dessous ne sont pas assez fins pour détecter la classe d’isomorphisme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' En fait dans le cas général on a l’équivalence suivante : Hodge(Λ(i)/Λ(i−2)) = (2, 0)g ⇔ mi ̸= 0 En d’autres termes le lieu de non annulation de l’invariant mi coïncide avec la strate maximale de la stratification par l’invariant Hodge(Λ(i)/Λ(i−2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Ce dernier est donc le bon invariant à considérer en dimension supérieure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' STRATIFICATION DES VARIÉTÉS DE HILBERT EN PRÉSENCE DE RAMIFICATION 48 Références [AG03] Fabrizio Andreatta and Eyal Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Goren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Geometry of Hilbert modular varieties over totally ramified primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' International Mathematics Research Notices, 2003 :1785–1835, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [BBM82] Pierre Berthelot, Lawrence Breen, and William Messing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Théorie de Dieudonné Cristalline II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Springer Berlin Heidelberg, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [BH16] Stephane Bijakowski and Valentin Hernandez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Groupes p-divisibles avec condition de Pappas- Rapoport et invariants de Hasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Journal de l’Ecole Polytechnique - Mathematiques, 4, 11 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [BH22a] Stéphane Bijakowski and Valentin Hernandez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On the geometry of the Pappas-Rapoport models in the (AR) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Preprint on webpage at http://stephane-bijakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='perso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='fr/AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [BH22b] Stéphane Bijakowski and Valentin Hernandez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' On The geometry of the Pappas-Rapoport mo- dels for PEL shimura varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Journal of the Institute of Mathematics of Jussieu, page 1–43, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [Bij16] Stéphane Bijakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' The compatibility with the duality for partial Hasse invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 03 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [Bij22] Stéphane Bijakowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Hodge Stratification in low dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [BM79] Pierre Berthelot and William Messing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Théorie de Dieudonné cristalline I.' metadata={'source': 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l’École Normale Supérieure, 7(1) :53–88, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' [dJ91] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' de Jong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' he Moduli Spaces of Principally Polarized Abelian Varieties with Γ0(p)-level Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Rijksuniversiteit Utrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} +page_content=' Mathematisch 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE4T4oBgHgl3EQfbwzK/content/2301.05078v1.pdf'} diff --git a/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/2301.04425v1.pdf.txt b/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/2301.04425v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa2e4305fadc2149f0ee0081dbd8cae945d7d363 --- /dev/null +++ b/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/2301.04425v1.pdf.txt @@ -0,0 +1,3396 @@ +Ultrafast two-colour X-ray emission spectroscopy reveals excited +state landscape in a base metal dyad +M. Nowakowski1,†, M. Huber-Gedert1,†, H. Elgabarty1, J. Kubicki2, A. Kertem2, N. Lindner2, D. +Khakhulin,3 F. Lima3, T.-K. Choi3,4, M. Biednov3, N. Piergies5, P. Zalden3, K. Kubicek3, A. Ro- +driguez-Fernandez3, M. Alaraby Salem1,T. Kühne1, W. Gawelda2,6,7, M. Bauer1* +1 Chemistry Department and Center for Sustainable Systems Design (CSSD), Faculty of Science, Paderborn Uni- +versity, Warburger Straße 100, 33098 Paderborn, Germany +2 Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, Poznań, 61-614, Poland +3 European X-Ray Free-Electron Laser Facility GmbH, Holzkoppel 4, 22869 Schenefeld, Germany +4 PAL-XFEL, Jigok-ro 127-80, 37673 Pohang, Republic of Korea +5 Institute of Nuclear Physics Polish Academy of Sciences, Kraków, 31-342, Poland. +6 Departamento de Química, Universidad Autónoma de Madrid, Campus Cantoblanco, 28047 Madrid, Spain +7 IMDEA Nanociencia, Calle Faraday 9, 28049 Madrid, Spain + +ABSTRACT: Effective photoinduced charge transfer makes molecular bimetallic assemblies attractive for applications +as active light induced proton reduction systems. For a more sustainable future, development of competitive base metal +dyads is mandatory. However, the electron transfer mechanisms from the photosensitizer to the proton reduction catalyst +in base metal dyads remain so far unexplored. We study a Fe-Co dyad that exhibits photocatalytic H2 production activity +using femtosecond X-ray emission spectroscopy, complemented by ultrafast optical spectroscopy and theoretical time- +dependent DFT calculations, to understand the electronic and structural dynamics after photoexcitation and during the +subsequent charge transfer process from the FeII photosensitizer to the cobaloxime catalyst. Using this novel approach, +the simultaneous measurement of the transient K X-ray emission at the iron and cobalt K-edges in a two-colour exper- +iment is enabled making it possible to correlate the excited state dynamics to the electron transfer processes. The meth- +odology, therefore, provides a clear and direct spectroscopic evidence of the Fe→Co electron transfer responsible for the +proton reduction activity. +INTRODUCTION +FeII complexes can operate as light-harvesting compo- +nents in bimetallic molecular assemblies or dyads, to con- +vert solar to chemical energy by ultrafast charge transfer +(CT) to a second, catalyst metal for photocatalytic water +splitting.1 In terms of sustainability, the second metal +should be abundant. Cobaloxime fulfils this requirement.2– +4 Despite the reported short lifetimes of metal-to-ligand +charge transfer (MLCT) states in iron(II) photosensitizers, +photocatalytic proton reduction activity was reported for +FeII-CoIII dyads.5 However, its activity remains mysterious, +as no charge transfer from the Fe to the Co center could +be observed experimentally. Thus, rational improvement +of Fe-Co dyads requires a radically different approach to +understand the working principle. A major challenge are +the ultrafast photophysics at the FeII center,6 and the diffi- +culty to monitor the CT from the photosensitizer to the cat- +alyst with element specificity and in real-time.1,7,8 Upon +photoexcitation the excited state dynamics in dyads can +involve MLCT and ligand-to-metal charge transfer states +(LMCT), metal-centred (MC) and ligand-mediated metal- +to-metal charge-transfer states (M’MCT).9,10 The de-exci- +tation cascade is an interplay between MC and CT states, +modulated by intramolecular vibrational energy dissipa- +tion, strong spin-orbit coupling such as intersystem cross- +ings (ISC), and internal conversions (IC).8,11 The funda- +mental principles guiding the properties are typically iden- +tified using laser spectroscopy.12,13 Noble metal com- +plexes exhibit long-lived CT states, which can easily be +followed with optical spectroscopy, due to the associated +intense absorption bands in UV-Vis range.14 On the other +hand, in most iron photosensitizers, the smaller ligand +field splitting leads to an unfavoured energetic order of +E(MLCT) > E(MC).2 In addition, MC states are hardly ac- +cessible in UV-Vis spectral range.15 +Contrary, X-ray emission spectroscopy (XES) is very sen- +sitive to MC states due to the localized character of core +levels. Both K (2p→1s) and K (3p→1s) emission lines +provide characteristic signatures of the multiplicity of the +involved transient MC states.16 For monomeric iron car- +bene photosensitizers, femtosecond XES could uniquely +reveal details of the excited states.13,16,17 The excited +states of [Fe(bmip)2]2+ [bmip = 2,6-bis(3-methyl-imidaz- + +2 +ole-1-ylidine)-pyridine] show a complex branching pat- +tern. One path is dominated by a long-lived 3MLCT, while +the second includes a rapid hot MLCT* to 3MC transition- +connected to bond oscillations in form of wavepacket dy- +namics11,16,18 which are a stabilizing factor for long-lived +3MLCT states.19 Ultrafast X-ray absorption near edge +structure spectroscopy (XANES) on photoactive Fe-Co +Prussian blue analogues revealed that a spin transition at +the Co centre preceeded CT between the Fe and Co cen- +ter.20 More recently, the photoinduced M’MCT transition in +a bimetallic Fe-Ru assembly was shown to have a critical +impact on the solvent organization around the excited +molecule.21 +We demonstrate the unique potential of a two-colour X- +ray emission spectroscopy (2C-XES) in photocatalysis re- +search. It allows for simultaneous, ultrafast detection of +the Fe and Co Kα XES in a [Fe-BL-Co] assembly of a het- +eroleptic FeII photosensitizer with two different biscar- +bene-pyridine ligands (C^N^C) connected to a cobalox- +ime catalyst via a bridging ligand (BL).5 The dynamics of +the excited state decay are monitored at the Fe and Co +sites to follow the departure of the charge from the photo- +sensitizer and its arrival at the catalyst in real-time. As +such, our experimental approach eliminates uncertainties +related to the charge transfer event timescale and solves +the puzzle about a possible charge transfer in base metal +dyads. +RESULTS + The dyad is synthesized combining a heteroleptic tetra- +NHC FeII photosensitizer [Fe-BL] coordinated by a 2,6- +bis[3-(2,6-diisopropylphenyl)imidazol-2-ylidene]pyridine +and a 2,6-bis(3-methyl-imidazol-2-ylidene)-4,4'-bipyridine +ligand (BL) with a CoIII cobaloxime catalyst, as presented +in Fig. 1a.5 A 4,4′-bipyridine (bpy) linker connects both +metals with a distance of 11 Å.22,23 The ground state opti- +cal absorption spectra of the dyad in comparison to the +constituting components [Fe-BL] (red) and cobaloxime +(Co(dmgH)2Cl(py)=[Co], green) are shown in Fig. 1c. The +top panel shows two absorption bands for the photosen- +sitizer at 398 nm and 481 nm (Fig. 1c, top). By coordina- +tion of the cobaloxime in [Fe-BL-Co] (black), the UV-Vis +spectrum of the dyad changes in a distinct manner: The +398 nm band remains unchanged, but the 481 nm band +is shifted to 497 nm, and a new band appears at 444 nm +(Fig. 1c, top). Cobaloxime itself shows only a weak ab- +sorption around 400 nm. +Quantum chemical assignment of the optical absorp- +tion bands In order to understand the properties of the +electronic excited states of [Fe-BL-Co], the involved tran- +sitions must be identified. As a first step, we carefully +benchmarked TPSSh/TDDFT excited-state calculations +of the photosensitizer against CASSCF/NEVPT2 (SI, sec. +1b-c). The latter combination of a multireference wave- +function with a perturbative treatment of electron correla- +tion accounts for both static and dynamic electron corre- +lation effects, and has been shown to yield highly accurate +results, but computational cost steeply increase with the +number of correlated orbitals.24 Both techniques reveal +that the bands at 398 nm and 481 nm in [Fe-BL] (Fig.1c, +top, red) are a mixture of MLCT transitions from FeII to +both the terminal and the BL (SI, sec. 1a-b,e). Together +with the TPSSh/TDDFT computations of UV-Vis spec- +trum, this assignment is used to understand the absorp- +tion properties of [Fe-BL-Co] (SI, sec. 1c and 1f). The +lower panel of Fig. 1c shows the TDDFT spectrum of the +dyad. The experimentally observed 398 nm absorption is +described by transitions a and b (SI, sec. 1c). Like in the +photosensitizer, they are composed of MLCT transitions +from iron to the terminal and bridging ligand. Additionally, +the electron density is transferred from the FeII to the CoIII +center along the bridging ligand in the form of an M’MCT +transition (SI, sec. 1c). The donor-acceptor contributions +to the latter are shown in Fig. 1b. The absorption at 497 +nm is dominated by an MLCT transition to the bridging +Fig. 1. a) Structure of the [Fe-BL-Co] dyad; b) Fe → Co CT (M’MCT): grey color indicates holes, red– electrons c) UV-Vis +data for [Fe-BL], [Co] and [Fe-BL-Co] (top) overlapped with TAS results for selected delay times and TD-DFT UV-Vis spec- +trum for the dyad (bottom). +a +b +-0.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +335 +410 +485 +560 +635 +0.00 +0.04 +0.08 +DA + 0.50 ps + 1.00 ps + 5.00 ps + 10.0 ps + 15.0 ps + 30.0 ps + 60.0 ps +3MLCT +Ground state bleach + e ·10-6 / cm-1 M-1 + Co(dmgH)2Cl(py) + [Fe-BL-Co] + + [Fe-BL] +[Fe-BL-Co], TPSSh + SMD(MeCN) + def2-TZVP +wavelength / nm +osc. strength / a.u. +b +a +c +c + +PF3 +ligand together with a weak M’MCT contribution (transi- +tion c). The shift of the 497 nm band in the dyad spectrum +compared to the photosensitizer spectrum (481 nm) is +well reproduced by TDDFT (480.6 nm vs 446.0 nm, SI, +sec. 1b-c). It is due to an increased charge transfer to the +terminal pyridine ring (transition c in Fig. 1) and revealed +by charge transfer components obtained for the dyad (SI, +sec. 1f). Unfortunately, TDDFT could not resolve the 444 +nm band in the dyad spectrum. After re-evaluation of the +former interpretation5, it is assumed that it is also present +in the photosensitizer spectrum but overlaps with the 481 +nm band. +In conclusion, the nature of the transitions in the dyad is +similar to those of the photosensitizer. Most important +however, additional M’MCT contributions are found in all +bands. Due to the unchanged absorption band at 398 nm +in both compounds, only weak LMCT absorption of co- +baloxime2,25 and the M’MCT contribution in [Fe-BL-Co], an +excitation wavelength of 400 nm was chosen for ultrafast +transient spectroscopy. +Transient Absorption Spectroscopy Transient absorp- +tion spectroscopy (TAS) results for [Fe-BL-Co] are pre- +sented in Fig. 1c. The ground state bleach occurs at 370- +560 nm, an excited-state absorption is observed <370 nm +and >560 nm. The transient absorption >560 nm is as- +signed to a 3MLCT state5 and its kinetics are composed of +three time constants (SI, sec. 2). The first component +(<100 fs) in this model takes into account all coherent ar- +tefacts26 and possible 1MLCT contribution. The second +component (2=350 fs) can be ascribed to either the re- +laxation from the hot 3MLCT* to thermally relaxed +3MLCT27,28, or to a 1MLCT → 3MLCT transition.11,15–17 This +is supported by the excited-state TDDFT, where the first +acceptor state for the 400 nm excitation is a 1MLCT. The +longest component can be assigned to the lifetime of the +relaxed 3MLCT state.6,27,29,30 Due to the very similar results +obtained in each fit (SI, sec. 2), a reliable average value +of 12.8±1.2 ps for the lifetime of the 3MLCT state is ob- +tained. For the constituting photosensitizer [Fe-BL] a +3MLCT lifetime of 11.1±0.4 ps is found (SI, sec. 2). The +increased 3MLCT lifetime in [Fe-BL-Co] is interpreted as +an indirect signature of CT processes, leaking into the re- +laxation channel over the 3MLCT state. Yet, it does not +provide unequivocal proof for a Fe→Co charge transfer +due to a lack of direct spectroscopic signatures for altered +charge densities at both Fe and Co cobalt centers. This +gap can be closed by XES. +Fe Kα XES dynamics XES is governed by different se- +lection rules than the optical absorption. The XES signal +originates from localized core electrons, and through the +width of the Kα1 XES line, it is directly proportional to the +effective number of unpaired d-electrons31 and cova- +lency32. Using a von Hamos emission spectrometer,33 in a +2C-XES scheme34, spectra for both Fe and Co could be +collected truly simultaneously, without any ambiguity of +the time-zero on a femtosecond timescale.35 Fig. 2 shows +the early temporal evolution of the two XES signals and +their kinetic traces along with the selected integration +ranges for both elements. +In [Fe-BL-Co] three time constants of 1,FeCo<0.14 ps, +2,FeCo=10.38(40) ps and 3,FeCo=1.74(18) ps are obtained +from fitting of the transient kinetics at the iron Kα1 emis- +sion, while for [Fe-BL] 1,Fe~0.25 ps, 2,Fe=8.98(27) ps and +3,Fe=1.71(35) ps are found (SI, sec. 3). The most notable +difference is thus the increased longest lifetime 2 in the +dyad, which is similarly observed in TA measurements.5 +The difference of 2 ps is attributed to the different sensi- +tivity of TA and XES towards CT states - MC states are +“optically silent” in UV-Vis spectral range. +Both singlet and triplet 1/3MLCT states of Fe compounds +have near-identical Kα XES signatures, since both have a +single Fe-localized unpaired d-electron.7 Moreover, since +the coupling of the deep 2p core-hole with the 3d manifold +is weak, Kα XES has little sensitivity to ISC inside the +Fig. 2. Fe and Co Kα1,2 transient XES line intensities of [Fe- +BL-Co] for delay times of -0.2 – 1.25 ps. Top panel: transient +XES signals at 1 ps delay time with integration regions of in- +terest (ROIs) marked by vertical dashed lines. Right panel: +integrated area under transient XES Fe Kα1 and Co Kα1 main +feature in function of delay time (points) with corresponding +fitted model (lines, top). +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +6390 +6400 +6910 +6920 +6930 +Delay time / ps +Emission energy / eV +Laser OFF +Laser ON +Intrgr. Area + Co fit x4 + Fe fit + Co data x4 + Fe data +Norm. Int +a.u. +Integration ROI +Fe K1 +Fe K2 +Co K1 +Co K2 +Integration ROI +Fig. 3. Kinetics of the Co Kα emission in [Fe-BL-Co] and +cobaloxime after 400 nm excitation along with corre- +sponding fits. Differential signal: [Co]- [Fe-BL-Co] is +marked as blue lines (data+fit). Filled areas represent un- +certainties. +0.0 +0.5 +1.0 +-0.3 +0.0 +0.3 +0.6 +0.9 +1.2 +Norm. Int. Counts / a.u. +t - t0 / ps + cobaloxime + cobaloxime fit + diff. of data + diff. of fit + [Fe-BL-Co] + [Fe-BL-Co] fit + +4 +MLCT manifold. With regards to metal spin multiplicity, +there is a significant difference between the 3MLCT +(Sloc=1/2) and 3MC (Sloc=1) states. Any relaxation process, +involving either one of these states, to the singlet ground +state is thus visible in transient Kα XES experiments. In +agreement with previously reported values and the cur- +rent TA results, the XES time constants are assigned in +the following way: the shortest lifetime 1 in both [Fe-BL] +and [Fe-BL-Co] dyad corresponds to a 3MLCT*→3MC +channel.36 The time constant 3 can be attributed to the +3MC state decaying into the ground state.37,38 The value +of time constant 1 fully agrees with reports of +3MLCT*→3MC channels in Fe(II)-NHC complexes.37,38 +Moreover, TDDFT excited-state potential energy surfaces +indeed identify a 3MC surface that intersects both the +1MLCT and the 3MLCT close to the Franck-Condon re- +gion. The 3MC is identified, using a Mulliken electron-hole +population analysis, as a triplet state containing both a +hole and an additional electron on the Fe metal, because +of the Fe(dxy/dyz/dxz) → Fe(dx2-y2/dz2) transition (SI, sec. +1d). Time constant  is assigned to the 3MLCT state.36–38 +Co Kα XES dynamics The Co K transient kinetics of the +dyad [Fe-BL-Co] is constituted of three time constants +1,FeCo=0.25(1) ps, 2,FeCo=4.12(1.39) ps and 3,FeCo~23.39 +ps, a striking difference to pure cobaloxime [Co], where +two +time +constants +of +1,Co=2.76(31) +ps +and +2,Co=23.39(1.82) ps are found (SI, sec. 3a). This differ- +ence is also obvious in the kinetics of the Co Kα XES in +[Co] (green) and [Fe-BL-Co] (red) in Fig. 3, which differ +substantially in shape over the first 0.5 ps (SI, sec. 3a). +The difference is caused by the short decay constant +1,FeCo=0.25(1) ps that is not present in pure cobaloxime. +This time constant thus represents a new excited state +population channel, created by the formation of the dyad, +which at later timescales is indispensable for photocata- +lytic hydrogen generation and for which the optical ab- +sorption data shows Fe-Co M’MCT contribution. Conse- +quently, the differential signal in Fig. 3 is the real-time sig- +nature of CT from the Fe to the Co center in [Fe-BL-Co]. +The direct excitation of the cobaloxime reflected in the Co +kinetics of Fig. 3 (green) corresponds to an LMCT state. +According to our DFT calculations,2 the HOMO in co- +baloxime is composed of degenerated π orbitals of the +dmgH ligand, and the LUMO consists of the Co dz2 orbital +leading to a very weak LMCT absorption at 396 nm. The +observed low cross-section excitation populates this +LMCT state of Co with a lifetime of 4.12(1.39) ps in [Fe- +BL-Co]. +Since cobaloxime has a documented activity as a proton +reduction catalyst,1,39 the increased catalytic activity of +[Fe-BL-Co] compared to [Fe-BL] + cobaloxime originates +from the M’MCT states in [Fe-BL-Co]. Note, the signal we +observe originates from linear combination of differently +excited species, since M’MCT and LMCT states cannot +exist simultaneously in the same molecule. The result is +evident, despite a low CT yield for our prototype dyad. +Nuclear motion detected by Fe Kα XES Both [Fe-BL] +and [Fe-BL-Co] show a pronounced, but distinct, coherent +nuclear wavepacket signatures in the transient iron K1 +kinetics. For the photosensitizer, the oscillations could be +modelled by a single damped periodic function (Fig. 4, SI, +sec. 3c), while in case of the dyad, it is composed of two +contributions (Fig. 4 and SI, sec. 3c). [Fe-BL] and [Fe-BL- +Co] share a half-period of 0.28(2) ps and 0.26(3) ps, re- +spectively. Similar oscillation half-periods were observed +in other systems. 16,40 +An additional oscillation of 0.19(1) ps appears in the dyad +(SI, sec. 3c). The coherent oscillation detected in [Fe-BL- +Co] (Fig. 4a) is a combination of signals observed in the +photosensitizer and the additional oscillation (T1/2=0.19 +ps) related to the coordination of cobaloxime. The statisti- +cal significance of the superposition could be proven (sec. +3). Calculated excited state potential energy surfaces +show that the oscillations appear along the Fe-N bonds +with the equilibrium at 2.05 Å (3MLCT*/3MC crossing, see +Fig. 5a). TDDFT results indicate several vibrational fre- +quencies in the range around 175 cm-1, exhibiting a col- +lective twisting motion of the bridging ligand, accompa- +nied by a rotational distortion and slight stretching of the +Fe-N bond. Raman spectra also exhibit intense bands for +[Fe-Co-BL] in 175-225 cm-1 range, present neither in [Fe- +BL], nor in cobaloxime (SI, sec. 3c). While the 0.26 ps +half-period can be associated with the spin state transition +due to the 3MLCT*/3MC crossing, the 0.19 ps oscillation is +likely due to the rotation of the cobaloxime moiety around +the Fe-Co axis. This motion could affect the charge trans- +fer due to the rotation of the pyridine ring, and modulation +of the π* orbitals overlap. +The excited state landscape in the [Fe-BL-Co] dyad can +be substantiated with these results. Femtosecond XES +study on [Fe(bmip)2]2+ showed excited state branching, in +which a vibrational wavepacket nearly identical to the one +in [Fe-BL-Co] is observed.16 A 3MC is partially populated +from the vibrational excited 3MLCT* state. Since this +wavepacket motion is associated with the MC state,19 it is +not visible in optical TAS measurements. With the minimal +spectral difference between 1MLCT/3MLCT states both in +TAS and XES, the shortest time constant of 1,Fe/FeCo in +[Fe-BL] and [Fe-BL-Co] is associated with a transition +from the 3MLCT* to 3MC state. The longest time 2,Fe/FeCo +reflects the +3MLCT→3MC pathway. The remaining +3,Fe/FeCo is assigned to the 3MC→GS recovery.7,11,17,41,42 +Population analysis Kinetic modelling can facilitate the +interpretation of the obtained time constants by testing dif- +ferent reaction models. Details of the approach can be +found in the supplementary information. At the Fe center +in [Fe-BL] and [Fe-BL-Co], an additional time constant of +0.22(7) ps is obtained. According to our TDDFT calcula- +tions, this can be related to the 1MLCT→3MLCT transition +after a population of the first excited 1MLCT state (SI, sec. +3e). This short time constant includes IC and ISC.11,16–18,40 +It is also in good agreement with the 350 fs component +obtained via TA. According to the proposed reaction +scheme, the 3MLCT* state, which is populated during the +1MLCT→3MLCT*→3MLCT decay, branches into a α +(3MLCT*→3MLCT→3MC) and β channel (3MLCT*→3MC), +with contributions of 79(5) % and 21(5) %, respectively, +as shown in Fig. 5a. In [Fe-BL], the branching ratio is 83 +% to 17 %, respectively (SI, sec. 3f). The observed wave- +packet oscillations originate from the β pathway.40 + +5 +Most importantly, an additional deactivation channel, orig- +inating from the 3MLCT state in the form of an M’MCT +electron transfer in the [Fe-BL-Co] dyad, is resulting from +the kinetic fitting as well. This transfer is clearly visible +when the 3MLCT population of the pure photosensitizer +[Fe-BL] (SI, sec. 3f) and the dyad [Fe-BL-Co] (Fig. 5c) in +the short time window is compared. In the former, the rise +of the 3MLCT population is initially damped, while in the +latter, the population of 3MLCT rises. The obtained value +of the CT rate is very consistent with the magnitude of dif- +ferences observed for the excited state kinetics at the Co +center in [Fe-BL-Co] and cobaloxime (cf. Fig. 3). +A two-state model with subsequent decay is used for the +cobaloxime (SI, sec. 3f, Fig. S3.9-3.10), consisting of the +LMCT state directly populated upon 400 nm excitation +and decaying to a lower-level state within 2.78(3) ps. For +[Fe-BL-Co] (Fig.5a), an additional electron transfer-ac- +ceptor state (M’MCT) is compulsory from the experi- +mental results. The M’MCT decays in 0.25 ps,21 parallel +to the directly excited LMCT decay. The amplitude ratio +between the direct excitation and CT transfer yield is 43.0 +% to 57.0 %, close to the value obtained via cross-section +analysis (SI, sec. 3b, and sec. 3f) and in line with TDDFT +results. The kinetic fitting required an additional lowest ex- +cited state of this direct decay path, which is of unknown +nature so far. However, an MC character is most likely.43 +The lifetime of this state is estimated to be around 23-30 +ps based on the fit results for the pure cobaloxime. Data +quality for the dyad prevent accurate fitting of this contri- +bution to the fluorescence signal in [Fe-BL-Co]. +Fig. 5a summarizes the results and conclusions from the +observed time constants, literature,11,16,17,30,42 and TDDFT +potential energy surfaces calculations along two reaction +coordinates (Fe – N bite angle and distances). The popu- +lation analysis for [Fe-BL-Co] resulting from kinetic mod- +elling is shown in Fig. 5b-c (SI, sec. 3f and the corre- +sponding diagram for [Fe-BL]). +CONCLUSIONS +Photoactive base metal dyads appear as promising alter- +native, as compared to precious metals, for inexpensive +and sustainable molecular assemblies capable of direct +harvesting of light and photocatalytic hydrogen produc- +tion. This still heavily depends on the rational improve- +ment of their performance, which involves the interplay +between their molecular design and photocatalytic prop- +erties. Our study shows the tremendous potential of ultra- +fast 2C-XES for direct characterization of photoinduced +CT processes exemplified by the case of a noble metal +free dyad [Fe-BL-Co] used in hydrogen production. In +combination with ultrafast optical spectroscopy, TDDFT +and CASSCF/NEVPT2 calculations and excited state +modelling, a CT from the FeII photosensitizer to the co- +baloxime catalyst could be proven. It contributes as a +M’MCT state of 0.25 ps lifetime to the very complex ex- +cited state landscape. In addition, we can distinguish the +direct excitation into an LMCT state of Co, which accom- +panies the CT process between both metals. The une- +quivocal determination and visualization of the ultrafast +CT is only possible by the intrinsic temporal self-calibra- +tion of the Fe and Co Kα signals in the 2C-XES experi- +ment. +With the achieved results a multitude of strategies to im- +prove the photocatalytic activity of such base metal dyads +can be deduced. It is common knowledge that the lifetime +of the 3MLCT as the first charge separated state needs to +be increased for iron photosensitizers to be active. How- +ever, from Fig. 5 it is immediately clear, that this is even +more important here. A decreased 3MLCT energy would +reduce the contribution of the 3MLCT→3MC decay chan- +nel, potentially in favour of the population of the M’MCT +state. Another way of decreasing non-CT decay channels +would be a reduction of the 3MLCT*→3MC contribution. +Since this pathway is connected to the nuclear wave- +packet, the associated vibrational motions might play a +crucial role. Further restriction of Fe-N oscillations, either +via replacing N with C atom or construction of a more rigid +ligand structure could selectively increase the 3MC en- +ergy. Both Fe-N and Fe-BL-Co motions are involved here +according to the presented results, and substitution of the +pyridine by a cyclometalated ligand might be a suitable +exchange for Fe-N. +The presented results thus offer a first step towards a ra- +tional design of base metal dyads for photocatalytic pro- +ton reduction reactions by direct observation and quanti- +fication of CT process in functional bimetallic photosensi- +tizer-catalyst assembly by 2C-XES. + +Fig. 4. Coherent nuclear wavepacket signals (black), fit- +ted oscillatory functions (red) part, damping (grey) for: a) +Fe part of [Fe-BL-Co], where additionally a non-damped +parts are visible (blue, purple); b) same for [Fe-BL]. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.03 +0.00 +0.03 +Norm. Integr. Counts - fit / a.u. +Probe delay / ps + experimental data + fit + Damping function + 3.71 THz + 5.56 THz +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.10 +-0.05 +0.00 +0.05 +0.10 +Norm. Integr. Counts - fit / a.u. +Probe delay / ps + 3.49 THz + experimental data + fit + Damping function +a +b + +6 + + +METHODS +UV-Vis spectroscopy +The investigated complexes were dissolved in acetonitrile +(spectroscopic-grade, 2.5·10-4 mol/L). UV-Vis spectra +were measured in 0.1 cm quartz cuvettes on a Lambda +465 spectrophotometer from PerkinElmer (Waltham, +Massachusetts, USA). Cobaloxime (1·10-5 mol/L) was +measured with a Lambda 45 double-beam UV spectro- +photometer from Perkin Elmer (Waltham, Massachusetts, +USA). +TAS spectroscopy +The experimental setup was described elsewhere.5,44 +Femtosecond transient absorption dynamic studies of +[Fe-BL] and [Fe-BL-Co] were conducted using modified +commercial Helios spectrometer (Ultrafast Systems, Sar- +asota, Florida, USA) with the IRF value of 120 fs. TAS +spectra were recorded for the 400 nm excitation in 60 ps +temporal range. The laser pulse energy was 2 μJ. Con- +centrations were chosen to be identical to the time re- +solved X-ray experiments (10 mM, MeCN), which caused +high absorbance of the solutions. Therefore, optimized +signal transmission was ensured by a 0.12 μm flow cell +with CaF2 windows. The use of a micro annular gear pump +(~1 ml/s flow) guaranteed the excitation of fresh solution +per laser pulse and reduction of sample degradation. Sub- +traction of solvent response from each data set eliminated +the solvent contribution in the TAS data. +Transient X-ray emission spectroscopy +Simultaneous emission of Fe and Co Kα was measured +with 120 fs time resolution at the FXE instrument at +SASE1 branch of EuXFEL, Schenefeld, Germany (Fig. +S3.1).35 The [Fe-BL-Co] dyad in 10 mM solution of ace- +tonitrile (MeCN) was measured in a cylindrical liquid jet +(200 µm) and sample recirculation was provided by HPLC +pump. Sample was excited by 400 nm optical laser with +power in the range of 5.5 - 15 μJ/pulse and 50 fs pulse +length (FWHM = 83 μm and 34 μm for horizontal and ver- +tical directions, respectively) which translates to ~55 % of +excitation rate. Electronic configuration in ground and ex- +cited states were probed by the SASE X-Ray beam with +a central energy of 9.3 keV with 125 bunches per pulse +train at 0.564 MHz intra-train repetition rate (beam size +FWHM = 20 μm, pulse duration 100 fs, ~1012 pho- +tons/pulse). The X-Ray beam was operating at the stand- +ard EuXFEL mode of 10 Hz repetition rate per train and +the optical laser was at 5 Hz, meaning alternating +pumped/unpumped trains. The beams were crossed with +angle of c.a. 20°. Subsequent fluorescence emission was +collected using wavelength-dispersive 16-crystal von +Hamos XES spectrometer (Fe Kα and Co Kα with +Ge(440) and Si(531) analyzer crystal reflections at 75.4° +and 77°, respectively) and a 2D charge integrating gain- +switching Jungfrau 1M detector with matrix of 1024 x 1024 +Fig. 5. (a) Ground and excited state potential energy surfaces along the Fe-N distance (bottom x-axis) and the Fe-N bite angle +(top x-axis). Insert: state diagram for Co. State contributions at the (b) Fe in [Fe-BL-Co] (c) Co and fs-XES signal. + +a +Fe - N bite angle / +b +0.6 +101 +102 +103 +104 +105 +106 +107 +108 +4 += 0.5 +MLCT +3MLCT +a. +0.4 +0.3 +0.2 +3MC +~0.14ps +P +3 - +0.2ps +0.1 +3MLCT* +0.0 +0.20.00.20.40.60.81.0 +21 % +delay time / ps +ev +1MLCT +3MC +fit +(GS) +2 +3MLCT +gs +exp +c +79 % +1.2 +10.4 ps +- +E +Population / a.u. +1.7 ps +Co Decay: +0.8 +M'MCT +LMCT +ground +0.4 +4.12 ps +state +0.25 ps +0.0 +>23 ps +0.20.0 +0.2 +0.4 +0.6 +0.81.0 +ground state +0 +delay time / ps +1.9 +2.0 +2.1 +2.2 +2.3 +2.4 +MM'CT +exp +Fe - N distance / A +fit +LMCT +gs7 +pixels and repetition rate of 10 Hz. The timing jitter be- +tween X-Ray and optical pulses was ~70 fs FWHM. Signal +was integrated over 60 s (500 trains) per time point. For +different delay time windows, a set of data was acquired +with specified temporal step size: for -5ps -15 ps it was 1 +ps while for 1.2 ps – 3.3 ps and -1.0 ps – 1.5 ps it was 150 +fs. For single delay time measurements, signal was col- +lected for 60 s. For each measurement number of repeti- +tions was set individually to provide good S/N ratio. As a +reference also the catalyst cobaloxime and the photosen- +sitizer [Fe-BL] were measured separately in the same ex- +perimental conditions and concentrations. Due to limited +solubility, cobaloxime was measured at 5 mM. +Quantum chemical calculations +Unless otherwise stated, all calculations were carried out +with the ORCA 5.0.1 quantum chemistry package.45 +Throughout we have used the Alrich’s def2-TZVP46 basis +set, and employed the Split-RI-J method and chain of +spheres (RIJCOSX) approximation to accelerate the cal- +culation of the exchange and Coulomb terms, together +with the def2/C and def2/J auxiliary bases.47 Spin-orbit +coupling corrections were introduced using the spin-orbit +mean field method.48 Solvation of the compounds was in- +cluded via SMD49 (MeCN) and dispersion correction was +introduced via DFT-D3 with the Becke-Johnson damping +scheme (D3BJ).50,51 +Unconstrained DFT optimizations of the investigated +complexes were done with the PBEh-3c method.52–54 The +UV-Vis spectra of [Fe-BL] and [Fe-BL-Co] were calculated +using the hybrid meta-GGA functional TPSSh55, employ- +ing the Time-dependent DFT (TDDFT) and the Tamm- +Dancoff approximation. The adequacy of the method was +justified by our benchmark study on the photosensitizer +against CASSCF/NEVPT2 (SI, sec. 1a). The singlet en- +ergy transitions (60 states) have been subjected to +Gaussian broadening with a width of 0.2 eV (full width at +half-height) before converting to the nm scale and com- +pared to the experimental UV-Vis spectra of the investi- +gated complexes (cf. SI, sec. 1b-c, Fig. S1.3 and S1.4). +Donor and acceptor orbitals of selected transitions and +their spatial distribution were visualized using Avogadro +(cf. Table S1.1 and S1.2). Singlet and triplet excited state +potential energy surfaces were computed staring from the +optimized ground state geometry, by discretizing a geo- +metric pathway that involves a simultaneous stretching of +the Fe-N distances at steps 0.05 Å, and the Fe-N bite an- +gles at steps of 0.7 degrees. At each point along this path- +way, the 60 lowest lying singlet and triplet states were +computed (i.e. 120 states in total), again using the afore- +mentioned computational setup. In order to identify the +nature of any given excited state, whether it is a 1MLCT, +3MLCT, or 3MC, we have resorted to the Mulliken popula- +tion analysis coupled with an electron-hole analysis.56,57 +Because our TDDFT calculations are based on the singlet +ground state as the reference state, the spin populations +of all atoms are zero by symmetry. Instead of relying on +spin densities, we identify a 1MLCT/ 3MLCT as a sin- +glet/triplet excited state where the total Mulliken popula- +tion of the Fe atom is decreased by one electron, and that +of the ligand atoms is increased by one electron. A 3MC +state is a triplet excited state where both the hole and the +electron are localized on the Fe atom, corresponding to +an electron transfer from the occupied dxy/dyz/dxz orbitals +to the virtual dx2-y2/dz2 orbitals. In all cases, only excited +states that lie below the initially excited 1MLCT were con- +sidered. Fig. S1.5 depicts an example of this analysis. The +geometry of the identified 3MC state was optimized and its +vibrational normal modes were computed in Gaussian 16 +with the def2-SVP basis set.58 + +ASSOCIATED CONTENT +Any methods, additional references, Nature Research report- +ing summaries, source data, extended data, supplementary +information, acknowledgements, peer review information; +details of author contributions and competing interests; and +statements of data and code availability are available at: +REFERENCES +1. +Veldkamp, B. S. et al. 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(2016). + + +9 +ACKNOWLEDGEMENTS +The authors gratefully acknowledge European XFEL in +Schenefeld, Germany, for provision of X-ray free-electron +laser beamtime at FXE and would like to thank the instru- +ment group and facility staff for their expert assistance. +M.B. acknowledges funding by the German DFG in frame +of priority program SPP 2102 (Grant number BA 4467/7- +1) and the German BMBF (Grant numbers 05K19PP1 and +05K18PPA). W.G. acknowledges partial funding from +Narodowe Centrum Nauki through SONATA BIS 6 grant +(2016/22/E/ST4/00543). W.G. further acknowledges +funding from Spanish MIU through “Ayudas Beatriz +Galindo” (BEAGAL18/00092), Comunidad de Madrid and +Universidad Autónoma de Madrid through Proyecto de +I+D para Investigadores del Programa Beatriz Galindo +(SI2/PBG/2020-00003), Spanish MICIU through Proyecto +de I+D+i 2019 (PID2019-108678GB-I00) and IMDEA- +Nanociencia through Severo Ochoa Programme for Cen- +tres of Excellence in R&D (MINECO, CEX2020-001039- +S). M.H.-G. acknowledges grants by Fonds der Chemi- +schen Industrie and Studienstiftund des deutschen Vol- +kes. Generous grants of computer time at the Pader- +borner Center for Parallel Computing PC2 is gratefully +acknowledged. +AUTHOR CONTRIBUTIONS +Conceptualization: M.N., M.H.-G., M.B. ; Data curation: +M.N., M.H.-G., H.E., J.K.; Formal analysis: M.N., M.H.-G., +H.E., J.K, A.K.; Funding acquisition: M.B.; Investigation: +M.N., M.H.-G., H.E., J.K., A.K., N.L., D.K., F.L., M.Bv., +N.P., P.Z., K.K., A.F.-R., W.G., M.B.; Methodology: M.N., +M.H-G., W.G., T.K., M.B.; Project administration: M.B.; +Resources: M.H.-G., M.B.; Software: M.N., M.Bv., H.E.; +Supervision: W.G., T.K., M.B.; Validation: M.N., H.E., +W.G., M.B.; Visualization: M.N., M.H.-G., H.E., J.K., W.G., +M.B.; Writing – original draft: M.N., M.H.-G., H.E., J.K., +W.G.; Writing – review & editing: M.N., M.H.-G., H.E., +J.K., A.K., N.L., D.K., F.L., T.-K.C., M.Bv., N.P., P.Z., K.K., +A.F.-R., W.G.,T.K., M.B.. +DATA AVAILABILITY STATEMENT +The datasets generated during and/or analysed during +the current study are available from the corresponding au- +thor on reasonable request. +COMPETING INTERESTS +The authors declare no competing interests. +ADDITIONAL INFORMATION +Extended data is available for this paper at +Supplementary information is available for this paper at +Correspondence and requests for materials should be +addressed to M.B. +Reprints and permissions information is available at +www.nature.com/reprints + + + + + + + + +10 +Ultrafast two-colour X-ray emission spectroscopy reveals excited state +landscape in a base metal dyad +M. Nowakowski1†, M. Huber-Gedert1†, H. Elgabarty1, J. Kubicki2, A. Kertem2, N. Lindner2, D. Kha- +khulin,3 F. Lima3, T.-K. Choi3,4, M. Biednov3, N. Piergies5, P. Zalden3, K. Kubicek3, A. Rodriguez- +Fernandez3, M. Alaraby Salem1, T. Kühne1, W. Gawelda2,6,7, M. Bauer1* +1 Chemistry Department and Center for Sustainable Systems Design (CSSD), Faculty of Science, Paderborn University, +Warburger Straße 100, 33098 Paderborn, Germany +2 Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, Poznań, 61-614, Poland +3 European X-Ray Free-Electron Laser Facility GmbH, Holzkoppel 4, 22869 Schenefeld, Germany +4 PAL-XFEL, Jigok-ro 127-80, 37673 Pohang, Republic of Korea +5 Institute of Nuclear Physics Polish Academy of Sciences, Kraków, 31-342, Poland. +6 Departamento de Química, Universidad Autónoma de Madrid, Campus Cantoblanco, 28047 Madrid, Spain +7 IMDEA Nanociencia, Calle Faraday 9, 28049 Madrid, Spain +Supplementary Information +Table of contents: + +1) Quantum chemical calculations: +a) Benchmarking the TDDFT UV-Vis spectra of [Fe-BL] +b) Computed and experimental UV-Vis spectrum of [Fe-BL] +c) Computed and experimental UV-Vis spectrum of [Fe-BL-Co] +d) Mulliken population-based electron-hole analysis of excited states +e) Decomposing the UV-Vis spectrum of [Fe-BL] in terms of charge-transfer components +f) Decomposing the UV-Vis spectrum of [Fe-BL-Co] in terms of charge-transfer components +2) TA experimental analysis +3) XES data analysis: +a) Fluorescence fitting procedure and results +b) A direct and non-direct contribution to Kα XES +c) Wavepacket analysis +d) Co Kα1 kinetic signals for -5-15 ps time window +e) d'-d interactions in Co +f) Kinetic model and population analysis results + + + + +11 +1. Quantum chemical calculations + +a) Benchmarking the TDDFT UV-Vis spectra +In order to understand the optical spectrum and the nature of the optical excitation process, we have +resorted to quantum chemical calculations using time-dependent density functional theory (TDDFT). +It is generally true that the study of transition metal complexes is challenging because of dynamic +correlation effects, system size, state degeneracies or near-degeneracies, and relativistic effects on top +of the typically large system sizes. In particular, TDDFT is known to have difficulties with systems +having charge-transfer states, and with extended π-systems1,2, both are features of the dyad. However, +despite these well-known issues, TDDFT has nevertheless been successfully applied to study such +systems, including d6 transition metal complexes.3 These known issues mean however, that one should +not blindly trust TDDFT results without scrutiny. +To this end, we have benchmarked TDDFT UV-Vis electronic spectra, using both the hybrid-GGA +B3LYP functional and the hybrid-meta-GGA TPSSh functional, against CASSCF-NEVPT2 spectra. +While CASSCF/NEVPT2 is known to reliably yield reasonable accuracy,4 the dyad molecule is too +large for the method. The CASSCF/NEVPT2 method explicitly takes account of both static and dy- +namic correlation effects and is known to provide highly accurate spectra.5 In order to keep the size +of the active space manageable we have done the benchmarking against the photosensitizer without +the cobaloxime moiety. As explained later, the active space required to accurately compute the elec- +tronic spectrum of the photosensitizer included 14 electrons in 13 active orbitals. A +CASSCF/NEVPT2 of the dyad, including all the 12 d-electrons together with the interacting ligand +electrons was computationally unfeasible due to the large number of occupied orbitals in the active +space. +Our benchmark calculations show that the TPSSh functional yields qualitatively correct result and +accurately reproduces the spectrum with a slight tendency to over-estimate the frequency of the peaks, +in particular the lowest-frequency peak. To obtained better comparison of calculated spectra with ex- +perimental ones calculated spectra are broadened by convolution with a Gaussian function with a +width of 0.2 eV (full width at half-height), before converting the scale to nm. + + + + + +12 +Starting orbitals for CASSCF +The starting orbitals for the CASSCF calculation were taken from the TPSSh ground-state Kohn- +Sham orbitals at the equilibrium geometry. The TPSSh ground state has the close-lying (within 0.3 +eV) Fe dxy, dyz, and dxz orbitals as the three occupied frontier orbitals, these were naturally included +in the active space. The dz2 and the dx2-y2 orbitals were found to be strongly mixed with ligand orbitals, +consistent with the strongly σ-donating heterocyclic carbene ligand. Both the occupied (bonding) and +unoccupied (antibonding) orbitals involving Fe dx2-y2 and dz2 were included in the active space. In +addition to the full set of (ligand-mixed) Fe d-orbitals, the two highest lying occupied π-bonding or- +bitals were included in the active space, together with the four lowest unoccupied molecular orbitals +(LUMO to LUMO+3). The LUMO is a π* orbital extending over the bipyridine moiety, while the +other three orbitals are all π* orbitals extending on the CNC moieties. Thus, the final active space +included 14 electrons in seven occupied orbitals and six virtual orbitals. +It is worth mentioning here that the B3LYP Kohn-Sham orbitals were identical in character to the +TPSSh orbitals, in agreement with the benchmark results that we discuss below. + +Comparison of TDDFT UV-Vis spectra to CASSCF(14,13)/NEVPT2 +The obtained spectra, which are depicted in Fig. S1.1, show several interesting features. The +CASSCF/NEVPT2 spectrum closely follows the experimental one, with two major peaks at 456.2 and +389.0 nm. We believe that the major source of the shift from the experimental spectrum is the implicit +solvation model. Between these two major absorption peaks, there is a weak absorption peak at 422.1 +nm (~10% of the oscillator strength of the strong peaks). The TDDFT spectra, although blue-shifted, +still provide qualitatively correct results, except for the wrong trend in the peak intensities, with the +low-frequency peak having a lower amplitude than the high-frequency one. The TPSSh functional is +clearly performing better than B3LYP, with the TPSSh peaks appearing at 446.0, 415.2, and 395.9 +nm, compared to 411.9, 387.1, and 365.1 nm for B3LYP. The accuracy of TDDFT transition frequen- +cies, which we find here, is consistent with the expected accuracy range of the method, typically +within 0.1-0.5 eV.5,6 The lack of any peaks below 300 nm in the CASSCF/NEVPT2 spectrum is be- +cause here we have only calculated the twelve lowest-lying singlet states (compared to 60 states in +TDDFT). + + + +13 +Rather than the exact positions of the peaks, more important to our benchmark is the nature of the +underlying states. Here, we find very consistent behavior between TDDFT (both functionals) and +CASSCF. Both methods agree that the main transitions bear predominantly the MLCT character and +originate from the three frontier orbitals to the virtual orbitals in the range LUMO – LUMO+3. The +low-frequency peak is consistently the transition Fe dyz → LUMO with contribution from the transi- +tion Fe dxy → LUMO+2 orbital. Also, all the methods show that the higher frequency peak is mainly +dxy → LUMO+3 with a minor contribution from dxy → LUMO+1, and that the weak intermediate +frequency peak is a transition from the three frontier orbitals to the three virtual orbitals LUMO+1 – +LUMO+3 (for details see Table S1.1). + +Influence of spin-orbit coupling +In computing all the TDDFT spectra, we have included corrections due to spin-orbit coupling. It is +worth noting however, that this turned out to have very little influence on peak positions, with typical +shifts of less than 1 nm. As an example, Fig. S1.2 shows the influence of spin-orbit coupling on the +UV-Vis spectrum of the photosensitizer, as obtained with B3LYP/TDDFT. +Fig. S1.1. UV-Vis absorption spectrum of the photosensitizer in implicit acetonitrile solvation. Black: +TDDFT with B3LYP, red: TDDFT with TPSSh, dotted blue: CASSCF(14,13), blue: +CASSCF(14,13)/NEVPT2. All the spectra are broadened by convolution with a Gaussian function with +a width of 0.2 eV (full width at half-height), before converting the scale to nm. + +B3LYP +TPSSH +1.2 +CASSCF(14,13) +CASSCF(14,13)/NEVPT2 + strength +Normalized osc. +0.8 +0.6 +0.4 +0.2 +0 +250 +300 +350 +400 +450 +500 +550 +Wavelength / nm + +14 + +b) Computed and experimental UV-Vis spectrum of [Fe-BL] +Fig. S1.3 Experimental UV-Vis spectrum of [Fe-BL] in MeCN and time-dependent TDDFT spectrum with +TPSSh. + + + +Fig. S1.2. Influence of spin-orbit coupling (SOC) on the UV-Vis spectrum of the photosensitizer. +300 +400 +500 +600 +0 +10 +20 +30 +40 +500 +1 +2 +e ·10-4 / cm-1 M-1 + [Fe-BL] + TPSSh def2-TZVP SMD(MeCN) +osc. strength / a.u. +wavelength / nm +a +c +b + +1.2 +B3LVP +B3LYP with SOC +1 +Osc. strength (a.u.) +0.8 +0.6 +0.4 - +0.2 +0 +250 +300 +350 +400 +450 +500 +550 +Wavelength / nm + +15 +Table S1.1 Computed dominant singlet vertical excitations a-c of [Fe-BL]. Donor and acceptor orbitals are +listed together with their contribution to the transition. The main character of the transition is indicated. +Transition +(state) +Donor +Acceptor +Contri- +bution +Character +a (8) +395.9 nm +HOMO (237) + +LUMO+3 (241) + +0.73 +MLCT +HOMO-2 (235) + +LUMO+1 (239) + +0.10 +MLCT +b (6) +415.2nm +HOMO-2 (235) + +LUMO+1 (239) + +0.37 +MLCT +HOMO (237) + +LUMO+2 (240) + +0.36 +MLCT +HOMO (237) + +LUMO+3 (241) + +0.22 +MLCT +c (4) +HOMO-1 (236) +LUMO (238) +0.64 +MLCT + + + +16 +446.0 nm + + +HOMO (237) + +LUMO+2 (240) + +0.20 +MLCT +HOMO-2 (235) + +LUMO (238) + +0.09 +MLCT + +c) Computed and experimental UV-Vis spectrum of [Fe-BL-Co] +Fig. S1.4 Experimental UV-Vis spectrum of [Fe-BL-Co] in MeCN and time-dependent TDDFT spectrum +with TPSSh. + +250 +300 +350 +400 +450 +500 +550 +600 +650 +0 +10 +20 +30 +40 +50 +0.0 +0.5 +1.0 +1.5 +2.0 +e ·10-4 / cm-1 M-1 + [Fe-BL-Co] +wavelength / nm +osc. strength / a.u. + TPSSh def2-TZVP SMD(MeCN) +a +b +c + + + +17 +Table S1.2 Computed dominant singlet vertical excitations a-c of [Fe-BL-Co]. Donor and acceptor or- +bitals are listed together with their contribution to the transition. The main character of the transition is +indicated. +Transition +(state) +Donor +Acceptor +Contri- +bution +Character +a (17) +401.9 nm +HOMO (320) + +LUMO+7 (328) + +0.37 +MLCT +HOMO (320) + +LUMO+4 (325) + +0.29 +MLCT/ +MMCT +HOMO (320) + +LUMO+5 (326) + +0.21 +MLCT/ +MMCT +b (15) +413.1 nm +HOMO (320) + +LUMO+7 (328) + +0.38 +MLCT +HOMO-2 (318) + +LUMO+3 (324) + +0.37 +MLCT +HOMO (320) + +LUMO+5 (326) + +0.14 +MLCT/ +MMCT +HOMO-1 (319) +LUMO (321) +0.78 +MLCT + + + +18 +c (8) +480.6 nm + + +HOMO (320) + +LUMO+5 (326) + +0.08 +MLCT/ +MMCT + +The TDDFT calculation of the dyad indicates transitions with partial MMCT character for the UV- +Vis band around 400nm and 480nm. The charge transfer from iron to cobalt is further analyzed by the +charge transfer analysis in section 1f. + +d) Mulliken population-based electron-hole analysis of excited states + +Fig. S1.5. Mulliken population analysis of the triplet excited states that show the strongest metal- +centered character. State number 53 is the initially populated 1MLCT state. "Fe" and "Co" refer to the +Mulliken populations of the two metal atoms, "pyridine" is the total charge on the bridge pyridine +attached to the Fe, and "Fe-coord" is the octahedral coordination. +53(S) +3(T) +7(T) +10(T) +20(T) +22(T) +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +Dq relative to the ground +state / e- +State + Fe atom + [Co] + pyridine + [Fe] +T - triplet +S - singlet + + + +19 +In order to identify the nature of any given excited state as obtained from TDDFT, whether it is a +1MLCT, 3MLCT, or 3MC, we have resorted to the Mulliken population analysis coupled with an +electron-hole analysis.7 Because our TDDFT calculations are based on the singlet ground state as the +reference state, the spin populations of all atoms are zero by symmetry. Instead of relying on spin +densities, we identify a 1MLCT/ 3MLCT as a singlet/triplet excited state where the total Mulliken +population of the Fe atom is decreased by one electron, and that of the ligand atoms is increased by +one electron. A 3MC state is a triplet excited state where both the hole and the electron are localized +on the Fe atom, corresponding to an charge transfer from the occupied dxy/dyz/dxz orbitals to the virtual +dx2-y2/dz2 orbitals. In all cases, only excited states that lie below the initially excited 1MLCT were +considered. +Fig. S1.5 graphically depicts the outcome of such an analysis on the optimized geometry of the singlet +ground state. In this particular case, state 7(T) is readily identified as the lowest lying 3MC state. +Calculation of the Mulliken population contribution of the Fe atom to the hole and electron +redistribution confirms the identity of this state, with the Fe atom contributing 84.8% to the electron +hole (i.e. the excited electron originates from the Fe), and with 68% of the redistributed electron +density concomitantly residing on the Fe atom (i.e. the excited electron resides on the Fe). + +e) Decomposing the UV-Vis spectrum of [Fe-BL] in terms of charge-transfer + + components +This qualitative characterization of the MLCT charge-transfer nature of the main transitions in the +spectrum like the one in Fig. S1.1 can be put into more quantitative terms using a hole-electron anal- +ysis.7,8 The idea here is to start with the usual expression for the UV-Vis spectrum as obtained via +broadening the excitation energies of all excited states: +𝜀(𝐸) ∝ ∑ 𝑓𝑖 +𝑖 +𝐺(𝐸 − 𝐸𝑖 +𝑒𝑥𝑐.) +Where 𝐸𝑖 +𝑒𝑥𝑐. is an excitation energy, 𝑓𝑖 the corresponding oscillator strength, and G(…) denotes con- +volution with a lineshape function (A Gaussian function in this work). If we now subdivided the mol- +ecule into two mutually exclusive fragments A and B (generalization to more fragments is trivial), +then the excitation spectrum can be readily decomposed as: + + + +20 +𝜀(𝐸)𝐴,𝐵 ∝ ∑ 𝑓𝑖𝑄𝑖 +𝐴,𝐵 +𝑖 +𝐺(𝐸 − 𝐸𝑖 +𝑒𝑥𝑐.) +where 𝑄𝑖 +𝐴,𝐵 is the amount of charge transfer from A to B in excited state i as obtained, in this case, by +a Mulliken population analysis. Because the sum of all inter- and intra-fragment charge transfer terms +is unity, the partitioning is exact, and the total spectrum is exactly divided into two intra-fragment +(A→ A and B→ B) charge redistribution terms and two inter-fragment (A→ B and B→ A) charge +transfer terms. +To decompose the photosensitizer spectrum in Fig. S1.6, the structure was subdivided into three frag- +ments: the iron atom (fragment 1), the terminal bipyridine moiety (fragment 3), and the rest of the +molecule (fragment 2). Figure S1.6 depicts the decomposed TPSSh/TDDFT spectrum. The decom- +posed spectrum clearly reveals the nature of all the peaks in the spectrum. For instance, the low- +frequency peak involves mainly (50% of the total amplitude) a charge transfer from the iron to the +terminal bipyridine, where the LUMO orbital resides, but also includes important 1→ 2 and 1→ 3 +charge-transfer contributions. On the other hand, the 1→ 2 charge transfer spectrum is most prominent +in the peak close to 400 nm, but also the shoulder due to the contribution of the weak intermediate +peak centered at 415.2 nm is also clear. + +Fig. S1.6. Decomposition of total UV-Vis spectrum into intrafragment charge redistribution and +interfragment charge transfer contributions. Fragment 1 is the iron atom, fragment 3 is the terminal +pyridine moiety, and fragment 2 is the rest of the molecule. + + + +21 +f) Decomposing the UV-Vis spectrum of [Fe-BL-Co] in terms of charge-transfer + components +In an analogic way to the [Fe-BL] case, we decomposed the UV-Vis spectrum of the [Fe-BL-Co]. Fig. +S1.7 shows separate parts of the dyad considered in this analysis along with the color code. Table S1.3 +presents contributions to the total charge for all considered transitions for each of the molecular parts +shown in Fig. S1.7. + +Table S1.3. The fractional contribution of each fragment to the hole and the electron in each of the three major +transitions in the spectrum. +Wavelength +/ nm +oscillator +strength +hole(1) +electron(1) +hole(2) +electron(2) +hole(3) +electron(3) +480.6 +0.2949 +0.976 +0.560 +0.016 +0.414 +0.008 +0.026 +413.1 +0.0338 +1.000 +0.989 +0.000 +0.010 +0.000 +0.000 +401.9 +0.2259 +0.993 +0.898 +0.002 +0.073 +0.006 +0.029 + +Fig. S1.8 presents the charge transfer analysis. It reveals that the peak at ~400 nm has the same nature +as in the photosensitizer, with a small fraction of Fe → Co charge transfer (shown directly in Fig. 1e). +The low-frequency peak corresponds to considerably more charge transfer to the terminal pyridine +ring. +Fig. S1.7. Definition of the three fragments used to decompose the UV-Vis spectrum. Orange: fragment +1, red: fragment 2, green: fragment 3 + + + +22 + + + + +Fig. S1.8. Decomposition of total UV-Vis spectrum into intrafragment charge redistribution and +interfragment charge transfer contributions. Fragment 1 is the iron atom moiety, fragment 2 is the +terminal pyridine moiety, and fragment 2 is the cobalt atom moiety. + + + +23 +2. TAS data analysis +Fig. S2.1. TAS spectra recorded upon 400 nm excitation for the [Fe-BL]. + +Fig. S2.2. Kinetics recorded for [Fe-BL-Co] at 650 nm, 640 nm, 630 nm and 610 nm together with +the fitted model (red lines). +405 +475 +545 +615 +-0.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +DA +nm + 0.500 ps + 1.00 ps + 5.00 ps + 10.0 ps + 15.0 ps + 30.0 ps + 60.0 ps +3MLCT +Ground state + bleach + +0.016 +610 nm +0.016 - +630 nm +0.014 - +0.014 +0.012 +370 ± 20 fs +0.012 +0.010 +0.010 +330 ±20 fs +0.008 - +0.008 +0.006 - +0.008 +0.004 - +12.5± 1.3pS +0.004 - +12.8 ±1.2 ps +0.002 +0.002 +0.000 - +0.000 +-0.002 +0 +-0.002 +10 +20 30 40 50 60 70 80 90 100 +0 +10 +20 30 40 50 80 70 80 90 100 +time delay I ps +time delay / ps +0.016 - +640 nm +0.016 - +650 nm +0.014- +0.014 +0.012- +0.012- +0.010 - +330 ± 20 fs +0.010 +320 ± 20 fs +0.008- +0.008- +0.008 +0.008 +0.004 - +11.9 ± 1.1 ps +0.004 . +13.7 ± 0.9 ps +0.002. +0.002. +0.000 - +0.000 +-0.002 +-0.002 +0 +10 +20 30 40 50 80 70 80 90 100 +0 +5 +10 +1520 +40 +60 +80 +100 +time delay I ps +time delay / ps + +24 + +Fig. S2.3. Kinetics recorded for [Fe-BL-Co] at 510 nm and 535 nm presenting the temporal evolu- +tion of the recovery of the GS together with the fitted model (red lines). + +It is commonly accepted that global analysis is applied to reveal a temporal evolution of such com- +plexes but in this case as our model assumes that the ground state (GS) is repopulated by 3MC and not +the 3MLCT (here, see Fig. 5a) we did not use this approach. Therefore, the recovery of the GS is +expected to be slower than the decay of the 3MLCT state, although both relaxation channels are oc- +curring on the same order of magnitude, i.e., few tens of picoseconds. Kinetics in 510-535 nm spectral +range provide that the recovery of the GS takes place with a time constant in the range from 15.0 - +17.0 ps. The 3MLCT-related time constant changes a little (12.1 ps - 12.7 ps) depending on the wave- +length selected for the strongest GS bleach band (Figure S2.2). It is very likely due to the vibrational +cooling of the hot GS.9 Few picosecond longer recovery of GS than the decay of 3MLCT state is +consistent with the fact that the lifetime of 3MC state is of the order of 2 ps (vide infra).10–14 The same +effect can be observed in kinetics extracted for the [Fe-BL] 630 nm, 550 nm, and 480 nm (Fig. S2.4). +The comparison with the kinetic data between the photosensitizer and the dyad can be done only +qualitatively since upon formation of the bimetallic assembly the 470 nm feature in [Fe-BL] UV-Vis +spectrum relaxes to ~500 nm. + + +0 +10 +20 30 40 50 60 70 80 90 100 +-0.030 +-0.025 +-0.020 +-0.015 +-0.010 +-0.005 +0.000 +DA +time delay / ps +17.0 ± 0.7 ps +510 nm +0 +10 +20 30 40 50 60 70 80 90 100 +-0.018 +-0.016 +-0.014 +-0.012 +-0.010 +-0.008 +-0.006 +-0.004 +-0.002 +0.000 +0.002 +DA +time delay / ps +15.0 ± 0.4 ps +535 nm + + + +25 +Fig. +S2.4. Kinetics recorded for [Fe-BL] at 630 nm, 550 nm and 480 nm together with the fitted model +(red lines). + + + +0 +10 +20 30 40 50 60 70 80 90 100 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +0.014 +DA +time delay / ps +120 ± 10 fs +11.1 ± 0.4 ps +630 nm +0 +10 +20 30 40 50 60 70 80 90 100 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +0.012 +DA +time delay / ps +120 ± 10 fs +16.5 ± 0.7 ps +550 nm +0 +10 +20 30 40 50 60 70 80 90 100 +-0.030 +-0.025 +-0.020 +-0.015 +-0.010 +-0.005 +0.000 +DA +time delay / ps +15.5 ± 0.3 ps +480 nm + + + +26 +3. XES data analysis +a) Fluorescence fitting procedure and results +The time-resolved X-ray emission spectroscopy (TR-XES) data was obtained on a setup +scheme presented in Fig. S3.1. + +The energy scale for TR-XES data was obtained on a basis of the ground state XES (gs-XES) +measurements performed using von Hamos spectrometer at P64 beamline of Petra-3 synchrotron at +DESY (Hamburg). The gs-XES energy calibration was obtained by measuring Fe foil and adjusting +the first inflection point in XAS spectrum to 7112 eV. Initial data correction: empty pulse removal +and dark correction were conducted on-site, while data reduction and extraction were performed re- +motely on DESY Maxwell server with the use of self-written Python scripts. A set of data from the +experiment was sorted, background reduced, filtered, and normalized to obtain ON/OFF XES spectra +in respect to delay time between the optical pump and X-ray probe pulses. From that a series of XES +spectra, differential (transient spectra, ΔXES) spectra were calculated, as ΔXES(t) = XESON(t) - +XESOFF(t), both for Fe and Co Kα lines, examples are in Fig. S3.1. Progressing changes in the ΔXES +profile were represented in form of the integral of the selected feature over all delay times. Those +Fig. S3.1. Scheme of the experimental setup used at FXE beamline. 50 fs FWHM optical pulses +of 400 nm wavelength (blue) were synchronized with 100 fs FWHM X-ray pulses (purple) with +the timing jitter of ~70 fs. Fe and Co Kα X-ray fluorescence emitted from the liquid jet (green) +sample was analyzed by a 16-crystal array of von Hamos spectrometer and directed to the 2D +Jungfrau detector + +Co +Kα +Fe +Ka +0.5 MHz +Glass +200 ms +Optical +nozzle +pulse +100 ms +0.5 MHz +VonHamosX-rayemissionspectrometer +X-ray +laser-on +laser-off +laser-on +laser-off +pulse +train +train +train +train +Sample +jet + +27 +kinetics were subsequently fitted with fluorescence rise and i–exponential decay functions to give +decay rates 𝜏𝑖 15: + 𝑦 = 𝑦0 + ∑ 𝐴𝑖𝑔𝑖(𝑡) +𝑖 + (S3.a.1) + 𝑔𝑖(𝑡) = 1 2 +⁄ (1 + 𝑒𝑟𝑓 ( +𝑡−𝑡0 +𝜎 +𝐶 +2 − +𝜎 +2𝐶𝜏𝑖)) 𝑒 +𝜎2 +𝐶2𝜏𝑖 +2𝑒 +−𝑡−𝑡0 +𝜏𝑖 (S3.a.2) +where: +𝐶 = 2√𝑙𝑛(16) ; +𝐴𝑖 – amplitude for the i –th exponent; +𝑡0 – the time-zero constant value; +𝜎 – Gaussian broadening due to IRF function. Due to used setup the IRF was fixed to 0.28 ps; +𝑖 = 1,2,3 – the degree of exponential function. + + +The fitting procedure was carried out in two steps. First, the largest dominating contribution was fitted +to a kinetic trace in the time window of 20 ps and step size of 1 ps. Second, to fit smaller decay time +constants, kinetic traces in time windows of ~2 ps and step size of 50 fs with the largest time constant +were fixed. All fitting procedures were performed with a value of FWHM in pump-probe cross-cor- +relation function set to σ = 0.284 ps. The σ value was refined in the post-fitting verification. The +summarized fitting results are presented in Table S3.1 and in Figs. S3.2-3. + + + + + +6910 +6915 +6920 +6925 +6930 +6935 +6940 +6385 +6390 +6395 +6400 +6405 +6410 +Norm. Int. / au +Emission energy / eV + 0.45 ps + 3.0 ps + 15.0 ps +Norm. Int. / au +Emission energy / eV + 0.45 ps + 3.0 ps + 15.0 ps +6910 +6915 +6920 +6925 +6930 +6935 +6940 +6385 +6390 +6395 +6400 +6405 +6410 +Norm. Int. / au +Emission energy / eV + 0.45 ps + 3.0 ps + 15.0 ps +Norm. Int. / au +Emission energy / eV + 0.45 ps + 3.0 ps + 15.0 ps +a +b +Fig. S3.2. Kα1 transient XES for: a) Fe @ [Fe-BL-Co]; b) Co @ [Fe-BL-Co]. + + + +28 + +Table S3.1. Summary for fluorescence fitting to experimental data. +[Fe-BL] 15 ps +[Fe-BL] 1.2 ps +A1 +- +A1 +0,170(26) +τ1 [ps] +- +τ1 [ps] +0,245(42) +A2 +0,241(5) +A2 +0,538(7) +τ2 [ps] +8,984(273) +τ 2 [ps] +10,142(2,054) +A3 +0,079(12) +A3 +0,126(8) +τ 3 [ps] +1,705(348) +τ 3 [ps] +2,421(640) +t0 [ps] +0,368(31) +t0 [ps] +0,015(7) +𝑦0 +0,070(2) +𝑦0 +0,061(6) +FWHMa [ps] +0,289(61) +FWHMa [ps] +0,305(21) + + + + +Fe @ [Fe-BL-Co] 15 ps +Fe @ [Fe-BL-Co] 1.2 ps +A1 +- +A1 +0,109(20) +τ1 [ps] +- +τ1 [ps] +0,115(23) +A2 +0,00135(2) +A2 +0,483(3) +τ2 [ps] +10,381(242) +τ 2 [ps] +12,417(1,399) +A3 +- +A3 +0,111(4) +τ 3 [ps] +- +τ 3 [ps] +1,740(182) +t0 [ps] +-0,064(12) +t0 [ps] +0,010(3) +𝑦0 +4,922(56) ·10-4 +𝑦0 +0,024(2) +FWHMa [ps] +0,284(48) +FWHMa [ps] +0,275(10) + + + + +Co @ [Fe-BL-Co] 15 ps +Co @ [Fe-BL-Co] 1.2 ps +A1 +3.140 (421) ·10-4 +A1 +0,00257(13) +τ1 [ps] +0.25 (fixed) +τ1 [ps] +0,242(14) +A2 +1.249(175) ·10-4 +A2 +- +τ 2 [ps] +4.12(1.39) +τ 2 [ps] +- +A3 +1.381(149) ·10-4 +A3 +0,00187(37) +τ 3 [ps] +23.39 (fixed) +τ 3 [ps] +6,084(1,134) +t0 [ps] +-0,116(10) +t0 [ps] +0,047(7) +𝑦0 +1,049(31) ·10-4 +𝑦0 +8,061(282) ·10-4 +FWHMa [ps] +0,284(42) +FWHMa [ps] +0,280(19) + + + + +Cobaloxime 215 ps +Cobaloxime 1.2 ps +A1 +2.342(233) ·10-5 +A1 +4.011(110) ·10-5 +τ1 [ps] +2.764 (312) +τ1 [ps] +2.391(172) +A2 +3.391(78) ·10-5 +A2 +- +τ 2 [ps] +23.391(1.820) +τ 2 [ps] +- +t0 [ps] +-0.056(0.093) +t0 [ps] +-0,015(0.040) ·10-2 +𝑦0 +4.812(51) ·10-5 +𝑦0 +3.391(35) ·10-5 +FWHMa [ps] +0.273(50) +FWHMa [ps] +0.305(84) +a Post-fitting refinement + + + +29 + + + + + +-0.25 +0.00 +0.25 +0.50 +0.75 +1.00 + Fe K1 + Fe K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +Model +Conv_3exp (User) +Equation +exp1=0.5*(1+erf((x-t0)*sqrt(ln(16)) +/fwhm-fwhm/(2*tau1*sqrt(ln(16)))) +)*exp(fwhm^2/(4*tau1^2*ln(16)))*e +xp(-(x-t0)/tau1); +exp2=0.5*(1+erf((x-t0)*sqrt(ln(16)) +/fwhm-fwhm/(2*tau2*sqrt(ln(16)))) +)*exp(fwhm^2/(4*tau2^2*ln(16)))*e +xp(-(x-t0)/tau2); +exp3=0.5*(1+erf((x-t0)*sqrt(ln(16)) +/fwhm-fwhm/(2*tau3*sqrt(ln(16)))) +)*exp(fwhm^2/(4*tau3^2*ln(16)))*e +xp(-(x-t0)/tau3); +y=A1*exp1 + A2*exp2 + A3*exp3 ++ y0 +Plot +L +A1 +0.16845 ± 0 +tau1 +0.24508 ± 0 +A2 +0.53829 ± 0 +tau2 +10.1301 ± 0 +A3 +0.12741 ± 0 +tau3 +2.39459 ± 0 +y0 +0.06257 ± 0 +t0 +-0.01471 ± 0.00694 +fwhm +0.284 ± 0 +Reduced Chi-Sqr +3.43142 +R-Square (COD) +0.98085 +Adj. R-Square +0.98085 +[Fe-BL] +1 = 0.25(4) ps + = 10.13(2.05) ps +3 = 2.39(63) ps +-0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 + Fe K1 + Fe K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +Model +Conv_3exp (User) +Equation +exp1=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau1*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau1^2*ln( +16)))*exp(-(x-t0)/tau1); +exp2=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau2*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau2^2*ln( +16)))*exp(-(x-t0)/tau2); +exp3=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau3*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau3^2*ln( +16)))*exp(-(x-t0)/tau3); +y=A1*exp1 + A2*exp2 + A3*exp +3 + y0 +Plot +L +A1 +0.48247 ± 0 +tau1 +12.41756 ± 0 +A2 +0.11084 ± 0 +tau2 +1.74261 ± 0 +A3 +0.10739 ± 0 +tau3 +0.11547 ± 0 +y0 +0.02428 ± 0 +t0 +0.01043 ± 0.00329 +fwhm +0.284 ± 0 +Reduced Chi-Sqr +0.70578 +R-Square (COD) +0.99536 +Adj. R-Square +0.99536 +[Fe-BL-Co] + = 12.42(1.40) ps +3 = 1.74(18) ps +1 = 0.12(2) ps +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 + Fe K1 + Fe K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +[Fe-BL-Co] +1 = - +2 = 10.38(40) ps +3 = - +-2 +0 +2 +4 +6 +8 +10 +12 +14 + Fe K1 + Fe K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +[Fe-BL] +1 = - +3 = 1.71(35) ps +2 = 8.98(27) ps +Fig. S3.3. Fluorescence decay fitting results for: a) [Fe-BL], long-time window; b) [Fe-BL], +short-time window; c) Fe @ [Fe-BL-Co], long-time window; d) Fe @ [Fe-BL-Co], short- +time window. +a) +b) +c) +d) + + + +30 + +b) A direct and non-direct contribution to Kα XES +At the optical pump wavelength of 400 nm both Fe and Co centres were simultaneously excited +(although predominantly Fe site) and probed with 9.3 keV X-rays. Hence, we have carried out rigorous +and detailed analysis to distinguish the direct and non-direct contributions originating from the pho- +toexcitation at the Co center in the studied dyad. In this case, we have computed the X-ray and optical +convoluted cross-section relation Cσ of cobaloxime and Co part of the dyad as follows: + 𝐶𝜎~ 𝜎𝑋−𝑅𝑎𝑦 +𝑐𝑜𝑏𝑎𝑙𝑜𝑥𝑖𝑚𝑒𝜎𝑈𝑉−𝑉𝐼𝑆 +𝑐𝑜𝑏𝑎𝑙𝑜𝑥𝑖𝑚𝑒 +(𝜎𝑋−𝑅𝑎𝑦 +𝑑𝑦𝑎𝑑 − 𝜎𝑋−𝑅𝑎𝑦 +𝑃𝑆 +)(𝜎𝑈𝑉−𝑉𝐼𝑆 +𝑑𝑦𝑎𝑑 +− 𝜎𝑈𝑉−𝑉𝐼𝑆 +𝑃𝑆 +) +⁄ +≈ 0.59 +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 + Co K1 + Co K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +1 = 0.25 ps (fixed) +[Fe-BL-Co] + = 4.12(1.39) ps +3 = 23.39 ps (fixed) +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Norm. Integr. Counts +Probe delay / ps + Co K1 + Co K1 (fitting) +Cobaloxime +1 = 2.76(31) ps + = 23.39(1.82) ps +-1.5 +-1.2 +-0.9 +-0.6 +-0.3 +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +Norm. Integr. Counts +Probe delay / ps + Co K1 + Co K1 (fitting) +Cobaloxime + = 2.39(17) ps +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Norm. Integr. Counts +Probe delay [ps] + Co K1 + Co K1 (fitting) +[Fe-BL-Co] +1 = 6.07(1.13) ps +2 = 0.25(1) ps +Fig. S3.4. Fluorescence decay fitting results for: a) [Co] @ [Fe-BL-Co], long-time window; b) [Co] +@ [Fe-BL-Co], short-time window; c) cobaloxime, long-time window; d) cobaloxime, short-time +window. +a) +b) +c) +d) + + + +31 +This number can be compared to the average ratio between the intensity of cobaloxime kinetic trace +and [Co] part of dyad kinetic trace, which was estimated to be 0.88. Our analysis yields that approxi- +mately 59% of Co signal in dyad originates from the different electronic structure around Co site in +the dyad, as compared to isolated cobaloxime. The theoretical cross sections were computed using +values listed in the NIST database. Optical cross sections were taken from UV-Vis spectra shown in +Fig. 1 b. + + + + +32 +c) Nuclear wavepacket analysis +The oscillatory signals, previously extracted from the kinetic traces of both [Fe-BL] and [Fe-BL-Co] +shown in Fig. 4 of the main text, were analyzed in detail using the Fourier transform analysis and are +presented in Fig S3.4. + +For [Fe-BL], we found one dominating frequency of 3.49 THz (the corresponding half-period of 0.29 +ps) and for the [Fe-BL-Co], there are two dominating frequencies of 3.71 THz (0.27 ps) and 5.56 THz +(0.19 ps). These frequencies correspond to vibrational modes of 116.4 cm-1 (for [Fe-BL]), 123.8 cm-1 +and 185.5 cm-1 (for [Fe-BL-Co]), respectively, in the range of one degree of freedom for single bond +thermal oscillations at 25 °C (kbT). Noteworthy, similar frequencies could be resolved experimentally +0.0 +12.6 +25.2 +37.8 +50.4 +63.0 +75.6 +-200 +0 +200 +400 +600 +800 +Amplitude +5.56 THz +0.19 ps +3.71 THz +0.27 ps +Phase / deg +Frequency / THz +-202.5 deg +3.2 deg +100 +150 +200 +250 +300 +0 +10000 +20000 +30000 +40000 +Intensity / a.u. +Raman shift / cm-1 +785 nm laser + [Co] + [Fe-BL] + [Fe-BL-Co] + + + +a +b +c +Fig. S3.5. Fourier transform of the oscillatory parts: a) [Fe-BL]; b) Fe part of [Fe-BL-Co]. Raman +spectra of [Co], [Fe-BL], and [Fe-BL-Co] obtained with 785 nm laser (c)). + +3.49 THz +0.29ps +Amplitude +0 +Phase / deg +50 +95.44 deg +-100 +-150 +0 +2 +4 +6 +8 +10 +12 +Freguency I THz + +33 +by means of Raman spectroscopy (see spectra in Fig. S3.4c). Specifically, the difference between +116.4 cm-1 and 123.8 cm-1 (~1 meV) could be observed only in terms of the amplitude, while a band +around 185.5 cm-1 should appear only in [Fe-BL-Co]. It must be underlined, that in TR-XES such +differences would be difficult to be detected. Since the phase of Fourier transform can be affected by +the presence of a significant noise contribution, we decided to use an additional method of analysis. +The nuclear wavepacket motion parameters for [Fe-BL-Co] were refined with the use of the damped +oscillatory, function: + 𝑓(𝑡) = 𝑦0 + 𝑒 +−𝑡 𝑡0 +⁄ +(𝑏1𝑠𝑖𝑛 (𝜋 +𝑡−𝑡𝑐1 +𝑤1 ) + 𝑏2𝑠𝑖𝑛 (𝜋 +𝑡−𝑡𝑐2 +𝑤2 )) (S3.c.1) +where: +y0 – intensity offset in a.u.; tc1, tc2 – phase shifts in ps; w1, w2 – oscillation period in ps; +t0 – damping factor in ps; b1, b2 – initial amplitude of oscillations in a.u. +The results are presented below. In case of [Fe-BL], the b2 was set to 0. It is worth to mention that for +[Fe-BL-Co] the damping factor is much bigger, although the uncertainty of this parameter prevents to +derive any quantitative conclusions. Interestingly, the oscillation in [Fe-BL] changes phase upon co- +balt coordination. This may suggest, that after forming the dyad, the longer oscillation is partially +quenched and re-induced upon photoexcitation. The phase of oscillation in [Fe-BL] is ~ 9°, i.e. close +to sinusoidal, therefore induced by impulse-stimulated Raman scattering.16–18 On the other hand, the +0.26 ps component phase in [Fe-BL-Co] is ~51°, thus of mixed sine/cosine character and for 0.19 ps, +the phase with ~89° is of cosine character. The cosine type of oscillation was proven to be a direct +marker of the influence of the excited state generation upon photoexcitation and followed by coherent +vibrations generation 18,19. Thus at least one of the vibrations in [Fe-BL-Co] is related to the charge +transfer accompanied by the Fe-ligand bond stretching. +Table S3.2. Wavepacket analysis fitting results with equation (S3.c.1). +[Fe-BL] +[Fe-BL-Co] +b1 +0.064(176) +b1 +-0.008(4) +tc1 [ps] +0.046(17) +tc1 [ps] +2.632(275) +w1 [ps] +0.284(22) +w1 [ps] +0.255(30) +𝑦0 +0.003(4) +b2 +0.015(4) +t0 [ps] +0.402(150) +tc2 [ps] +8.425(370) + + +w2 [ps] +0.185(8) + + +𝑦0 +0.000(2) + + +t0 [ps] +1.632(1.270) + + + + + + + + +34 +Given the large experimental error bars, we wanted to discard the possibility that the additional oscil- +lation observed for the dyad could compensate for the high damping value. In other words, we have +also verified the scenario, in which the extracted oscillations could be described with a single sine +function. These fit results are shown in Fig. S3.5a and compared statistically with the double sine fit, +previously presented in Fig. 4a, using an F-test (Fig. S3.5b). The F-test is intended to compare two +models, where one of them contains less parameters and is nested into more complex one. The result +of the test indicates statistical significance of the more complex fit, especially when χ2 comparison is +not enough, thus preventing us from overfitting the data. In our specific case, the single-sine model +was described by 5 parameters and was nested into a more complex model described with 8 parame- +ters. The fitting procedure was carried out with experimental error bars as weights, and in F-value +calculation, χ2 values of 0.01338 and 0.00814 were found for single- and double-sine fit functions, +respectively. Since all fits were done with p = 0.05, the null hypothesis in this test stated that fitted +functions are not different in 95% probability. For such conditions, the critical F-value to reject null +at 95% probability was 3.6875, while the obtained value was 5.0053. Therefore, we can conclude that +both models are statistically different and thus confirm the presence of the second oscillation in [Fe- +BL-Co] kinetic traces. + + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +-0.04 +-0.02 +0.00 +0.02 +0.04 +Norm. Integr. Counts - fit / a.u. +Probe delay / ps + 3.49 THz + experimental data + fit + Damping function +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +Density +Probe delay / ps +F distribution for two models + with 5 and 8 parameters +F = 5.0053 +Fig. S3.6. a) Single oscillatory fit to wavepacket signal; b) F distribution (black) for nested 5- +parameter model in 8-parameter model along with F-value in this study (blue). The vertical line +indicates critical F-value of 3.6875 above which the null hypothesis can be rejected in the current +conditions at p level of 0.05. +a +b + + + +35 +d) Co Kα1 kinetic signals for -5-15 ps time window +The statistics of the Co Kα1 long-time kinetic signal in [Fe-BL-Co] is substantially worse than for +pure cobaloxime measurement, owing mainly to the two factors. First of all, due to substantially dif- +ferent absorption cross sections in the UV-Vis range, upon the photoexcitation of the dyad, the [Fe- +BL] part is predominantly excited, while the direct excitation of the Co part is nearly completely +avoided. Secondly, the X-ray beam intensity is distributed over two metal centers. Since with the +measurement of cobaloxime alone, the aim was to detect excited states formed upon direct photoex- +citation, and no [Fe-BL] was present. Still, the long kinetic trace for [Fe-BL-Co] substantially differs +in shape as compared to pure cobaloxime, especially in the initial 5-6 ps range (sec. 3a, Fig. S3.3 a, +and c). Consequently, although a possible long-lived component in Co moiety could not be excluded, +it is visible only after the first ~7 ps of the [Fe-BL-Co] Co Kα1 kinetics, where the signal almost +reaches the background level. It would not affect the presence of 1,FeCo, and short-time kinetic trace, +since contributions from the shortest time constants appear at the beginning of the kinetic evolution. +Moreover, the step size in this measurement was equal to 1 ps, which corresponds to approx. 2 data +points that represent time constant of 1 ps. The initial fitting results of the kinetic traces for Co Kα +fluorescence in [Fe-BL-Co] in 15 ps time window are shown in Fig. S3.3 a. All fit parameters were +left as free and as a result we obtained good fit results with 2 time constants of 1.4 and 17 ps, respec- +tively. The result is shown in Fig. S3.5 and the corresponding parameters are summarized in Table +S3.3 (column A). Interestingly, the fitting of Co Kα fluorescence to the 1.2 ps kinetic trace revealed +unambiguous presence of another ultrashort contribution of 0.25 ps. The 1 ps time constant in the 15 +ps kinetic fit, was statistically represented by a single point, therefore it could be an artefact. To verify +this, we repeated the fitting of the Co Kα fluorescence to the 15 ps kinetic trace using a fixed time +constant of 0.25 ps and keeping all other fit conditions the same. However, the fit did not converge to +any reasonable result, and therefore we extended the fit model with an additional time constant. First, +we fitted all 3 decays, and the fitting procedure produced a very large time constant and high uncer- +tainties. This contribution was interpreted as a representation of an electronic state with decay signif- +icantly longer than the time window of measurement. Therefore, the large value was fixed, and data +were re-fitted. The result is shown in Fig. S3.5 b and corresponding parameters are available in Table +S3.3 under column B. The results again exhibit 1 ps time constant (and did not require the shortest +0.25 ps time constant), which was interpreted as an artefact and a time constant of 7 ps with a very +high uncertainty. For purpose of testing this hypothesis further, we employed a fit procedure with 3 +time constants, of which two: 0.25 ps and infinite were fixed. The result is shown in Fig. S3.5 c and + + + +36 +corresponding parameters are available in Table S3.3 under column C. The results concluded that the +1 ps value from the first fitting attempt was indeed an artefact due to too low temporal step size of the +measurement. Earlier reports suggest that a two-exponential, sequential decay for cobaloxime (one +fast and second around 20 ps), and we assumed the same scenario for our longer time constants.20 +Therefore, A2, τ2, A3, and τ3 represented a decay of the LMCT state to the ground state through the +MC state upon direct photoexcitation, while τ1 and A1 represented M’MCT transition. This implied, +that the corresponding amplitudes of the A2 and A3 in the model from Table S3.3 C will be similar, +because both concern excited states in the same simple decay pathway. However, the difference be- +tween them is around a factor of 2. Due to the fact, that the LMCT excitation is still present, we +assumed that the related decay pathway will correspond to the one in an isolated cobaloxime complex, +especially for the lowest-lying state. A final fitting attempt was conducted on the model described in +Table S3.3 C, with τ3 fixed at the value obtained from cobaloxime, namely 23.39 ps. The result is +presented in Fig. S3.3A and Table S3.1. Notably, the relation between A2 and A3 is almost 1. The time +constant of 4.1 ps is discussed in the main text. The value of τ3 was later re-evaluated with other time +constants fixed, and a value of 29.39(14.46) ps was obtained. +Table S3.3. Co Kα fluorescence decay fitting results for [Fe-BL-Co] with different assumptions. +A +B +C +A1 +4.453(1.535) ·10-4 +A1 +4.101(844) ·10-4 +A1 +3.160(116) ·10-3 +τ1 [ps] +1.36(25) +τ1 [ps] +1.16(50) +τ1 [ps] +0.25 (fixed) +A2 +1.914(103) ·10-4 +A2 +1.874(757) ·10-4 +A2 +1.967(114) ·10-4 +τ 2 [ps] +16.74(3.54) +τ 2 [ps] +7.11(5.67) +τ 2 [ps] +5.97(1.19) +t0 [ps] +-0.56(7) +A3 +5.993(4.267) ·10-5 +A3 +6.393 (117) ·10-5 +𝑦0 +1.041(48) ·10-5 +τ 3 [ps] +infinite (fixed) +τ 3 [ps] +infinite (fixed) +FWHMa [ps] +0.284 (fixed) +t0 [ps] +-0.57(5) +t0 [ps] +-0.01(2) + + +𝑦0 +1.025(25) ·10-4 +𝑦0 +1.024(25) ·10-4 + + +FWHMa [ps] +0.284 (fixed) +FWHMa [ps] +0.284 (fixed) + + + + +37 + +e) d-d interactions in Co +The d-d interactions can occur when different valence d orbitals in a metal complex are not fully +occupied. In octahedral symmetry, the selection rules forbid d-d transitions to occur and thus they can +be observed when the symmetry of the complex is significantly distorted, for example by the Jahn- +Teller effect. Even though the symmetry of Co(dmgH)2Cl(py) complex is significantly distorted, thus +allowing weak d-d transitions to occur, the UV-Vis spectrum of pure cobaloxime does not contain the +characteristic d-d bands (Fig. 1 b) observed for another family of cobaloximes with axial alkyl and +amino ligands [Co(dmgH)2(Alkyl)(Base)]21. Therefore, the low-lying acceptor state for LMCT deex- +citation is not populated due to the d-d transition. In order to assign the time constants obtained via +the fitting of TR-XES kinetic traces to possible transitions in our Co complex, the 1,Co was tentatively +assigned as LMCT → MC decay, while 2,Co represent the MC relaxation in this model. We want to +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 + Co K1 + Co K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +1 = 1.36(25) ps +[Fe-BL-Co] + = 16.74(3.41) ps +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 + Co K1 + Co K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +1 = 0.25 ps (fixed) + = 5.97(1.14) ps +3 = ∞ (fixed) +Model +Conv_3exp (User) +Equation +exp1=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau1*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau1^2*ln( +16)))*exp(-(x-t0)/tau1); +exp2=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau2*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau2^2*ln( +16)))*exp(-(x-t0)/tau2); +exp3=0.5*(1+erf((x-t0)*sqrt(ln(1 +6))/fwhm-fwhm/(2*tau3*sqrt(ln(1 +6)))))*exp(fwhm^2/(4*tau3^2*ln( +16)))*exp(-(x-t0)/tau3); +y=A1*exp1 + A2*exp2 + A3*exp +3 + y0 +Plot +D +A1 +6.39296E-5 ± 0 +tau1 +1000 ± 0 +A2 +1.96616E-4 ± 0 +tau2 +5.97151 ± 0 +A3 +0.00208 ± 0 +tau3 +0.25 ± 0 +y0 +1.02384E-4 ± 2.51058E-6 +t0 +0.00873 ± 0.02733 +fwhm +1.2 ± 0 +Reduced Chi-Sqr +0.07862 +R-Square (COD) +0.98572 +Adj. R-Square +0.98497 +[Fe-BL-Co] +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 + Co K1 + Co K1 (fitting) +Norm. Integr. Counts +Probe delay / ps +1 = 1.16(50) ps + = 7.11(3.67) ps +3 = ∞ (fixed) +[Fe-BL-Co] +b) +c) +Fig. S3.7. Co Kα fluorescence decay fitting results for [Fe-BL-Co] 15 ps kinetics: a) with 2 time +constants; b) with 3 time constants and very long time constant fixed; c) with 3 time constants and +0.25 ps and infinite time constants fixed. +a) + + + +38 +underline, that the MC nature of the second state must be independently confirmed, yet the LMCT +state was also confirmed by DFT results, making MC state an obvious candidate as an acceptor for +the LMCT decay. The proposed state diagram for the relaxation on the Co site of the dyad is also +analogical to numerous Fe polypyridyl complexes with long-lived MC states.22,23 A dedicated Co Kβ +TR-XES experiment would further confirm the nature of the assigned states involved in the decay +process, which was beyond the scope of the present study. + +f) Kinetic model for XES fluorescence kinetic traces. +Fe Kα XES. For Fe in [Fe-BL-Co], two decay channels were proposed (Fig. 5): +1) alpha (α) channel: 1/3MLCT* +𝑘1→ 3MLCT +𝑘2→ 3MC +𝑘3→ gs; +2) beta (β) channel: 1/3MLCT* +𝑘4→ 3MC +𝑘3→ gs. +They were described by a system of differential equations: + +𝑑 +𝑀𝐿𝐶𝑇 +1 +𝑑𝑡 += −𝑘1 𝑀𝐿𝐶𝑇 +1 +− 𝑘4 𝑀𝐿𝐶𝑇 +1 + (S3.1.a) + +𝑑 +𝑀𝐿𝐶𝑇 +3 +𝑑𝑡 += 𝑘1 𝑀𝐿𝐶𝑇 +1 +− 𝑘2 𝑀𝐿𝐶𝑇 +3 +− 𝑘𝑐𝑡 𝑀𝐿𝐶𝑇 +3 + (S3.1.b) + +𝑑 +𝑀𝐶 +3 +𝑑𝑡 += 𝑘4 𝑀𝐿𝐶𝑇 +1 ++ 𝑘2 𝑀𝐿𝐶𝑇 +3 +− 𝑘3 𝑀𝐶 +3 + (S3.1.c) + 𝑔𝑠 = 𝑀 − +𝑀𝐿𝐶𝑇 +1 +− +𝑀𝐿𝐶𝑇 +3 +− +𝑀𝐶 +3 + (S3.1.d) + 𝑀𝐿𝐶𝑇(𝑡 = 0) +1 += 𝑀 (S3.1.e) + 𝑀𝐿𝐶𝑇(𝑡 = 0) +3 += 0 (S3.1.f) + 𝑀𝐶(𝑡 = 0) +3 += 0 (S3.1.g) +where: M – initial excited state fraction. For the modelling purposes, an equal relation between con- +centration and signal strength was assumed. The system of differential equations above was solved +numerically in Mathematica 11 software and all solutions were broadened by Heaviside step function +under the convoluted with normalized Gaussian function to model rise time of electronic state: + 𝑔𝑏𝑟𝑜𝑎𝑑𝑒𝑛𝑒𝑑(𝑡) = +1 +2𝜎√2𝜋 ∫ 𝑒− 𝑦2 +2𝜎2 ℎ(𝑡 − 𝑡0 − 𝑦)𝑔(𝑡)𝑑𝑦 (S3.3) +where: +𝑔(𝑡) – broadened function defined by one of eq. S3.1.a – S3.1.g; +ℎ(𝑡 − 𝑡0 − 𝑦) – Heaviside step function; +σ – Gaussian standard deviation function. +The final fitted function was as follows: + 𝑓𝐹𝑒(𝑡) = 𝑦0 + +𝑀𝐿𝐶𝑇 +1 ++ +𝑀𝐿𝐶𝑇 +3 ++ +𝑀𝐶 +3 + (S3.4) + + + +39 +where: +𝑦0 – vertical offset. + +Co Kα XES. There were two decay paths identified in [Co]: +1) M’MCT +𝑘7→ gs; +2) LMCT +𝑘5→ MC +𝑘6→ gs. +The charge transfer (CT) was treated as instantaneous, therefore it is completed within the IRF func- +tion. The M’MCT state was acting as an acceptor of CT from bridging ligand BL, while LMCT state +was representing direct optical excitation and was described by an independent k5 rate constant. The +analogical transition was observed in pure cobaloxime kinetic data with 2.76 ps time constant. The +differential formula system with boundary conditions was as follows: + +𝑑𝑀′𝑀𝐶𝑇 +𝑑𝑡 += −𝑘7𝑀′𝑀𝐶𝑇 (S3.5.a) + +𝑑𝐿𝑀𝐶𝑇 +𝑑𝑡 += −𝑘5𝐿𝑀𝐶𝑇 (S3.5.b) + +𝑑𝑀𝐶 +𝑑𝑡 = 𝑘5𝐿𝑀𝐶𝑇 - 𝑘6𝑀𝐶 (S3.5.c) + 𝑔𝑠𝐶𝑜 = 𝑀′𝑀𝐶𝑇0 + 𝐿𝑀𝐶𝑇0 − 𝑀′𝑀𝐶𝑇 − 𝐿𝑀𝐶𝑇 (S3.5.d) + 𝑀′𝑀𝐶𝑇(𝑡 = 0) = 𝑀′𝑀𝐶𝑇0 (S3.5.e) + 𝐿𝑀𝐶𝑇(𝑡 = 0) = 𝐿𝑀𝐶𝑇0 (S3.5.f) +The final fitting function was: + 𝑓𝐶𝑜(𝑡) = 𝑦0 + 𝑀′𝑀𝐶𝑇 + 𝐿𝑀𝐶𝑇 + 𝑀𝐶 (S3.6) +The 𝑀′𝑀𝐶𝑇 and 𝐿𝑀𝐶𝑇 were also broadened by the function described in eq. 3. +All fitting results are summarized in Table S3.4. In the first approach, decay constant values obtained +from the fluorescence decay formula fitting were fixed. The amplitudes and offsets were fitted. After- +ward obtained values were fixed to refine the rate constants. + +Table S3.4. Summary for fitting of kinetic equations to experimental data. +Fe in [Fe-BL] // Fe in [Fe-BL-Co] +[Co] in [Fe-BL-Co] dyad // cobaloxime + 𝑀 +0.736(19) +0.608(9) +𝑀′𝑀𝐶𝑇 0 0.966(160) +- + + + +𝐿𝑀𝐶𝑇 0 +0.730(64) +0.925(68) +t0 [ps] +-0.034(8) +0.002(0.0005) +t0 [ps] +-0.005(0.015) +-0.018(0.021) +𝑦0 +0.047(14) +0.027(0.007) +𝑦0 +0.002(0.032) +0.004(0.048) +k1 / τ1 +[ps-1 /ps] +4.527(1.495) / +0.221(72) +4.527 (fixed) / +0.221 (fixed) +k5 / τ5 +[ps-1/ ps] +4.12 (fixed) + +2.76 (fixed) + + + + + + +𝑘𝑒𝑡/ τet +[ps-1 /ps] +- + +-0.294(35) / - +3.404(405) +k6 / τ6 +[ps-1 /ps] +3.659(1.246) / +0.273(93) +- + + + +40 + +The k2, k3, k4, and k5 rate constants were calculated from k = 1/τ relation. Decay time constants +τ were taken from fluorescence fitting results. In total, five parameters were fitted to the experimental +data: M, y0, t0, k1 and ket. Unlike for the FWHM value used in the fluorescence fitting, the σ parameter +here is a standard deviation of Gaussian in the IRF function. The re-evaluated value of σ = 0.106(5) +ps corresponds to 0.25(1) ps from the fluorescence fitting, which clearly resembles the FWHM of IRF +function (0.28 ps). The t0 was set as a free parameter to ensure fit convergence. The k1 decay rate +represents a 1MLCT->3MLCT transition, and the inclusion of this decay rate is necessary due to nu- +clear wavepacket motion shown in Fig. 4 and Fig. S3.4. According to DFT results, eventual charge +transfer should go through BL ligand acceptor state. The CT in the [Fe-BL] part of the dyad was +considered to originate from two possible states: the 3MLCT or 𝑀𝐿𝐶𝑇 +3 +. In both cases, to model CT, +an additional element of –kct 𝑀𝐿𝐶𝑇∗ +3 + or –kctMLCT was added into differential equations S3.1.b and +S3.1.a respectively. In the first case, any fitting attempts were unsuccessful. For CT from 3MLCT +state, a two-step analysis was applied. First, for [Fe-BL] a rate constant k1 for 1MLCT→ 3MLCT +transition was evaluated. For CT from 3MLCT in [Fe-BL-Co] this value was fixed, and the fit was +conducted with the free kct parameter. Results for global kinetic fitting are presented in Figs. 5, S3.6 +and S3.7. + +In the case of cobaloxime analysis, a two-state decay model was used to match the fluores- +cence decay results. Two scenarios were included, where states either decay in parallel or in a hierar- +chical way. Only the hierarchical model was reproducing the data with the states diagram (Fig. S3.8) +and fitted kinetic traces presented in Fig. S3.7. In early kinetics (-1-1 ps), the signal dynamics can be +described by 2.76 decay time, while a longer period requires the inclusion of a second, infinite time +constant. This model was the basis for [Fe-BL-Co] analysis with 4.12 ps time fixed in a short-time +window (Fig. S3.6). Importantly, the dynamics in short- and long-time windows could be reproduced +with 2 kinetic constants. The results of kinetic equation fitting (Fig 5 c, Fig. S3.6 b) confirmed the +fluorescence kinetic trace analysis for Co in [Fe-BL-Co]. For the sake of precision, two models were +also tested: the M’MCT and LMCT states decaying in a hierarchical and parallel way. Only the model +presented in Fig. 5 a reproduced data with satisfactory quality. Due to a very low Co signal intensity +in [Fe-BL-Co] the estimated uncertainties are high. + + + +41 + +-5.0 +-2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +0.0001 +0.0002 +0.0003 +0.0004 +0.0005 +0.0006 +Population / a.u. +Delay time / ps + M'MCT + LMCT + MC + gs + exp + fit +1 = 0.25 ps +2 = 4.12 ps +-0,2 +0,0 +0,2 +0,4 +0,6 +0,8 +1,0 +0,0 +0,1 +0,2 +0,3 +0,4 +0,5 +0,6 +0,7 +0,8 +Population / a.u. +delay time / ps + 1MLCT + 3MLCT + 3MC + gs + fit + exp +-2 +0 +2 +4 +6 +8 +10 +12 +14 +0,1 +0,2 +0,3 +0,4 +Populations / a.u. +delay time / ps + 1MLCT + 3MLCT + 3MC + gs + exp + fit +Fig. S3.8. Total kinetic traces for: a) [Fe-BL], short-time window; b) Co @ [Fe-BL-Co], long- +time window; c) [Fe-BL], long-time window; d) Fe @ [Fe-BL-Co], long-time window. +a) +b) +c) +d) +-0,5 +0,0 +0,5 +1,0 +1,5 +0,00003 +0,00004 +0,00005 +0,00006 +0,00007 +Fraction of absorbers +Delay time / ps + LMCT + gs + exp +0,0 +2,5 +5,0 +7,5 +10,0 +0,00005 +0,00006 +0,00007 +0,00008 +0,00009 +0,00010 +0,00011 +Fraction of absorbers +Delay time / ps + LMCT + MC + gs + exp + fit + + +Fig. S3.9. Populations for short- and long-time window kinetic traces for cobaloxime: a) +long-time window; b) short-time window. + +0,6 +MLCT* +MLCT +0,5 +MC +Fraction of absorbers +gs +exp +0,4 +fit +0.3 +0,2 +0.1 +0,0 +-5.0 +2,5 +0.0 +2,5 +5,0 +7.5 +10,0 +12,5 +15.0 +Delay time I ps + +42 + + + +References: + +1. +Dreuw, A. & Head-Gordon, M. 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Tuning excited state electron transfer in Fe tetracyano-polypyridyl +complexes. arXiv 1–21 (2019). + + + diff --git a/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/load_file.txt b/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5596660389465dfbfc2228821f2d7ff9efa04042 --- /dev/null +++ b/8NE3T4oBgHgl3EQfRwmH/content/tmp_files/load_file.txt @@ -0,0 +1,2585 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf,len=2584 +page_content='Ultrafast two-colour X-ray emission spectroscopy reveals excited state landscape in a base metal dyad M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Nowakowski1,†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Huber-Gedert1,†, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Elgabarty1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kubicki2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kertem2, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lindner2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Khakhulin,3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lima3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-K.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 61-614,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Poland 3 European X-Ray Free-Electron Laser Facility GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Holzkoppel 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 22869 Schenefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Germany 4 PAL-XFEL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Jigok-ro 127-80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 37673 Pohang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Republic of Korea 5 Institute of Nuclear Physics Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kraków,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 31-342,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 6 Departamento de Química, Universidad Autónoma de Madrid, Campus Cantoblanco, 28047 Madrid, Spain 7 IMDEA Nanociencia, Calle Faraday 9, 28049 Madrid, Spain ABSTRACT: Effective photoinduced charge transfer makes molecular bimetallic assemblies attractive for applications as active light induced proton reduction systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For a more sustainable future, development of competitive base metal dyads is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' However, the electron transfer mechanisms from the photosensitizer to the proton reduction catalyst in base metal dyads remain so far unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' We study a Fe-Co dyad that exhibits photocatalytic H2 production activity using femtosecond X-ray emission spectroscopy, complemented by ultrafast optical spectroscopy and theoretical time- dependent DFT calculations, to understand the electronic and structural dynamics after photoexcitation and during the subsequent charge transfer process from the FeII photosensitizer to the cobaloxime catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Using this novel approach, the simultaneous measurement of the transient K\uf061 X-ray emission at the iron and cobalt K-edges in a two-colour exper- iment is enabled making it possible to correlate the excited state dynamics to the electron transfer processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The meth- odology, therefore, provides a clear and direct spectroscopic evidence of the Fe→Co electron transfer responsible for the proton reduction activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' INTRODUCTION FeII complexes can operate as light-harvesting compo- nents in bimetallic molecular assemblies or dyads, to con- vert solar to chemical energy by ultrafast charge transfer (CT) to a second, catalyst metal for photocatalytic water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 In terms of sustainability, the second metal should be abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Cobaloxime fulfils this requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2– 4 Despite the reported short lifetimes of metal-to-ligand charge transfer (MLCT) states in iron(II) photosensitizers, photocatalytic proton reduction activity was reported for FeII-CoIII dyads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 However, its activity remains mysterious, as no charge transfer from the Fe to the Co center could be observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Thus, rational improvement of Fe-Co dyads requires a radically different approach to understand the working principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A major challenge are the ultrafast photophysics at the FeII center,6 and the diffi- culty to monitor the CT from the photosensitizer to the cat- alyst with element specificity and in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1,7,8 Upon photoexcitation the excited state dynamics in dyads can involve MLCT and ligand-to-metal charge transfer states (LMCT), metal-centred (MC) and ligand-mediated metal- to-metal charge-transfer states (M’MCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9,10 The de-exci- tation cascade is an interplay between MC and CT states, modulated by intramolecular vibrational energy dissipa- tion, strong spin-orbit coupling such as intersystem cross- ings (ISC), and internal conversions (IC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8,11 The funda- mental principles guiding the properties are typically iden- tified using laser spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12,13 Noble metal com- plexes exhibit long-lived CT states, which can easily be followed with optical spectroscopy, due to the associated intense absorption bands in UV-Vis range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='14 On the other hand, in most iron photosensitizers, the smaller ligand field splitting leads to an unfavoured energetic order of E(MLCT) > E(MC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 In addition, MC states are hardly ac- cessible in UV-Vis spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='15 Contrary, X-ray emission spectroscopy (XES) is very sen- sitive to MC states due to the localized character of core levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Both K\uf061 (2p→1s) and K\uf062 (3p→1s) emission lines provide characteristic signatures of the multiplicity of the involved transient MC states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16 For monomeric iron car- bene photosensitizers, femtosecond XES could uniquely reveal details of the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='13,16,17 The excited states of [Fe(bmip)2]2+ [bmip = 2,6-bis(3-methyl-imidaz- 2 ole-1-ylidine)-pyridine] show a complex branching pat- tern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' One path is dominated by a long-lived 3MLCT, while the second includes a rapid hot MLCT* to 3MC transition- connected to bond oscillations in form of wavepacket dy- namics11,16,18 which are a stabilizing factor for long-lived 3MLCT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 Ultrafast X-ray absorption near edge structure spectroscopy (XANES) on photoactive Fe-Co Prussian blue analogues revealed that a spin transition at the Co centre preceeded CT between the Fe and Co cen- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20 More recently, the photoinduced M’MCT transition in a bimetallic Fe-Ru assembly was shown to have a critical impact on the solvent organization around the excited molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='21 We demonstrate the unique potential of a two-colour X- ray emission spectroscopy (2C-XES) in photocatalysis re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It allows for simultaneous, ultrafast detection of the Fe and Co Kα XES in a [Fe-BL-Co] assembly of a het- eroleptic FeII photosensitizer with two different biscar- bene-pyridine ligands (C^N^C) connected to a cobalox- ime catalyst via a bridging ligand (BL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 The dynamics of the excited state decay are monitored at the Fe and Co sites to follow the departure of the charge from the photo- sensitizer and its arrival at the catalyst in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' As such, our experimental approach eliminates uncertainties related to the charge transfer event timescale and solves the puzzle about a possible charge transfer in base metal dyads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=" RESULTS The dyad is synthesized combining a heteroleptic tetra- NHC FeII photosensitizer [Fe-BL] coordinated by a 2,6- bis[3-(2,6-diisopropylphenyl)imidazol-2-ylidene]pyridine and a 2,6-bis(3-methyl-imidazol-2-ylidene)-4,4'-bipyridine ligand (BL) with a CoIII cobaloxime catalyst, as presented in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 A 4,4′-bipyridine (bpy) linker connects both metals with a distance of 11 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='22,23 The ground state opti- cal absorption spectra of the dyad in comparison to the constituting components [Fe-BL] (red) and cobaloxime (Co(dmgH)2Cl(py)=[Co], green) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The top panel shows two absorption bands for the photosen- sitizer at 398 nm and 481 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' By coordina- tion of the cobaloxime in [Fe-BL-Co] (black), the UV-Vis spectrum of the dyad changes in a distinct manner: The 398 nm band remains unchanged, but the 481 nm band is shifted to 497 nm, and a new band appears at 444 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Cobaloxime itself shows only a weak ab- sorption around 400 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Quantum chemical assignment of the optical absorp- tion bands In order to understand the properties of the electronic excited states of [Fe-BL-Co], the involved tran- sitions must be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' As a first step, we carefully benchmarked TPSSh/TDDFT excited-state calculations of the photosensitizer against CASSCF/NEVPT2 (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The latter combination of a multireference wave- function with a perturbative treatment of electron correla- tion accounts for both static and dynamic electron corre- lation effects, and has been shown to yield highly accurate results, but computational cost steeply increase with the number of correlated orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='24 Both techniques reveal that the bands at 398 nm and 481 nm in [Fe-BL] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1c, top, red) are a mixture of MLCT transitions from FeII to both the terminal and the BL (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1a-b,e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Together with the TPSSh/TDDFT computations of UV-Vis spec- trum, this assignment is used to understand the absorp- tion properties of [Fe-BL-Co] (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c and 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c shows the TDDFT spectrum of the dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The experimentally observed 398 nm absorption is described by transitions a and b (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Like in the photosensitizer, they are composed of MLCT transitions from iron to the terminal and bridging ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Additionally, the electron density is transferred from the FeII to the CoIII center along the bridging ligand in the form of an M’MCT transition (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The donor-acceptor contributions to the latter are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The absorption at 497 nm is dominated by an MLCT transition to the bridging Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) Structure of the [Fe-BL-Co] dyad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) Fe → Co CT (M’MCT): grey color indicates holes, red– electrons c) UV-Vis data for [Fe-BL], [Co] and [Fe-BL-Co] (top) overlapped with TAS results for selected delay times and TD-DFT UV-Vis spec- trum for the dyad (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 335 410 485 560 635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='08 DA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='50 ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 ps 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 ps 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 3MLCT Ground state bleach e ·10-6 / cm-1 M-1 Co(dmgH)2Cl(py) [Fe-BL-Co] [Fe-BL] [Fe-BL-Co], TPSSh SMD(MeCN) def2-TZVP wavelength / nm osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' strength / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b a c c PF3 ligand together with a weak M’MCT contribution (transi- tion c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The shift of the 497 nm band in the dyad spectrum compared to the photosensitizer spectrum (481 nm) is well reproduced by TDDFT (480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 nm vs 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 nm, SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is due to an increased charge transfer to the terminal pyridine ring (transition c in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1) and revealed by charge transfer components obtained for the dyad (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Unfortunately, TDDFT could not resolve the 444 nm band in the dyad spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' After re-evaluation of the former interpretation5, it is assumed that it is also present in the photosensitizer spectrum but overlaps with the 481 nm band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In conclusion, the nature of the transitions in the dyad is similar to those of the photosensitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Most important however, additional M’MCT contributions are found in all bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Due to the unchanged absorption band at 398 nm in both compounds, only weak LMCT absorption of co- baloxime2,25 and the M’MCT contribution in [Fe-BL-Co], an excitation wavelength of 400 nm was chosen for ultrafast transient spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Transient Absorption Spectroscopy Transient absorp- tion spectroscopy (TAS) results for [Fe-BL-Co] are pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The ground state bleach occurs at 370- 560 nm, an excited-state absorption is observed <370 nm and >560 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The transient absorption >560 nm is as- signed to a 3MLCT state5 and its kinetics are composed of three time constants (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The first component (<100 fs) in this model takes into account all coherent ar- tefacts26 and possible 1MLCT contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The second component (\uf0742=350 fs) can be ascribed to either the re- laxation from the hot 3MLCT* to thermally relaxed 3MLCT27,28, or to a 1MLCT → 3MLCT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11,15–17 This is supported by the excited-state TDDFT, where the first acceptor state for the 400 nm excitation is a 1MLCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The longest component can be assigned to the lifetime of the relaxed 3MLCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6,27,29,30 Due to the very similar results obtained in each fit (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2), a reliable average value of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps for the lifetime of the 3MLCT state is ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For the constituting photosensitizer [Fe-BL] a 3MLCT lifetime of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 ps is found (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The increased 3MLCT lifetime in [Fe-BL-Co] is interpreted as an indirect signature of CT processes, leaking into the re- laxation channel over the 3MLCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Yet, it does not provide unequivocal proof for a Fe→Co charge transfer due to a lack of direct spectroscopic signatures for altered charge densities at both Fe and Co cobalt centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This gap can be closed by XES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fe Kα XES dynamics XES is governed by different se- lection rules than the optical absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The XES signal originates from localized core electrons, and through the width of the Kα1 XES line, it is directly proportional to the effective number of unpaired d-electrons31 and cova- lency32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Using a von Hamos emission spectrometer,33 in a 2C-XES scheme34, spectra for both Fe and Co could be collected truly simultaneously, without any ambiguity of the time-zero on a femtosecond timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2 shows the early temporal evolution of the two XES signals and their kinetic traces along with the selected integration ranges for both elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In [Fe-BL-Co] three time constants of \uf0741,FeCo<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='14 ps, \uf0742,FeCo=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='38(40) ps and \uf0743,FeCo=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='74(18) ps are obtained from fitting of the transient kinetics at the iron Kα1 emis- sion, while for [Fe-BL] \uf0741,Fe~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps, \uf0742,Fe=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98(27) ps and \uf0743,Fe=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='71(35) ps are found (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The most notable difference is thus the increased longest lifetime \uf0742 in the dyad, which is similarly observed in TA measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 The difference of 2 ps is attributed to the different sensi- tivity of TA and XES towards CT states - MC states are “optically silent” in UV-Vis spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Both singlet and triplet 1/3MLCT states of Fe compounds have near-identical Kα XES signatures, since both have a single Fe-localized unpaired d-electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 Moreover, since the coupling of the deep 2p core-hole with the 3d manifold is weak, Kα XES has little sensitivity to ISC inside the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fe and Co Kα1,2 transient XES line intensities of [Fe- BL-Co] for delay times of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Top panel: transient XES signals at 1 ps delay time with integration regions of in- terest (ROIs) marked by vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Right panel: integrated area under transient XES Fe Kα1 and Co Kα1 main feature in function of delay time (points) with corresponding fitted model (lines, top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 6390 6400 6910 6920 6930 Delay time / ps Emission energy / eV Laser OFF Laser ON Intrgr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Area Co fit x4 Fe fit Co data x4 Fe data Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integration ROI Fe K\uf0611 Fe K\uf0612 Co K\uf0611 Co K\uf0612 Integration ROI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kinetics of the Co Kα emission in [Fe-BL-Co] and cobaloxime after 400 nm excitation along with corre- sponding fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Differential signal: [Co]- [Fe-BL-Co] is marked as blue lines (data+fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Filled areas represent un- certainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' t - t0 / ps cobaloxime cobaloxime fit diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' of data diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' of fit [Fe-BL-Co] [Fe-BL-Co] fit 4 MLCT manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' With regards to metal spin multiplicity, there is a significant difference between the 3MLCT (Sloc=1/2) and 3MC (Sloc=1) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Any relaxation process, involving either one of these states, to the singlet ground state is thus visible in transient Kα XES experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In agreement with previously reported values and the cur- rent TA results, the XES time constants are assigned in the following way: the shortest lifetime \uf0741 in both [Fe-BL] and [Fe-BL-Co] dyad corresponds to a 3MLCT*→3MC channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='36 The time constant \uf0743 can be attributed to the 3MC state decaying into the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='37,38 The value of time constant \uf0741 fully agrees with reports of 3MLCT*→3MC channels in Fe(II)-NHC complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='37,38 Moreover, TDDFT excited-state potential energy surfaces indeed identify a 3MC surface that intersects both the 1MLCT and the 3MLCT close to the Franck-Condon re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The 3MC is identified, using a Mulliken electron-hole population analysis, as a triplet state containing both a hole and an additional electron on the Fe metal, because of the Fe(dxy/dyz/dxz) → Fe(dx2-y2/dz2) transition (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Time constant \uf074\uf032 is assigned to the 3MLCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='36–38 Co Kα XES dynamics The Co K\uf061 transient kinetics of the dyad [Fe-BL-Co] is constituted of three time constants \uf0741,FeCo=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25(1) ps, \uf0742,FeCo=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39) ps and \uf0743,FeCo~23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39 ps, a striking difference to pure cobaloxime [Co], where two time constants of \uf0741,Co=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='76(31) ps and \uf0742,Co=23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='82) ps are found (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This differ- ence is also obvious in the kinetics of the Co Kα XES in [Co] (green) and [Fe-BL-Co] (red) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3, which differ substantially in shape over the first 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 ps (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The difference is caused by the short decay constant \uf0741,FeCo=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25(1) ps that is not present in pure cobaloxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This time constant thus represents a new excited state population channel, created by the formation of the dyad, which at later timescales is indispensable for photocata- lytic hydrogen generation and for which the optical ab- sorption data shows Fe-Co M’MCT contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Conse- quently, the differential signal in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3 is the real-time sig- nature of CT from the Fe to the Co center in [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The direct excitation of the cobaloxime reflected in the Co kinetics of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3 (green) corresponds to an LMCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' According to our DFT calculations,2 the HOMO in co- baloxime is composed of degenerated π orbitals of the dmgH ligand, and the LUMO consists of the Co dz2 orbital leading to a very weak LMCT absorption at 396 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The observed low cross-section excitation populates this LMCT state of Co with a lifetime of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39) ps in [Fe- BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since cobaloxime has a documented activity as a proton reduction catalyst,1,39 the increased catalytic activity of [Fe-BL-Co] compared to [Fe-BL] + cobaloxime originates from the M’MCT states in [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Note, the signal we observe originates from linear combination of differently excited species, since M’MCT and LMCT states cannot exist simultaneously in the same molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result is evident, despite a low CT yield for our prototype dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Nuclear motion detected by Fe Kα XES Both [Fe-BL] and [Fe-BL-Co] show a pronounced, but distinct, coherent nuclear wavepacket signatures in the transient iron K\uf0611 kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For the photosensitizer, the oscillations could be modelled by a single damped periodic function (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4, SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3c), while in case of the dyad, it is composed of two contributions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4 and SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' [Fe-BL] and [Fe-BL- Co] share a half-period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='28(2) ps and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='26(3) ps, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Similar oscillation half-periods were observed in other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 16,40 An additional oscillation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19(1) ps appears in the dyad (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The coherent oscillation detected in [Fe-BL- Co] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4a) is a combination of signals observed in the photosensitizer and the additional oscillation (T1/2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 ps) related to the coordination of cobaloxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The statisti- cal significance of the superposition could be proven (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Calculated excited state potential energy surfaces show that the oscillations appear along the Fe-N bonds with the equilibrium at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05 Å (3MLCT*/3MC crossing, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TDDFT results indicate several vibrational fre- quencies in the range around 175 cm-1, exhibiting a col- lective twisting motion of the bridging ligand, accompa- nied by a rotational distortion and slight stretching of the Fe-N bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Raman spectra also exhibit intense bands for [Fe-Co-BL] in 175-225 cm-1 range, present neither in [Fe- BL], nor in cobaloxime (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' While the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='26 ps half-period can be associated with the spin state transition due to the 3MLCT*/3MC crossing, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 ps oscillation is likely due to the rotation of the cobaloxime moiety around the Fe-Co axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This motion could affect the charge trans- fer due to the rotation of the pyridine ring, and modulation of the π* orbitals overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The excited state landscape in the [Fe-BL-Co] dyad can be substantiated with these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Femtosecond XES study on [Fe(bmip)2]2+ showed excited state branching, in which a vibrational wavepacket nearly identical to the one in [Fe-BL-Co] is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16 A 3MC is partially populated from the vibrational excited 3MLCT* state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since this wavepacket motion is associated with the MC state,19 it is not visible in optical TAS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' With the minimal spectral difference between 1MLCT/3MLCT states both in TAS and XES, the shortest time constant of \uf0741,Fe/FeCo in [Fe-BL] and [Fe-BL-Co] is associated with a transition from the 3MLCT* to 3MC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The longest time \uf0742,Fe/FeCo reflects the 3MLCT→3MC pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The remaining \uf0743,Fe/FeCo is assigned to the 3MC→GS recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7,11,17,41,42 Population analysis Kinetic modelling can facilitate the interpretation of the obtained time constants by testing dif- ferent reaction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Details of the approach can be found in the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' At the Fe center in [Fe-BL] and [Fe-BL-Co], an additional time constant of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='22(7) ps is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' According to our TDDFT calcula- tions, this can be related to the 1MLCT→3MLCT transition after a population of the first excited 1MLCT state (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This short time constant includes IC and ISC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11,16–18,40 It is also in good agreement with the 350 fs component obtained via TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' According to the proposed reaction scheme, the 3MLCT* state, which is populated during the 1MLCT→3MLCT*→3MLCT decay, branches into a α (3MLCT*→3MLCT→3MC) and β channel (3MLCT*→3MC), with contributions of 79(5) % and 21(5) %, respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In [Fe-BL], the branching ratio is 83 % to 17 %, respectively (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The observed wave- packet oscillations originate from the β pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='40 5 Most importantly, an additional deactivation channel, orig- inating from the 3MLCT state in the form of an M’MCT electron transfer in the [Fe-BL-Co] dyad, is resulting from the kinetic fitting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This transfer is clearly visible when the 3MLCT population of the pure photosensitizer [Fe-BL] (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3f) and the dyad [Fe-BL-Co] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5c) in the short time window is compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In the former, the rise of the 3MLCT population is initially damped, while in the latter, the population of 3MLCT rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The obtained value of the CT rate is very consistent with the magnitude of dif- ferences observed for the excited state kinetics at the Co center in [Fe-BL-Co] and cobaloxime (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A two-state model with subsequent decay is used for the cobaloxime (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3f, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10), consisting of the LMCT state directly populated upon 400 nm excitation and decaying to a lower-level state within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='78(3) ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For [Fe-BL-Co] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5a), an additional electron transfer-ac- ceptor state (M’MCT) is compulsory from the experi- mental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The M’MCT decays in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps,21 parallel to the directly excited LMCT decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The amplitude ratio between the direct excitation and CT transfer yield is 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 % to 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 %, close to the value obtained via cross-section analysis (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3b, and sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3f) and in line with TDDFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The kinetic fitting required an additional lowest ex- cited state of this direct decay path, which is of unknown nature so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' However, an MC character is most likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='43 The lifetime of this state is estimated to be around 23-30 ps based on the fit results for the pure cobaloxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Data quality for the dyad prevent accurate fitting of this contri- bution to the fluorescence signal in [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5a summarizes the results and conclusions from the observed time constants, literature,11,16,17,30,42 and TDDFT potential energy surfaces calculations along two reaction coordinates (Fe – N bite angle and distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The popu- lation analysis for [Fe-BL-Co] resulting from kinetic mod- elling is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5b-c (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3f and the corre- sponding diagram for [Fe-BL]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' CONCLUSIONS Photoactive base metal dyads appear as promising alter- native, as compared to precious metals, for inexpensive and sustainable molecular assemblies capable of direct harvesting of light and photocatalytic hydrogen produc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This still heavily depends on the rational improve- ment of their performance, which involves the interplay between their molecular design and photocatalytic prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Our study shows the tremendous potential of ultra- fast 2C-XES for direct characterization of photoinduced CT processes exemplified by the case of a noble metal free dyad [Fe-BL-Co] used in hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In combination with ultrafast optical spectroscopy, TDDFT and CASSCF/NEVPT2 calculations and excited state modelling, a CT from the FeII photosensitizer to the co- baloxime catalyst could be proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It contributes as a M’MCT state of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps lifetime to the very complex ex- cited state landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In addition, we can distinguish the direct excitation into an LMCT state of Co, which accom- panies the CT process between both metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The une- quivocal determination and visualization of the ultrafast CT is only possible by the intrinsic temporal self-calibra- tion of the Fe and Co Kα signals in the 2C-XES experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' With the achieved results a multitude of strategies to im- prove the photocatalytic activity of such base metal dyads can be deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is common knowledge that the lifetime of the 3MLCT as the first charge separated state needs to be increased for iron photosensitizers to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' How- ever, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5 it is immediately clear, that this is even more important here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A decreased 3MLCT energy would reduce the contribution of the 3MLCT→3MC decay chan- nel, potentially in favour of the population of the M’MCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Another way of decreasing non-CT decay channels would be a reduction of the 3MLCT*→3MC contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since this pathway is connected to the nuclear wave- packet, the associated vibrational motions might play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Further restriction of Fe-N oscillations, either via replacing N with C atom or construction of a more rigid ligand structure could selectively increase the 3MC en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Both Fe-N and Fe-BL-Co motions are involved here according to the presented results, and substitution of the pyridine by a cyclometalated ligand might be a suitable exchange for Fe-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The presented results thus offer a first step towards a ra- tional design of base metal dyads for photocatalytic pro- ton reduction reactions by direct observation and quanti- fication of CT process in functional bimetallic photosensi- tizer-catalyst assembly by 2C-XES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Coherent nuclear wavepacket signals (black), fit- ted oscillatory functions (red) part, damping (grey) for: a) Fe part of [Fe-BL-Co], where additionally a non-damped parts are visible (blue, purple);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) same for [Fe-BL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='03 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts - fit / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Probe delay / ps experimental data fit Damping function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='71 THz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='56 THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts - fit / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Probe delay / ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='49 THz experimental data fit Damping function a b 6 METHODS UV-Vis spectroscopy The investigated complexes were dissolved in acetonitrile (spectroscopic-grade, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5·10-4 mol/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' UV-Vis spectra were measured in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 cm quartz cuvettes on a Lambda 465 spectrophotometer from PerkinElmer (Waltham, Massachusetts, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Cobaloxime (1·10-5 mol/L) was measured with a Lambda 45 double-beam UV spectro- photometer from Perkin Elmer (Waltham, Massachusetts, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TAS spectroscopy The experimental setup was described elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5,44 Femtosecond transient absorption dynamic studies of [Fe-BL] and [Fe-BL-Co] were conducted using modified commercial Helios spectrometer (Ultrafast Systems, Sar- asota, Florida, USA) with the IRF value of 120 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TAS spectra were recorded for the 400 nm excitation in 60 ps temporal range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The laser pulse energy was 2 μJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Con- centrations were chosen to be identical to the time re- solved X-ray experiments (10 mM, MeCN), which caused high absorbance of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Therefore, optimized signal transmission was ensured by a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12 μm flow cell with CaF2 windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The use of a micro annular gear pump (~1 ml/s flow) guaranteed the excitation of fresh solution per laser pulse and reduction of sample degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Sub- traction of solvent response from each data set eliminated the solvent contribution in the TAS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Transient X-ray emission spectroscopy Simultaneous emission of Fe and Co Kα was measured with 120 fs time resolution at the FXE instrument at SASE1 branch of EuXFEL, Schenefeld, Germany (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='35 The [Fe-BL-Co] dyad in 10 mM solution of ace- tonitrile (MeCN) was measured in a cylindrical liquid jet (200 µm) and sample recirculation was provided by HPLC pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Sample was excited by 400 nm optical laser with power in the range of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 - 15 μJ/pulse and 50 fs pulse length (FWHM = 83 μm and 34 μm for horizontal and ver- tical directions, respectively) which translates to ~55 % of excitation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Electronic configuration in ground and ex- cited states were probed by the SASE X-Ray beam with a central energy of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 keV with 125 bunches per pulse train at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='564 MHz intra-train repetition rate (beam size FWHM = 20 μm, pulse duration 100 fs, ~1012 pho- tons/pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The X-Ray beam was operating at the stand- ard EuXFEL mode of 10 Hz repetition rate per train and the optical laser was at 5 Hz, meaning alternating pumped/unpumped trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The beams were crossed with angle of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 20°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Subsequent fluorescence emission was collected using wavelength-dispersive 16-crystal von Hamos XES spectrometer (Fe Kα and Co Kα with Ge(440) and Si(531) analyzer crystal reflections at 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4° and 77°, respectively) and a 2D charge integrating gain- switching Jungfrau 1M detector with matrix of 1024 x 1024 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' (a) Ground and excited state potential energy surfaces along the Fe-N distance (bottom x-axis) and the Fe-N bite angle (top x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Insert: state diagram for Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' State contributions at the (b) Fe in [Fe-BL-Co] (c) Co and fs-XES signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a Fe - N bite angle / b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 101 102 103 104 105 106 107 108 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 MLCT 3MLCT a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 3MC ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='14ps P 3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 3MLCT* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 21 % delay time / ps ev 1MLCT 3MC fit (GS) 2 3MLCT gs exp c 79 % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 ps E Population / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 ps Co Decay: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content="8 M'MCT LMCT ground 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12 ps state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 >23 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ground state 0 delay time / ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content="4 MM'CT exp Fe - N distance / A fit LMCT gs7 pixels and repetition rate of 10 Hz." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The timing jitter be- tween X-Ray and optical pulses was ~70 fs FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Signal was integrated over 60 s (500 trains) per time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For different delay time windows, a set of data was acquired with specified temporal step size: for -5ps -15 ps it was 1 ps while for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 ps and -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 ps it was 150 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For single delay time measurements, signal was col- lected for 60 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For each measurement number of repeti- tions was set individually to provide good S/N ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' As a reference also the catalyst cobaloxime and the photosen- sitizer [Fe-BL] were measured separately in the same ex- perimental conditions and concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Due to limited solubility, cobaloxime was measured at 5 mM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Quantum chemical calculations Unless otherwise stated, all calculations were carried out with the ORCA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 quantum chemistry package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='45 Throughout we have used the Alrich’s def2-TZVP46 basis set, and employed the Split-RI-J method and chain of spheres (RIJCOSX) approximation to accelerate the cal- culation of the exchange and Coulomb terms, together with the def2/C and def2/J auxiliary bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='47 Spin-orbit coupling corrections were introduced using the spin-orbit mean field method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='48 Solvation of the compounds was in- cluded via SMD49 (MeCN) and dispersion correction was introduced via DFT-D3 with the Becke-Johnson damping scheme (D3BJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='50,51 Unconstrained DFT optimizations of the investigated complexes were done with the PBEh-3c method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='52–54 The UV-Vis spectra of [Fe-BL] and [Fe-BL-Co] were calculated using the hybrid meta-GGA functional TPSSh55, employ- ing the Time-dependent DFT (TDDFT) and the Tamm- Dancoff approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The adequacy of the method was justified by our benchmark study on the photosensitizer against CASSCF/NEVPT2 (SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The singlet en- ergy transitions (60 states) have been subjected to Gaussian broadening with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 eV (full width at half-height) before converting to the nm scale and com- pared to the experimental UV-Vis spectra of the investi- gated complexes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' SI, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1b-c, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Donor and acceptor orbitals of selected transitions and their spatial distribution were visualized using Avogadro (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Singlet and triplet excited state potential energy surfaces were computed staring from the optimized ground state geometry, by discretizing a geo- metric pathway that involves a simultaneous stretching of the Fe-N distances at steps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05 Å, and the Fe-N bite an- gles at steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' At each point along this path- way, the 60 lowest lying singlet and triplet states were computed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 120 states in total), again using the afore- mentioned computational setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In order to identify the nature of any given excited state, whether it is a 1MLCT, 3MLCT, or 3MC, we have resorted to the Mulliken popula- tion analysis coupled with an electron-hole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='56,57 Because our TDDFT calculations are based on the singlet ground state as the reference state, the spin populations of all atoms are zero by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Instead of relying on spin densities, we identify a 1MLCT/ 3MLCT as a sin- glet/triplet excited state where the total Mulliken popula- tion of the Fe atom is decreased by one electron, and that of the ligand atoms is increased by one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A 3MC state is a triplet excited state where both the hole and the electron are localized on the Fe atom, corresponding to an electron transfer from the occupied dxy/dyz/dxz orbitals to the virtual dx2-y2/dz2 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In all cases, only excited states that lie below the initially excited 1MLCT were con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 depicts an example of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The geometry of the identified 3MC state was optimized and its vibrational normal modes were computed in Gaussian 16 with the def2-SVP basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='58 ASSOCIATED CONTENT Any methods, additional references, Nature Research report- ing summaries, source data, extended data, supplementary information, acknowledgements, peer review information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' details of author contributions and competing interests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' and statements of data and code availability are 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 91, 3–6 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' & Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' An sp-hybridized all-carboatomic ring, cyclo[18]carbon: Electronic structure, electronic spectrum, and optical nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Carbon N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 165, 461– 467 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' & Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Multiwfn: A multifunctional wavefunction analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 33, 580–592 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Frisch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Gaussian 16, Revision C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 9 ACKNOWLEDGEMENTS The authors gratefully acknowledge European XFEL in Schenefeld, Germany, for provision of X-ray free-electron laser beamtime at FXE and would like to thank the instru- ment group and facility staff for their expert assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' acknowledges funding by the German DFG in frame of priority program SPP 2102 (Grant number BA 4467/7- 1) and the German BMBF (Grant numbers 05K19PP1 and 05K18PPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' acknowledges partial funding from Narodowe Centrum Nauki through SONATA BIS 6 grant (2016/22/E/ST4/00543).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' further acknowledges funding from Spanish MIU through “Ayudas Beatriz Galindo” (BEAGAL18/00092), Comunidad de Madrid and Universidad Autónoma de Madrid through Proyecto de I+D para Investigadores del Programa Beatriz Galindo (SI2/PBG/2020-00003), Spanish MICIU through Proyecto de I+D+i 2019 (PID2019-108678GB-I00) and IMDEA- Nanociencia through Severo Ochoa Programme for Cen- tres of Excellence in R&D (MINECO, CEX2020-001039- S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' acknowledges grants by Fonds der Chemi- schen Industrie and Studienstiftund des deutschen Vol- kes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Generous grants of computer time at the Pader- borner Center for Parallel Computing PC2 is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS Conceptualization: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Data curation: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Formal analysis: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Funding acquisition: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Investigation: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Methodology: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Project administration: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Resources: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Software: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Supervision: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Validation: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Bv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=',T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='. DATA AVAILABILITY STATEMENT The datasets generated during and/or analysed during the current study are available from the corresponding au- thor on reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' COMPETING INTERESTS The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ADDITIONAL INFORMATION Extended data is available for this paper at Supplementary information is available for this paper at Correspondence and requests for materials should be addressed to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Reprints and permissions information is available at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='com/reprints 10 Ultrafast two-colour X-ray emission spectroscopy reveals excited state landscape in a base metal dyad M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Nowakowski1†, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Huber-Gedert1†, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Elgabarty1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kubicki2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kertem2, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lindner2, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kha- khulin,3 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Lima3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Choi3,4, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Biednov3, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Piergies5, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Zalden3, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kubicek3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Rodriguez- Fernandez3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Alaraby Salem1, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kühne1, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Gawelda2,6,7, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Bauer1* 1 Chemistry Department and Center for Sustainable Systems Design (CSSD),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Faculty of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Paderborn University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Warburger Straße 100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 33098 Paderborn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Germany 2 Faculty of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Adam Mickiewicz University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Uniwersytetu Poznańskiego 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Poznań,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 61-614,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Poland 3 European X-Ray Free-Electron Laser Facility GmbH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Holzkoppel 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 22869 Schenefeld,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Germany 4 PAL-XFEL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Jigok-ro 127-80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 37673 Pohang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Republic of Korea 5 Institute of Nuclear Physics Polish Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kraków,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 31-342,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 6 Departamento de Química,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Universidad Autónoma de Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Campus Cantoblanco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 28047 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Spain 7 IMDEA Nanociencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Calle Faraday 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 28049 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Spain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Supplementary Information ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='Table of contents: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1) Quantum chemical calculations: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a) Benchmarking the TDDFT UV-Vis spectra of [Fe-BL] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='b) Computed and experimental UV-Vis spectrum of [Fe-BL] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c) Computed and experimental UV-Vis spectrum of [Fe-BL-Co] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='d) Mulliken population-based electron-hole analysis of excited states ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e) Decomposing the UV-Vis spectrum of [Fe-BL] in terms of charge-transfer components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='f) Decomposing the UV-Vis spectrum of [Fe-BL-Co] in terms of charge-transfer components ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2) TA experimental analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3) XES data analysis: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a) Fluorescence fitting procedure and results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='b) A direct and non-direct contribution to Kα XES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c) Wavepacket analysis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='d) Co Kα1 kinetic signals for -5-15 ps time window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content="e) d'-d interactions in Co " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='f) Kinetic model and population analysis results ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Quantum chemical calculations a) Benchmarking the TDDFT UV-Vis spectra In order to understand the optical spectrum and the nature of the optical excitation process, we have resorted to quantum chemical calculations using time-dependent density functional theory (TDDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is generally true that the study of transition metal complexes is challenging because of dynamic correlation effects, system size, state degeneracies or near-degeneracies, and relativistic effects on top of the typically large system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In particular, TDDFT is known to have difficulties with systems having charge-transfer states, and with extended π-systems1,2, both are features of the dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' However, despite these well-known issues, TDDFT has nevertheless been successfully applied to study such systems, including d6 transition metal complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 These known issues mean however, that one should not blindly trust TDDFT results without scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' To this end, we have benchmarked TDDFT UV-Vis electronic spectra, using both the hybrid-GGA B3LYP functional and the hybrid-meta-GGA TPSSh functional, against CASSCF-NEVPT2 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' While CASSCF/NEVPT2 is known to reliably yield reasonable accuracy,4 the dyad molecule is too large for the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The CASSCF/NEVPT2 method explicitly takes account of both static and dy- namic correlation effects and is known to provide highly accurate spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 In order to keep the size of the active space manageable we have done the benchmarking against the photosensitizer without the cobaloxime moiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' As explained later, the active space required to accurately compute the elec- tronic spectrum of the photosensitizer included 14 electrons in 13 active orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A CASSCF/NEVPT2 of the dyad, including all the 12 d-electrons together with the interacting ligand electrons was computationally unfeasible due to the large number of occupied orbitals in the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Our benchmark calculations show that the TPSSh functional yields qualitatively correct result and accurately reproduces the spectrum with a slight tendency to over-estimate the frequency of the peaks, in particular the lowest-frequency peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' To obtained better comparison of calculated spectra with ex- perimental ones calculated spectra are broadened by convolution with a Gaussian function with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 eV (full width at half-height), before converting the scale to nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 12 Starting orbitals for CASSCF The starting orbitals for the CASSCF calculation were taken from the TPSSh ground-state Kohn- Sham orbitals at the equilibrium geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The TPSSh ground state has the close-lying (within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 eV) Fe dxy, dyz, and dxz orbitals as the three occupied frontier orbitals, these were naturally included in the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The dz2 and the dx2-y2 orbitals were found to be strongly mixed with ligand orbitals, consistent with the strongly σ-donating heterocyclic carbene ligand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Both the occupied (bonding) and unoccupied (antibonding) orbitals involving Fe dx2-y2 and dz2 were included in the active space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In addition to the full set of (ligand-mixed) Fe d-orbitals, the two highest lying occupied π-bonding or- bitals were included in the active space, together with the four lowest unoccupied molecular orbitals (LUMO to LUMO+3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The LUMO is a π* orbital extending over the bipyridine moiety, while the other three orbitals are all π* orbitals extending on the CNC moieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Thus, the final active space included 14 electrons in seven occupied orbitals and six virtual orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is worth mentioning here that the B3LYP Kohn-Sham orbitals were identical in character to the TPSSh orbitals, in agreement with the benchmark results that we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Comparison of TDDFT UV-Vis spectra to CASSCF(14,13)/NEVPT2 The obtained spectra, which are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1, show several interesting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The CASSCF/NEVPT2 spectrum closely follows the experimental one, with two major peaks at 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 and 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' We believe that the major source of the shift from the experimental spectrum is the implicit solvation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Between these two major absorption peaks, there is a weak absorption peak at 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 nm (~10% of the oscillator strength of the strong peaks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The TDDFT spectra, although blue-shifted, still provide qualitatively correct results, except for the wrong trend in the peak intensities, with the low-frequency peak having a lower amplitude than the high-frequency one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The TPSSh functional is clearly performing better than B3LYP, with the TPSSh peaks appearing at 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0, 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2, and 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 nm, compared to 411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9, 387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1, and 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 nm for B3LYP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The accuracy of TDDFT transition frequen- cies, which we find here, is consistent with the expected accuracy range of the method, typically within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5,6 The lack of any peaks below 300 nm in the CASSCF/NEVPT2 spectrum is be- cause here we have only calculated the twelve lowest-lying singlet states (compared to 60 states in TDDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 13 Rather than the exact positions of the peaks, more important to our benchmark is the nature of the underlying states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Here, we find very consistent behavior between TDDFT (both functionals) and CASSCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Both methods agree that the main transitions bear predominantly the MLCT character and originate from the three frontier orbitals to the virtual orbitals in the range LUMO – LUMO+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The low-frequency peak is consistently the transition Fe dyz → LUMO with contribution from the transi- tion Fe dxy → LUMO+2 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Also, all the methods show that the higher frequency peak is mainly dxy → LUMO+3 with a minor contribution from dxy → LUMO+1, and that the weak intermediate frequency peak is a transition from the three frontier orbitals to the three virtual orbitals LUMO+1 – LUMO+3 (for details see Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Influence of spin-orbit coupling In computing all the TDDFT spectra, we have included corrections due to spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is worth noting however, that this turned out to have very little influence on peak positions, with typical shifts of less than 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' As an example, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 shows the influence of spin-orbit coupling on the UV-Vis spectrum of the photosensitizer, as obtained with B3LYP/TDDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' UV-Vis absorption spectrum of the photosensitizer in implicit acetonitrile solvation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Black: TDDFT with B3LYP, red: TDDFT with TPSSh, dotted blue: CASSCF(14,13), blue: CASSCF(14,13)/NEVPT2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' All the spectra are broadened by convolution with a Gaussian function with a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 eV (full width at half-height), before converting the scale to nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' B3LYP TPSSH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 CASSCF(14,13) CASSCF(14,13)/NEVPT2 strength Normalized osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0 250 300 350 400 450 500 550 Wavelength / nm 14 b) Computed and experimental UV-Vis spectrum of [Fe-BL] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 Experimental UV-Vis spectrum of [Fe-BL] in MeCN and time-dependent TDDFT spectrum with TPSSh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Influence of spin-orbit coupling (SOC) on the UV-Vis spectrum of the photosensitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 300 400 500 600 0 10 20 30 40 500 1 2 e ·10-4 / cm-1 M-1 [Fe-BL] TPSSh def2-TZVP SMD(MeCN) osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' strength / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' wavelength / nm a c b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 B3LVP B3LYP with SOC 1 Osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' strength (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0 250 300 350 400 450 500 550 Wavelength / nm 15 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 Computed dominant singlet vertical excitations a-c of [Fe-BL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Donor and acceptor orbitals are listed together with their contribution to the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The main character of the transition is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Transition (state) Donor Acceptor Contri- bution Character a (8) 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 nm HOMO (237) LUMO+3 (241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='73 MLCT HOMO-2 (235) LUMO+1 (239) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10 MLCT b (6) 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2nm HOMO-2 (235) LUMO+1 (239) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='37 MLCT HOMO (237) LUMO+2 (240) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='36 MLCT HOMO (237) LUMO+3 (241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='22 MLCT c (4) HOMO-1 (236) LUMO (238) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='64 MLCT 16 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 nm HOMO (237) LUMO+2 (240) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20 MLCT HOMO-2 (235) LUMO (238) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='09 MLCT c) Computed and experimental UV-Vis spectrum of [Fe-BL-Co] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 Experimental UV-Vis spectrum of [Fe-BL-Co] in MeCN and time-dependent TDDFT spectrum with TPSSh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 250 300 350 400 450 500 550 600 650 0 10 20 30 40 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 e ·10-4 / cm-1 M-1 [Fe-BL-Co] wavelength / nm osc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' strength / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TPSSh def2-TZVP SMD(MeCN) a b c 17 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 Computed dominant singlet vertical excitations a-c of [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Donor and acceptor or- bitals are listed together with their contribution to the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The main character of the transition is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Transition (state) Donor Acceptor Contri- bution Character a (17) 401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 nm HOMO (320) LUMO+7 (328) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='37 MLCT HOMO (320) LUMO+4 (325) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='29 MLCT/ MMCT HOMO (320) LUMO+5 (326) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='21 MLCT/ MMCT b (15) 413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 nm HOMO (320) LUMO+7 (328) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='38 MLCT HOMO-2 (318) LUMO+3 (324) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='37 MLCT HOMO (320) LUMO+5 (326) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='14 MLCT/ MMCT HOMO-1 (319) LUMO (321) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='78 MLCT 18 c (8) 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 nm HOMO (320) LUMO+5 (326) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='08 MLCT/ MMCT The TDDFT calculation of the dyad indicates transitions with partial MMCT character for the UV- Vis band around 400nm and 480nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The charge transfer from iron to cobalt is further analyzed by the charge transfer analysis in section 1f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' d) Mulliken population-based electron-hole analysis of excited states Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Mulliken population analysis of the triplet excited states that show the strongest metal- centered character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' State number 53 is the initially populated 1MLCT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' "Fe" and "Co" refer to the Mulliken populations of the two metal atoms, "pyridine" is the total charge on the bridge pyridine attached to the Fe, and "Fe-coord" is the octahedral coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 53(S) 3(T) 7(T) 10(T) 20(T) 22(T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 Dq relative to the ground state / e- State Fe atom [Co] pyridine [Fe] T - triplet S - singlet 19 In order to identify the nature of any given excited state as obtained from TDDFT, whether it is a 1MLCT, 3MLCT, or 3MC, we have resorted to the Mulliken population analysis coupled with an electron-hole analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 Because our TDDFT calculations are based on the singlet ground state as the reference state, the spin populations of all atoms are zero by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Instead of relying on spin densities, we identify a 1MLCT/ 3MLCT as a singlet/triplet excited state where the total Mulliken population of the Fe atom is decreased by one electron, and that of the ligand atoms is increased by one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A 3MC state is a triplet excited state where both the hole and the electron are localized on the Fe atom, corresponding to an charge transfer from the occupied dxy/dyz/dxz orbitals to the virtual dx2-y2/dz2 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In all cases, only excited states that lie below the initially excited 1MLCT were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 graphically depicts the outcome of such an analysis on the optimized geometry of the singlet ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In this particular case, state 7(T) is readily identified as the lowest lying 3MC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Calculation of the Mulliken population contribution of the Fe atom to the hole and electron redistribution confirms the identity of this state, with the Fe atom contributing 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8% to the electron hole (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' the excited electron originates from the Fe), and with 68% of the redistributed electron density concomitantly residing on the Fe atom (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' the excited electron resides on the Fe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' e) Decomposing the UV-Vis spectrum of [Fe-BL] in terms of charge-transfer components This qualitative characterization of the MLCT charge-transfer nature of the main transitions in the spectrum like the one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 can be put into more quantitative terms using a hole-electron anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7,8 The idea here is to start with the usual expression for the UV-Vis spectrum as obtained via broadening the excitation energies of all excited states: 𝜀(𝐸) ∝ ∑ 𝑓𝑖 𝑖 𝐺(𝐸 − 𝐸𝑖 𝑒𝑥𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=') Where 𝐸𝑖 𝑒𝑥𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' is an excitation energy, 𝑓𝑖 the corresponding oscillator strength, and G(…) denotes con- volution with a lineshape function (A Gaussian function in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' If we now subdivided the mol- ecule into two mutually exclusive fragments A and B (generalization to more fragments is trivial), then the excitation spectrum can be readily decomposed as: 20 𝜀(𝐸)𝐴,𝐵 ∝ ∑ 𝑓𝑖𝑄𝑖 𝐴,𝐵 𝑖 𝐺(𝐸 − 𝐸𝑖 𝑒𝑥𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=') where 𝑄𝑖 𝐴,𝐵 is the amount of charge transfer from A to B in excited state i as obtained, in this case, by a Mulliken population analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Because the sum of all inter- and intra-fragment charge transfer terms is unity, the partitioning is exact, and the total spectrum is exactly divided into two intra-fragment (A→ A and B→ B) charge redistribution terms and two inter-fragment (A→ B and B→ A) charge transfer terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' To decompose the photosensitizer spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6, the structure was subdivided into three frag- ments: the iron atom (fragment 1), the terminal bipyridine moiety (fragment 3), and the rest of the molecule (fragment 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 depicts the decomposed TPSSh/TDDFT spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The decom- posed spectrum clearly reveals the nature of all the peaks in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For instance, the low- frequency peak involves mainly (50% of the total amplitude) a charge transfer from the iron to the terminal bipyridine, where the LUMO orbital resides, but also includes important 1→ 2 and 1→ 3 charge-transfer contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' On the other hand, the 1→ 2 charge transfer spectrum is most prominent in the peak close to 400 nm, but also the shoulder due to the contribution of the weak intermediate peak centered at 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 nm is also clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Decomposition of total UV-Vis spectrum into intrafragment charge redistribution and interfragment charge transfer contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fragment 1 is the iron atom, fragment 3 is the terminal pyridine moiety, and fragment 2 is the rest of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 21 f) Decomposing the UV-Vis spectrum of [Fe-BL-Co] in terms of charge-transfer components In an analogic way to the [Fe-BL] case, we decomposed the UV-Vis spectrum of the [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 shows separate parts of the dyad considered in this analysis along with the color code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 presents contributions to the total charge for all considered transitions for each of the molecular parts shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The fractional contribution of each fragment to the hole and the electron in each of the three major transitions in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Wavelength / nm oscillator strength hole(1) electron(1) hole(2) electron(2) hole(3) electron(3) 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='029 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 presents the charge transfer analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It reveals that the peak at ~400 nm has the same nature as in the photosensitizer, with a small fraction of Fe → Co charge transfer (shown directly in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The low-frequency peak corresponds to considerably more charge transfer to the terminal pyridine ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Definition of the three fragments used to decompose the UV-Vis spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Orange: fragment 1, red: fragment 2, green: fragment 3 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Decomposition of total UV-Vis spectrum into intrafragment charge redistribution and interfragment charge transfer contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fragment 1 is the iron atom moiety, fragment 2 is the terminal pyridine moiety, and fragment 2 is the cobalt atom moiety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TAS data analysis Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' TAS spectra recorded upon 400 nm excitation for the [Fe-BL].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kinetics recorded for [Fe-BL-Co] at 650 nm, 640 nm, 630 nm and 610 nm together with the fitted model (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 405 475 545 615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 DA nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='500 ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 ps 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 ps 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 3MLCT Ground state bleach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='016 610 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='016 - 630 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012 370 ± 20 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 330 ±20 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='006 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3pS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 80 70 80 90 100 time delay I ps time delay / ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='016 - 640 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='016 - 650 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 - 330 ± 20 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 320 ± 20 fs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0 10 20 30 40 50 80 70 80 90 100 0 5 10 1520 40 60 80 100 time delay I ps time delay / ps 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kinetics recorded for [Fe-BL-Co] at 510 nm and 535 nm presenting the temporal evolu- tion of the recovery of the GS together with the fitted model (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is commonly accepted that global analysis is applied to reveal a temporal evolution of such com- plexes but in this case as our model assumes that the ground state (GS) is repopulated by 3MC and not the 3MLCT (here, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5a) we did not use this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Therefore, the recovery of the GS is expected to be slower than the decay of the 3MLCT state, although both relaxation channels are oc- curring on the same order of magnitude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=', few tens of picoseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kinetics in 510-535 nm spectral range provide that the recovery of the GS takes place with a time constant in the range from 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 - 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The 3MLCT-related time constant changes a little (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 ps - 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 ps) depending on the wave- length selected for the strongest GS bleach band (Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is very likely due to the vibrational cooling of the hot GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 Few picosecond longer recovery of GS than the decay of 3MLCT state is consistent with the fact that the lifetime of 3MC state is of the order of 2 ps (vide infra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10–14 The same effect can be observed in kinetics extracted for the [Fe-BL] 630 nm, 550 nm, and 480 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The comparison with the kinetic data between the photosensitizer and the dyad can be done only qualitatively since upon formation of the bimetallic assembly the 470 nm feature in [Fe-BL] UV-Vis spectrum relaxes to ~500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 DA time delay / ps 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 ps 510 nm 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 DA time delay / ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 ps 535 nm 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kinetics recorded for [Fe-BL] at 630 nm, 550 nm and 480 nm together with the fitted model (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='014 DA time delay / ps 120 ± 10 fs 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 ps 630 nm 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='012 DA time delay / ps 120 ± 10 fs 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7 ps 550 nm 0 10 20 30 40 50 60 70 80 90 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000 DA time delay / ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 ps 480 nm 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' XES data analysis a) Fluorescence fitting procedure and results The time-resolved X-ray emission spectroscopy (TR-XES) data was obtained on a setup scheme presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The energy scale for TR-XES data was obtained on a basis of the ground state XES (gs-XES) measurements performed using von Hamos spectrometer at P64 beamline of Petra-3 synchrotron at DESY (Hamburg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The gs-XES energy calibration was obtained by measuring Fe foil and adjusting the first inflection point in XAS spectrum to 7112 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Initial data correction: empty pulse removal and dark correction were conducted on-site, while data reduction and extraction were performed re- motely on DESY Maxwell server with the use of self-written Python scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A set of data from the experiment was sorted, background reduced, filtered, and normalized to obtain ON/OFF XES spectra in respect to delay time between the optical pump and X-ray probe pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' From that a series of XES spectra, differential (transient spectra, ΔXES) spectra were calculated, as ΔXES(t) = XESON(t) - XESOFF(t), both for Fe and Co Kα lines, examples are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Progressing changes in the ΔXES profile were represented in form of the integral of the selected feature over all delay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Those Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Scheme of the experimental setup used at FXE beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 50 fs FWHM optical pulses of 400 nm wavelength (blue) were synchronized with 100 fs FWHM X-ray pulses (purple) with the timing jitter of ~70 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fe and Co Kα X-ray fluorescence emitted from the liquid jet (green) sample was analyzed by a 16-crystal array of von Hamos spectrometer and directed to the 2D Jungfrau detector Co Kα Fe Ka 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 MHz Glass 200 ms Optical nozzle pulse 100 ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 MHz VonHamosX-rayemissionspectrometer X-ray laser-on laser-off laser-on laser-off pulse train train train train Sample jet 27 kinetics were subsequently fitted with fluorescence rise and i–exponential decay functions to give decay rates 𝜏𝑖 15: 𝑦 = 𝑦0 + ∑ 𝐴𝑖𝑔𝑖(𝑡) 𝑖 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1) 𝑔𝑖(𝑡) = 1 2 ⁄ (1 + 𝑒𝑟𝑓 ( 𝑡−𝑡0 𝜎 𝐶 2 − 𝜎 2𝐶𝜏𝑖)) 𝑒 𝜎2 𝐶2𝜏𝑖 2𝑒 −𝑡−𝑡0 𝜏𝑖 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2) where: 𝐶 = 2√𝑙𝑛(16) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 𝐴𝑖 – amplitude for the i –th exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 𝑡0 – the time-zero constant value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 𝜎 – Gaussian broadening due to IRF function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Due to used setup the IRF was fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='28 ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 𝑖 = 1,2,3 – the degree of exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The fitting procedure was carried out in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' First, the largest dominating contribution was fitted to a kinetic trace in the time window of 20 ps and step size of 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Second, to fit smaller decay time constants, kinetic traces in time windows of ~2 ps and step size of 50 fs with the largest time constant were fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' All fitting procedures were performed with a value of FWHM in pump-probe cross-cor- relation function set to σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The σ value was refined in the post-fitting verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The summarized fitting results are presented in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 and in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 6910 6915 6920 6925 6930 6935 6940 6385 6390 6395 6400 6405 6410 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' / au Emission energy / eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='45 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' / au Emission energy / eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='45 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 6910 6915 6920 6925 6930 6935 6940 6385 6390 6395 6400 6405 6410 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' / au Emission energy / eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='45 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' / au Emission energy / eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='45 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 ps a b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Kα1 transient XES for: a) Fe @ [Fe-BL-Co];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) Co @ [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 28 Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Summary for fluorescence fitting to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' [Fe-BL] 15 ps [Fe-BL] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps A1 A1 0,170(26) τ1 [ps] τ1 [ps] 0,245(42) A2 0,241(5) A2 0,538(7) τ2 [ps] 8,984(273) τ 2 [ps] 10,142(2,054) A3 0,079(12) A3 0,126(8) τ 3 [ps] 1,705(348) τ 3 [ps] 2,421(640) t0 [ps] 0,368(31) t0 [ps] 0,015(7) 𝑦0 0,070(2) 𝑦0 0,061(6) FWHMa [ps] 0,289(61) FWHMa [ps] 0,305(21) Fe @ [Fe-BL-Co] 15 ps Fe @ [Fe-BL-Co] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps A1 A1 0,109(20) τ1 [ps] τ1 [ps] 0,115(23) A2 0,00135(2) A2 0,483(3) τ2 [ps] 10,381(242) τ 2 [ps] 12,417(1,399) A3 A3 0,111(4) τ 3 [ps] τ 3 [ps] 1,740(182) t0 [ps] 0,064(12) t0 [ps] 0,010(3) 𝑦0 4,922(56) ·10-4 𝑦0 0,024(2) FWHMa [ps] 0,284(48) FWHMa [ps] 0,275(10) Co @ [Fe-BL-Co] 15 ps Co @ [Fe-BL-Co] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps A1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='140 (421) ·10-4 A1 0,00257(13) τ1 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 (fixed) τ1 [ps] 0,242(14) A2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='249(175) ·10-4 A2 τ 2 [ps] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39) τ 2 [ps] A3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='381(149) ·10-4 A3 0,00187(37) τ 3 [ps] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39 (fixed) τ 3 [ps] 6,084(1,134) t0 [ps] 0,116(10) t0 [ps] 0,047(7) 𝑦0 1,049(31) ·10-4 𝑦0 8,061(282) ·10-4 FWHMa [ps] 0,284(42) FWHMa [ps] 0,280(19) Cobaloxime 215 ps Cobaloxime 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps A1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='342(233) ·10-5 A1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='011(110) ·10-5 τ1 [ps] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='764 (312) τ1 [ps] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='391(172) A2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='391(78) ·10-5 A2 τ 2 [ps] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='391(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='820) τ 2 [ps] t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='056(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='093) t0 [ps] 0,015(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='040) ·10-2 𝑦0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='812(51) ·10-5 𝑦0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='391(35) ·10-5 FWHMa [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='273(50) FWHMa [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='305(84) a Post-fitting refinement 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 Fe K\uf0611 Fe K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps Model Conv_3exp (User) Equation exp1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(16)) /fwhm-fwhm/(2*tau1*sqrt(ln(16)))) )*exp(fwhm^2/(4*tau1^2*ln(16)))*e xp(-(x-t0)/tau1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(16)) /fwhm-fwhm/(2*tau2*sqrt(ln(16)))) )*exp(fwhm^2/(4*tau2^2*ln(16)))*e xp(-(x-t0)/tau2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp3=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(16)) /fwhm-fwhm/(2*tau3*sqrt(ln(16)))) )*exp(fwhm^2/(4*tau3^2*ln(16)))*e xp(-(x-t0)/tau3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' y=A1*exp1 + A2*exp2 + A3*exp3 + y0 Plot L A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16845 ± 0 tau1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='24508 ± 0 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='53829 ± 0 tau2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1301 ± 0 A3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12741 ± 0 tau3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39459 ± 0 y0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='06257 ± 0 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01471 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00694 fwhm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 ± 0 Reduced Chi-Sqr 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='43142 R-Square (COD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98085 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98085 [Fe-BL] \uf0741 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25(4) ps \uf074\uf032 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='13(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05) ps \uf0743 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39(63) ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 Fe K\uf0611 Fe K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps Model Conv_3exp (User) Equation exp1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau1*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau1^2*ln( 16)))*exp(-(x-t0)/tau1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau2*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau2^2*ln( 16)))*exp(-(x-t0)/tau2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp3=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau3*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau3^2*ln( 16)))*exp(-(x-t0)/tau3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' y=A1*exp1 + A2*exp2 + A3*exp 3 + y0 Plot L A1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='48247 ± 0 tau1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='41756 ± 0 A2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11084 ± 0 tau2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='74261 ± 0 A3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='10739 ± 0 tau3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11547 ± 0 y0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02428 ± 0 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01043 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00329 fwhm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 ± 0 Reduced Chi-Sqr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='70578 R-Square (COD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='99536 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='99536 [Fe-BL-Co] \uf074\uf032 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='42(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='40) ps \uf0743 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='74(18) ps \uf0741 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12(2) ps 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Fe K\uf0611 Fe K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps [Fe-BL-Co] \uf0741 = - \uf0742 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='38(40) ps \uf0743 = - 2 0 2 4 6 8 10 12 14 Fe K\uf0611 Fe K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps [Fe-BL] \uf0741 = - \uf0743 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='71(35) ps \uf0742 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98(27) ps Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fluorescence decay fitting results for: a) [Fe-BL], long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) [Fe-BL], short-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' c) Fe @ [Fe-BL-Co], long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' d) Fe @ [Fe-BL-Co], short- time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) b) c) d) 30 b) A direct and non-direct contribution to Kα XES At the optical pump wavelength of 400 nm both Fe and Co centres were simultaneously excited (although predominantly Fe site) and probed with 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 keV X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Hence, we have carried out rigorous and detailed analysis to distinguish the direct and non-direct contributions originating from the pho- toexcitation at the Co center in the studied dyad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In this case, we have computed the X-ray and optical convoluted cross-section relation Cσ of cobaloxime and Co part of the dyad as follows: 𝐶𝜎~ 𝜎𝑋−𝑅𝑎𝑦 𝑐𝑜𝑏𝑎𝑙𝑜𝑥𝑖𝑚𝑒𝜎𝑈𝑉−𝑉𝐼𝑆 𝑐𝑜𝑏𝑎𝑙𝑜𝑥𝑖𝑚𝑒 (𝜎𝑋−𝑅𝑎𝑦 𝑑𝑦𝑎𝑑 − 𝜎𝑋−𝑅𝑎𝑦 𝑃𝑆 )(𝜎𝑈𝑉−𝑉𝐼𝑆 𝑑𝑦𝑎𝑑 − 𝜎𝑈𝑉−𝑉𝐼𝑆 𝑃𝑆 ) ⁄ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='59 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Co K\uf0611 Co K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps \uf0741 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps (fixed) [Fe-BL-Co] \uf074\uf032 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39) ps \uf0743 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39 ps (fixed) 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps Co K\uf0611 Co K\uf0611 (fitting) Cobaloxime \uf0741 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='76(31) ps \uf074\uf032 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='82) ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps Co K\uf0611 Co K\uf0611 (fitting) Cobaloxime \uf074 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39(17) ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay [ps] Co K\uf0611 Co K\uf0611 (fitting) [Fe-BL-Co] \uf0741 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='07(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='13) ps \uf0742 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25(1) ps Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fluorescence decay fitting results for: a) [Co] @ [Fe-BL-Co], long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) [Co] @ [Fe-BL-Co], short-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' c) cobaloxime, long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' d) cobaloxime, short-time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) b) c) d) 31 This number can be compared to the average ratio between the intensity of cobaloxime kinetic trace and [Co] part of dyad kinetic trace, which was estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Our analysis yields that approxi- mately 59% of Co signal in dyad originates from the different electronic structure around Co site in the dyad, as compared to isolated cobaloxime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The theoretical cross sections were computed using values listed in the NIST database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Optical cross sections were taken from UV-Vis spectra shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 32 c) Nuclear wavepacket analysis The oscillatory signals, previously extracted from the kinetic traces of both [Fe-BL] and [Fe-BL-Co] shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4 of the main text, were analyzed in detail using the Fourier transform analysis and are presented in Fig S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For [Fe-BL], we found one dominating frequency of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='49 THz (the corresponding half-period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='29 ps) and for the [Fe-BL-Co], there are two dominating frequencies of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='71 THz (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='27 ps) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='56 THz (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' These frequencies correspond to vibrational modes of 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 cm-1 (for [Fe-BL]), 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 cm-1 and 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 cm-1 (for [Fe-BL-Co]), respectively, in the range of one degree of freedom for single bond thermal oscillations at 25 °C (kbT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Noteworthy, similar frequencies could be resolved experimentally 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 200 0 200 400 600 800 Amplitude 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='56 THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='71 THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='27 ps Phase / deg Frequency / THz 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 deg 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 deg 100 150 200 250 300 0 10000 20000 30000 40000 Intensity / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Raman shift / cm-1 785 nm laser [Co] [Fe-BL] [Fe-BL-Co] a b c Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fourier transform of the oscillatory parts: a) [Fe-BL];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) Fe part of [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Raman spectra of [Co], [Fe-BL], and [Fe-BL-Co] obtained with 785 nm laser (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='49 THz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='29ps Amplitude 0 Phase / deg 50 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='44 deg 100 150 0 2 4 6 8 10 12 Freguency I THz 33 by means of Raman spectroscopy (see spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Specifically, the difference between 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 cm-1 and 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 cm-1 (~1 meV) could be observed only in terms of the amplitude, while a band around 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 cm-1 should appear only in [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It must be underlined, that in TR-XES such differences would be difficult to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since the phase of Fourier transform can be affected by the presence of a significant noise contribution, we decided to use an additional method of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The nuclear wavepacket motion parameters for [Fe-BL-Co] were refined with the use of the damped oscillatory, function: 𝑓(𝑡) = 𝑦0 + 𝑒 −𝑡 𝑡0 ⁄ (𝑏1𝑠𝑖𝑛 (𝜋 𝑡−𝑡𝑐1 𝑤1 ) + 𝑏2𝑠𝑖𝑛 (𝜋 𝑡−𝑡𝑐2 𝑤2 )) (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1) where: y0 – intensity offset in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' tc1, tc2 – phase shifts in ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' w1, w2 – oscillation period in ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' t0 – damping factor in ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b1, b2 – initial amplitude of oscillations in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The results are presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In case of [Fe-BL], the b2 was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It is worth to mention that for [Fe-BL-Co] the damping factor is much bigger, although the uncertainty of this parameter prevents to derive any quantitative conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Interestingly, the oscillation in [Fe-BL] changes phase upon co- balt coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This may suggest, that after forming the dyad, the longer oscillation is partially quenched and re-induced upon photoexcitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The phase of oscillation in [Fe-BL] is ~ 9°, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' close to sinusoidal, therefore induced by impulse-stimulated Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16–18 On the other hand, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='26 ps component phase in [Fe-BL-Co] is ~51°, thus of mixed sine/cosine character and for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19 ps, the phase with ~89° is of cosine character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The cosine type of oscillation was proven to be a direct marker of the influence of the excited state generation upon photoexcitation and followed by coherent vibrations generation 18,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Thus at least one of the vibrations in [Fe-BL-Co] is related to the charge transfer accompanied by the Fe-ligand bond stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Wavepacket analysis fitting results with equation (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' [Fe-BL] [Fe-BL-Co] b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='064(176) b1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='008(4) tc1 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='046(17) tc1 [ps] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='632(275) w1 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284(22) w1 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='255(30) 𝑦0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='003(4) b2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='015(4) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='402(150) tc2 [ps] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='425(370) w2 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='185(8) 𝑦0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='000(2) t0 [ps] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='632(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='270) 34 Given the large experimental error bars, we wanted to discard the possibility that the additional oscil- lation observed for the dyad could compensate for the high damping value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In other words, we have also verified the scenario, in which the extracted oscillations could be described with a single sine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' These fit results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5a and compared statistically with the double sine fit, previously presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4a, using an F-test (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The F-test is intended to compare two models, where one of them contains less parameters and is nested into more complex one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result of the test indicates statistical significance of the more complex fit, especially when χ2 comparison is not enough, thus preventing us from overfitting the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In our specific case, the single-sine model was described by 5 parameters and was nested into a more complex model described with 8 parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The fitting procedure was carried out with experimental error bars as weights, and in F-value calculation, χ2 values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01338 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00814 were found for single- and double-sine fit functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since all fits were done with p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05, the null hypothesis in this test stated that fitted functions are not different in 95% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For such conditions, the critical F-value to reject null at 95% probability was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6875, while the obtained value was 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Therefore, we can conclude that both models are statistically different and thus confirm the presence of the second oscillation in [Fe- BL-Co] kinetic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='04 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts - fit / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Probe delay / ps 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='49 THz experimental data fit Damping function 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 Density Probe delay / ps F distribution for two models with 5 and 8 parameters F = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0053 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) Single oscillatory fit to wavepacket signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) F distribution (black) for nested 5- parameter model in 8-parameter model along with F-value in this study (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The vertical line indicates critical F-value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6875 above which the null hypothesis can be rejected in the current conditions at p level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a b 35 d) Co Kα1 kinetic signals for -5-15 ps time window The statistics of the Co Kα1 long-time kinetic signal in [Fe-BL-Co] is substantially worse than for pure cobaloxime measurement, owing mainly to the two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' First of all, due to substantially dif- ferent absorption cross sections in the UV-Vis range, upon the photoexcitation of the dyad, the [Fe- BL] part is predominantly excited, while the direct excitation of the Co part is nearly completely avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Secondly, the X-ray beam intensity is distributed over two metal centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Since with the measurement of cobaloxime alone, the aim was to detect excited states formed upon direct photoex- citation, and no [Fe-BL] was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Still, the long kinetic trace for [Fe-BL-Co] substantially differs in shape as compared to pure cobaloxime, especially in the initial 5-6 ps range (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 a, and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Consequently, although a possible long-lived component in Co moiety could not be excluded, it is visible only after the first ~7 ps of the [Fe-BL-Co] Co Kα1 kinetics, where the signal almost reaches the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' It would not affect the presence of \uf0741,FeCo, and short-time kinetic trace, since contributions from the shortest time constants appear at the beginning of the kinetic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Moreover, the step size in this measurement was equal to 1 ps, which corresponds to approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2 data points that represent time constant of 1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The initial fitting results of the kinetic traces for Co Kα fluorescence in [Fe-BL-Co] in 15 ps time window are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' All fit parameters were left as free and as a result we obtained good fit results with 2 time constants of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4 and 17 ps, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 and the corresponding parameters are summarized in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 (column A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Interestingly, the fitting of Co Kα fluorescence to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ps kinetic trace revealed unambiguous presence of another ultrashort contribution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The 1 ps time constant in the 15 ps kinetic fit, was statistically represented by a single point, therefore it could be an artefact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' To verify this, we repeated the fitting of the Co Kα fluorescence to the 15 ps kinetic trace using a fixed time constant of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps and keeping all other fit conditions the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' However, the fit did not converge to any reasonable result, and therefore we extended the fit model with an additional time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' First, we fitted all 3 decays, and the fitting procedure produced a very large time constant and high uncer- tainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This contribution was interpreted as a representation of an electronic state with decay signif- icantly longer than the time window of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Therefore, the large value was fixed, and data were re-fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 b and corresponding parameters are available in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 under column B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The results again exhibit 1 ps time constant (and did not require the shortest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps time constant), which was interpreted as an artefact and a time constant of 7 ps with a very high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For purpose of testing this hypothesis further, we employed a fit procedure with 3 time constants, of which two: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps and infinite were fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 c and 36 corresponding parameters are available in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 under column C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The results concluded that the 1 ps value from the first fitting attempt was indeed an artefact due to too low temporal step size of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Earlier reports suggest that a two-exponential, sequential decay for cobaloxime (one fast and second around 20 ps), and we assumed the same scenario for our longer time constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='20 Therefore, A2, τ2, A3, and τ3 represented a decay of the LMCT state to the ground state through the MC state upon direct photoexcitation, while τ1 and A1 represented M’MCT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This implied, that the corresponding amplitudes of the A2 and A3 in the model from Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 C will be similar, because both concern excited states in the same simple decay pathway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' However, the difference be- tween them is around a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Due to the fact, that the LMCT excitation is still present, we assumed that the related decay pathway will correspond to the one in an isolated cobaloxime complex, especially for the lowest-lying state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A final fitting attempt was conducted on the model described in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 C, with τ3 fixed at the value obtained from cobaloxime, namely 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The result is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3A and Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Notably, the relation between A2 and A3 is almost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The time constant of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 ps is discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The value of τ3 was later re-evaluated with other time constants fixed, and a value of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39(14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='46) ps was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Co Kα fluorescence decay fitting results for [Fe-BL-Co] with different assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' A B C A1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='453(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='535) ·10-4 A1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='101(844) ·10-4 A1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='160(116) ·10-3 τ1 [ps] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='36(25) τ1 [ps] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16(50) τ1 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 (fixed) A2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='914(103) ·10-4 A2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='874(757) ·10-4 A2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='967(114) ·10-4 τ 2 [ps] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='74(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='54) τ 2 [ps] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='67) τ 2 [ps] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='97(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='19) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='56(7) A3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='993(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='267) ·10-5 A3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='393 (117) ·10-5 𝑦0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='041(48) ·10-5 τ 3 [ps] infinite (fixed) τ 3 [ps] infinite (fixed) FWHMa [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 (fixed) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='57(5) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='01(2) 𝑦0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='025(25) ·10-4 𝑦0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='024(25) ·10-4 FWHMa [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 (fixed) FWHMa [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='284 (fixed) 37 e) d-d interactions in Co The d-d interactions can occur when different valence d orbitals in a metal complex are not fully occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In octahedral symmetry, the selection rules forbid d-d transitions to occur and thus they can be observed when the symmetry of the complex is significantly distorted, for example by the Jahn- Teller effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Even though the symmetry of Co(dmgH)2Cl(py) complex is significantly distorted, thus allowing weak d-d transitions to occur, the UV-Vis spectrum of pure cobaloxime does not contain the characteristic d-d bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 1 b) observed for another family of cobaloximes with axial alkyl and amino ligands [Co(dmgH)2(Alkyl)(Base)]21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Therefore, the low-lying acceptor state for LMCT deex- citation is not populated due to the d-d transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In order to assign the time constants obtained via the fitting of TR-XES kinetic traces to possible transitions in our Co complex, the \uf0741,Co was tentatively assigned as LMCT → MC decay, while \uf0742,Co represent the MC relaxation in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' We want to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Co K\uf0611 Co K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps \uf0741 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='36(25) ps [Fe-BL-Co] \uf074\uf032 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='74(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='41) ps 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Co K\uf0611 Co K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps \uf0741 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps (fixed) \uf074\uf032 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='97(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='14) ps \uf0743 = ∞ (fixed) Model Conv_3exp (User) Equation exp1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau1*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau1^2*ln( 16)))*exp(-(x-t0)/tau1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau2*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau2^2*ln( 16)))*exp(-(x-t0)/tau2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' exp3=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5*(1+erf((x-t0)*sqrt(ln(1 6))/fwhm-fwhm/(2*tau3*sqrt(ln(1 6)))))*exp(fwhm^2/(4*tau3^2*ln( 16)))*exp(-(x-t0)/tau3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' y=A1*exp1 + A2*exp2 + A3*exp 3 + y0 Plot D A1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='39296E-5 ± 0 tau1 1000 ± 0 A2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='96616E-4 ± 0 tau2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='97151 ± 0 A3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00208 ± 0 tau3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ± 0 y0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02384E-4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='51058E-6 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='00873 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='02733 fwhm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='2 ± 0 Reduced Chi-Sqr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='07862 R-Square (COD) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98572 Adj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' R-Square 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='98497 [Fe-BL-Co] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Co K\uf0611 Co K\uf0611 (fitting) Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Integr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Counts Probe delay / ps \uf0741 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='16(50) ps \uf074\uf032 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='11(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='67) ps \uf0743 = ∞ (fixed) [Fe-BL-Co] b) c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Co Kα fluorescence decay fitting results for [Fe-BL-Co] 15 ps kinetics: a) with 2 time constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) with 3 time constants and very long time constant fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' c) with 3 time constants and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps and infinite time constants fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) 38 underline, that the MC nature of the second state must be independently confirmed, yet the LMCT state was also confirmed by DFT results, making MC state an obvious candidate as an acceptor for the LMCT decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The proposed state diagram for the relaxation on the Co site of the dyad is also analogical to numerous Fe polypyridyl complexes with long-lived MC states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='22,23 A dedicated Co Kβ TR-XES experiment would further confirm the nature of the assigned states involved in the decay process, which was beyond the scope of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' f) Kinetic model for XES fluorescence kinetic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fe Kα XES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For Fe in [Fe-BL-Co], two decay channels were proposed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5): 1) alpha (α) channel: 1/3MLCT* 𝑘1→ 3MLCT 𝑘2→ 3MC 𝑘3→ gs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2) beta (β) channel: 1/3MLCT* 𝑘4→ 3MC 𝑘3→ gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' They were described by a system of differential equations: 𝑑 𝑀𝐿𝐶𝑇 1 𝑑𝑡 = −𝑘1 𝑀𝐿𝐶𝑇 1 − 𝑘4 𝑀𝐿𝐶𝑇 1 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a) 𝑑 𝑀𝐿𝐶𝑇 3 𝑑𝑡 = 𝑘1 𝑀𝐿𝐶𝑇 1 − 𝑘2 𝑀𝐿𝐶𝑇 3 − 𝑘𝑐𝑡 𝑀𝐿𝐶𝑇 3 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='b) 𝑑 𝑀𝐶 3 𝑑𝑡 = 𝑘4 𝑀𝐿𝐶𝑇 1 + 𝑘2 𝑀𝐿𝐶𝑇 3 − 𝑘3 𝑀𝐶 3 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c) 𝑔𝑠 = 𝑀 − 𝑀𝐿𝐶𝑇 1 − 𝑀𝐿𝐶𝑇 3 − 𝑀𝐶 3 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='d) 𝑀𝐿𝐶𝑇(𝑡 = 0) 1 = 𝑀 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e) 𝑀𝐿𝐶𝑇(𝑡 = 0) 3 = 0 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='f) 𝑀𝐶(𝑡 = 0) 3 = 0 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='g) where: M – initial excited state fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For the modelling purposes, an equal relation between con- centration and signal strength was assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The system of differential equations above was solved numerically in Mathematica 11 software and all solutions were broadened by Heaviside step function under the convoluted with normalized Gaussian function to model rise time of electronic state: 𝑔𝑏𝑟𝑜𝑎𝑑𝑒𝑛𝑒𝑑(𝑡) = 1 2𝜎√2𝜋 ∫ 𝑒− 𝑦2 2𝜎2 ℎ(𝑡 − 𝑡0 − 𝑦)𝑔(𝑡)𝑑𝑦 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3) where: 𝑔(𝑡) – broadened function defined by one of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a – S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ℎ(𝑡 − 𝑡0 − 𝑦) – Heaviside step function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' σ – Gaussian standard deviation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The final fitted function was as follows: 𝑓𝐹𝑒(𝑡) = 𝑦0 + 𝑀𝐿𝐶𝑇 1 + 𝑀𝐿𝐶𝑇 3 + 𝑀𝐶 3 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4) 39 where: 𝑦0 – vertical offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Co Kα XES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' There were two decay paths identified in [Co]: 1) M’MCT 𝑘7→ gs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2) LMCT 𝑘5→ MC 𝑘6→ gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The charge transfer (CT) was treated as instantaneous, therefore it is completed within the IRF func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The M’MCT state was acting as an acceptor of CT from bridging ligand BL, while LMCT state was representing direct optical excitation and was described by an independent k5 rate constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The analogical transition was observed in pure cobaloxime kinetic data with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='76 ps time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The differential formula system with boundary conditions was as follows: 𝑑𝑀′𝑀𝐶𝑇 𝑑𝑡 = −𝑘7𝑀′𝑀𝐶𝑇 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a) 𝑑𝐿𝑀𝐶𝑇 𝑑𝑡 = −𝑘5𝐿𝑀𝐶𝑇 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='b) 𝑑𝑀𝐶 𝑑𝑡 = 𝑘5𝐿𝑀𝐶𝑇 - 𝑘6𝑀𝐶 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='c) 𝑔𝑠𝐶𝑜 = 𝑀′𝑀𝐶𝑇0 + 𝐿𝑀𝐶𝑇0 − 𝑀′𝑀𝐶𝑇 − 𝐿𝑀𝐶𝑇 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='d) 𝑀′𝑀𝐶𝑇(𝑡 = 0) = 𝑀′𝑀𝐶𝑇0 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='e) 𝐿𝑀𝐶𝑇(𝑡 = 0) = 𝐿𝑀𝐶𝑇0 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='f) The final fitting function was: 𝑓𝐶𝑜(𝑡) = 𝑦0 + 𝑀′𝑀𝐶𝑇 + 𝐿𝑀𝐶𝑇 + 𝑀𝐶 (S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6) The 𝑀′𝑀𝐶𝑇 and 𝐿𝑀𝐶𝑇 were also broadened by the function described in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' All fitting results are summarized in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In the first approach, decay constant values obtained from the fluorescence decay formula fitting were fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The amplitudes and offsets were fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' After- ward obtained values were fixed to refine the rate constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Summary for fitting of kinetic equations to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Fe in [Fe-BL] // Fe in [Fe-BL-Co] [Co] in [Fe-BL-Co] dyad // cobaloxime 𝑀 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='736(19) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='608(9) 𝑀′𝑀𝐶𝑇 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='966(160) 𝐿𝑀𝐶𝑇 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='730(64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='925(68) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='034(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0005) t0 [ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='005(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='015) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='018(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='021) 𝑦0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='047(14) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='027(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='007) 𝑦0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='002(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='032) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='004(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='048) k1 / τ1 [ps-1 /ps] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='527(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='495) / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='221(72) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='527 (fixed) / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='221 (fixed) k5 / τ5 [ps-1/ ps] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12 (fixed) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='76 (fixed) 𝑘𝑒𝑡/ τet [ps-1 /ps] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='294(35) / - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='404(405) k6 / τ6 [ps-1 /ps] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='659(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='246) / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='273(93) 40 The k2, k3, k4, and k5 rate constants were calculated from k = 1/τ relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Decay time constants τ were taken from fluorescence fitting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In total, five parameters were fitted to the experimental data: M, y0, t0, k1 and ket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Unlike for the FWHM value used in the fluorescence fitting, the σ parameter here is a standard deviation of Gaussian in the IRF function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The re-evaluated value of σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='106(5) ps corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25(1) ps from the fluorescence fitting, which clearly resembles the FWHM of IRF function (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='28 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The t0 was set as a free parameter to ensure fit convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The k1 decay rate represents a 1MLCT->3MLCT transition, and the inclusion of this decay rate is necessary due to nu- clear wavepacket motion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' According to DFT results, eventual charge transfer should go through BL ligand acceptor state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The CT in the [Fe-BL] part of the dyad was considered to originate from two possible states: the 3MLCT or 𝑀𝐿𝐶𝑇 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In both cases, to model CT, an additional element of –kct 𝑀𝐿𝐶𝑇∗ 3 or –kctMLCT was added into differential equations S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='b and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='a respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In the first case, any fitting attempts were unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For CT from 3MLCT state, a two-step analysis was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' First, for [Fe-BL] a rate constant k1 for 1MLCT→ 3MLCT transition was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For CT from 3MLCT in [Fe-BL-Co] this value was fixed, and the fit was conducted with the free kct parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Results for global kinetic fitting are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5, S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In the case of cobaloxime analysis, a two-state decay model was used to match the fluores- cence decay results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Two scenarios were included, where states either decay in parallel or in a hierar- chical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Only the hierarchical model was reproducing the data with the states diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8) and fitted kinetic traces presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' In early kinetics (-1-1 ps), the signal dynamics can be described by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='76 decay time, while a longer period requires the inclusion of a second, infinite time constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' This model was the basis for [Fe-BL-Co] analysis with 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12 ps time fixed in a short-time window (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Importantly, the dynamics in short- and long-time windows could be reproduced with 2 kinetic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' The results of kinetic equation fitting (Fig 5 c, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='6 b) confirmed the fluorescence kinetic trace analysis for Co in [Fe-BL-Co].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' For the sake of precision, two models were also tested: the M’MCT and LMCT states decaying in a hierarchical and parallel way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Only the model presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 5 a reproduced data with satisfactory quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Due to a very low Co signal intensity in [Fe-BL-Co] the estimated uncertainties are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0006 Population / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=" Delay time / ps M'MCT LMCT MC gs exp fit \uf0741 = 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='25 ps \uf0742 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='12 ps 0,2 0,0 0,2 0,4 0,6 0,8 1,0 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 Population / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' delay time / ps 1MLCT 3MLCT 3MC gs fit exp 2 0 2 4 6 8 10 12 14 0,1 0,2 0,3 0,4 Populations / a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' delay time / ps 1MLCT 3MLCT 3MC gs exp fit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Total kinetic traces for: a) [Fe-BL], short-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) Co @ [Fe-BL-Co], long- time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' c) [Fe-BL], long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' d) Fe @ [Fe-BL-Co], long-time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' a) b) c) d) 0,5 0,0 0,5 1,0 1,5 0,00003 0,00004 0,00005 0,00006 0,00007 Fraction of absorbers Delay time / ps LMCT gs exp 0,0 2,5 5,0 7,5 10,0 0,00005 0,00006 0,00007 0,00008 0,00009 0,00010 0,00011 Fraction of absorbers Delay time / ps LMCT MC gs exp fit Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Populations for short- and long-time window kinetic traces for cobaloxime: a) long-time window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' b) short-time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 0,6 MLCT* MLCT 0,5 MC Fraction of absorbers gs exp 0,4 fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='3 0,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='1 0,0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2,5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 2,5 5,0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='5 10,0 12,5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content='0 Delay time I ps 42 References: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Dreuw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' & Head-Gordon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin and Bacteriochlorophyll-Spheroidene Complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 126, 4007–4016 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Autschbach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Charge-transfer excitations and time-dependent density functional theory: Problems and some proposed solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' ChemPhysChem 10, 1757–1760 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Vlček, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' & Záliš, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Modeling of charge-transfer transitions and excited states in d6 transition metal complexes by DFT techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfRwmH/content/2301.04425v1.pdf'} +page_content=' Chem.' metadata={'source': 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sha256:c77afd40c9fc02ff40b1eaec37a85cf2f989d2bb5398352f3cc08141b19b6968 +size 2097197 diff --git a/A9AzT4oBgHgl3EQf__9t/vector_store/index.faiss b/A9AzT4oBgHgl3EQf__9t/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e7b09f267a25b2db25c4313ae393369161750ee7 --- /dev/null +++ b/A9AzT4oBgHgl3EQf__9t/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b6a4b9cd936edc21b43ddd4098ea8c7fbd2adee9b8840a958fe27fa231563ba6 +size 6553645 diff --git a/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/2301.02647v1.pdf.txt b/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/2301.02647v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f99fec754207cc9d16af6481096fe05b03ed45b7 --- /dev/null +++ b/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/2301.02647v1.pdf.txt @@ -0,0 +1,2056 @@ +Universal adaptive optics for microscopy through +embedded neural network control +Qi Hu1, Martin Hailstone2, Jingyu Wang1, Matthew Wincott1, Danail Stoychev2, Huriye +Atilgan3, Dalia Gala2, Tai Chaiamarit2, Richard M. Parton2, Jacopo Antonello1, Adam M. +Packer3, Ilan Davis2, and Martin J. Booth1,* +1Department of Engineering Science, University of Oxford +2Department of Biochemistry, University of Oxford +3Department of Physiology, Anatomy, and Genetics, University of Oxford +*martin.booth@eng.ox.ac.uk +ABSTRACT +The resolution and contrast of microscope imaging is often affected by aberrations introduced by imperfect optical systems +and inhomogeneous refractive structures in specimens. Adaptive optics (AO) compensates these aberrations and restores +diffraction limited performance. A wide range of AO solutions have been introduced, often tailored to a specific microscope type +or application. Until now, a universal AO solution – one that can be readily transferred between microscope modalities – has +not been deployed. We propose versatile and fast aberration correction using a physics-based machine learning (ML) assisted +wavefront-sensorless AO control method. Unlike previous ML methods, we used a bespoke neural network (NN) architecture, +designed using physical understanding of image formation, that was embedded in the control loop of the microscope. The +approach means that not only is the resulting NN orders of magnitude simpler than previous NN methods, but the concept is +translatable across microscope modalities. We demonstrated the method on a two-photon, a three-photon and a widefield +three-dimensional (3D) structured illumination microscope. Results showed that the method outperformed commonly-used +modal-based sensorless AO methods. We also showed that our ML-based method was robust in a range of challenging +imaging conditions, such as extended 3D sample structures, specimen motion, low signal to noise ratio and activity-induced +fluorescence fluctuations. Moreover, as the bespoke architecture encapsulated physical understanding of the imaging process, +the internal NN configuration was no-longer a “black box”, but provided physical insights on internal workings, which could +influence future designs. +Introduction +The imaging quality of high-resolution optical microscopes is often detrimentally affected by aberrations which result in +compromised scientific information in the images. These aberrations can arise from imperfections in the optical design of the +microscope, but are most commonly due to inhomogeneous refractive index structures within the specimen. Adaptive optics +(AO) has been built into many microscopes, restoring image quality through aberration correction by reconfigurable elements, +such as deformable mirrors (DMs) or liquid crystal spatial light modulators (LC-SLMs).1–6 Applications of AO-enabled +microscopes have ranged from deep tissue imaging in multiphoton microscopy through to the ultra-high resolution required for +optical nanoscopy. This range of applications has led to a wide variety of AO solutions that have invariably been tailored to a +specific microscope modality or application. +There are two main classes AO operation: in one case, a wavefront sensor measures aberrations; in the other case, +aberrations are inferred from images – so called “wavefront sensorless AO”, or “sensorless AO” for short. For operations +with a wavefront sensor, phase aberrations are measured directly by wavefront sensors such as a Shack-Hartmann sensor7,8 or +an interferometer9–11. Such operations are direct and fast but also have intrinsic disadvantages such as requiring a complex +optical design and suffering from non-common path errors. Furthermore, such wavefront sensors often have limitations and +are less versatile. For example, an interferometer requires a coherent source and all such methods suffer from problems due +to out-of-focus light. On the other hand, sensorless AO methods normally function with a simpler optical design and thus +are more easily adaptable for a wide range of imaging applications. However, sensorless AO methods are based on iterative +deductions of phase aberrations and thus tend to be more time consuming; this is coupled with repeated and prolonged sample +exposures, which inevitably lead to photo-damage or motion related errors. +There have been many developments in AO technology, and in particular sensorless AO methods. Conventionally, sensorless +AO operates based on the principle that the optimal image quality corresponds to the best aberration correction12,13. A suitably +defined metric, such as the total signal intensity14–27 or a spatial frequency based sharpness metric28–33, is used to quantify the +arXiv:2301.02647v1 [eess.IV] 6 Jan 2023 + +image quality. Phase is modulated by the AO while this quality metric reading is measured and optimised. There have been +discussions on how the phase should be modulated12,24,34,35 and how the optimisation algorithm should be designed21,36–38. +However, as mentioned before, such “conventional” sensorless AO methods depend on iterative optimisation of a scalar metric, +where all image information is condensed into a single number, and the optimisation process is usually through mode by mode +adjustment. Such methods were thus not the most efficient approach to solving this multi-dimensional optimisation problem and +the effective range of correction was limited. While a higher dimensional metric was considered to extract more information +from images39, the optimisation of such a vector metric was not straightforward. +While the utility of each of these conventional sensorless AO methods has been demonstrated separately, each method had +been defined for a particular microscope type and application. Until now, no such AO solution has been introduced that can be +universally transferred between microscope modalities and applications. +We propose in this article a new approach to sensorless AO that addresses the limitations of previous methods and provides +a route to a universal AO solution that is applicable to any form of microscopy. This solution is constructed around a physics- +based machine learning (ML) framework that incorporates novel neural network (NN) architectures with carefully crafted +training procedures, in addition to data pre-processing that is informed by knowledge of the image formation process of the +microscope. The resulting NN is embedded into the control of the microscope, improving the efficiency and range of sensorless +AO estimation beyond that possible with conventional methods. This approach delivers versatile aberration measurement and +correction that can be adapted to the application, such as the correction of different types of aberration, over an increased range +of aberration size, across different microscope modalities and specimens. +In recent years, machine learning (ML) has been trialled in AO for its great computational capability to extract and +process information. However, many of these approaches required access to point spread functions (PSFs) or experimentally +acquired bead images40–46 ; these requirements limited the translatability of these methods to a wider range of applications. +Reinforcement learning was applied to correct for phase aberrations when imaging non point-like objects47; however, the +method still involved iterative corrections and was not advantageous in terms of its correction efficiency, accuracy and correction +working range compared to conventional sensorless AO algorithms. Untrained neural networks (NN) were used to determine +wavefront phase and were demonstrated on non point-like objects48,49; however, such methods were reported to normally +require a few minutes of network convergence, which limits their potential in live imaging applications. +Our new approach differs considerably from previous ML assisted aberration estimation, as previous methods mostly +employed standard deep NN architectures that used raw images as the input data. Our method builds upon physical knowledge +of the imaging process and is designed around the abilities of the AO to introduce aberration biases, which improve the +information content of the NN input data. This approach means that the resulting NN is orders of magnitude simpler, in terms +of trainable parameters, than previous NN methods (See Table S1 in supplemental document). Furthermore, our method is +readily translatable across microscope modalities. As NN training is carried out on a synthetic data set, adaptation for a different +modality simply requires regeneration of the image data using a new imaging model. The NN architecture and training process +are otherwise similar. +To illustrate the versatility of this concept, we have demonstrated the method on three different types of fluorescence +microscopes with different forms of AO corrector: a two-photon (2-P) microscope using a SLM, a three-photon (3-P) intravital +microscope using a DM, and a widefield three dimensional (3-D) SIM microscope using a DM. In all cases, we showed that the +new method outperformed commonly used conventional sensorless AO methods. The results further showed that the ML-based +method was robust in a range of challenging imaging conditions, such as specimen motion, low signal to noise ratio, and +fluorescence fluctuations. Moreover, as the bespoke architecture encapsulated into its design physical understanding of the +imaging process, there was a link between the weights in the trained NN and physical properties of the imaging process. This +means that the internal NN configuration needs no-longer to be considered as a “black box”, but can be used to provide physical +insights on internal workings and how information about aberrations is encoded into images. +Concept and implementation +The overall MLAO concept is illustrated in Figure 1. The experimental application follows closely the concept of modal +sensorless AO, whereby a sequence of images are taken, each with a different bias aberration applied using the adaptive element. +The set of images are then used as the input to the ML-enabled estimator, which replaces the previous conventional method +of optimisation of an image quality metric. The estimated correction aberration is then applied to the adaptive element. If +necessary, the process can be iterated for refined correction. The significant advantage of the new method is the way in which +the estimator can more efficiently use image information to determine the aberration correction. +The concept has been designed in order to achieve particular capabilities that extend beyond those of conventional sensorless +AO. The new method should ideally achieve more efficient aberration estimation from fewer images, to reduce time and +exposure of measurement. It should operate over a larger range of aberration amplitudes, compared to previous methods. A +particular estimator should be robust to variations between similar microscopes and the concept should be translatable across +2/16 + +(a) +Microscope +Adaptive +element +Initial +Corrected +Image2 +Image1 +Efficient MLAO estimation +Corrected image +Aberrated image +Iterative correction +(b) +Versatile +Network +training +(c) +F += +F +Image2 +Image1 +Aberration +correction +Maximum +pixel +reading +Small +scale +feature +Large +scale +feature +Shape +bespoke CNN +FCL +F -1 +Image pre-processing +Convolution ++ +local +maxpooling +Global +maxpooling +Fully +connected +Pseudo-PSF +Image variations +Synthesised images +Image = Object ∗ PSF + Noise +microscope imaging model +PSF2-P, PSF3-P, PSFWF +Fluorescence +fluctuations +Spatial +sampling +size +Sample +motion +Sample +structure/ +sparsity +Aberration +and +brightness +Detector, +photon and +structured +noise +3-D +structures/ +background +Figure 1. The MLAO concept. (a) Overview of the AO correction process. A minimum of two bias aberrations were +introduced by the adaptive element; corresponding images of the same field were captured. The images were passed to the +MLAO estimator, which determined the Zernike coefficients for correction. The correction speed was limited only by the speed +of image acquisition, not by computation. (b) Training data generation. A range of image variations were included in the +synthetic data set for NN training to cope with variations in real experimental scenarios. The data was a combination of +artificial and real microscope images, chosen to model a wide range of realistic specimen structures. Images were created +through convolution of specimen structures with an appropriate PSF, generated for the specific microscope modality, +incorporating aberrations. (c) Image pre-processing and NN architecture. Images were pre-processed to compute pseudo-PSFs, +which were predominantly independent of specimen structure. F and F −1 represent the forward and inverse Fourier +transform, respectively. A central cropped region of the pseudo-PSF images was used as the inputs to a CNN. The CNN was +designed and trained specifically for aberration determination. The output from the overall network was the correction +coefficients for the Zernike modes. The NN architecture was such that the convolutional layer outputs could be correlated with +spatial scales of the aberration effects on the pseudo-PSFs and hence the imaging process. Hence, the distribution of weights in +the network had physical relevance. +3/16 + +3 +2 +1 +0 +-1 +-2 +-3different microscope types and applications. From a practical perspective, it is also important that training can be performed on +synthetic data, as it would be impractical to obtain the vast data set necessary for training from experimentally obtained images. +An essential step towards efficient use of image data is the image pre-processing before they are presented to the NN. Rather +than taking raw image data as the inputs, the NN receives pre-processed data calculated from pairs of biased images, which we +term a “pseudo-PSF”, as shown in Fig. 1 and explained in the methods section. This pseudo-PSF contains information about +the input aberration and is mostly independent of the unknown specimen structure. By removing the specimen information at +this stage, we can reduce the demands on the subsequent NN, hence vastly simplifying the architecture required to retrieve the +aberration information. +As most of the useful information related to aberrations was contained within the central pixels of the pseudo-PSF, a region +of 32×32 pixels was extracted as the input to the NN. The first section of the NN was a bespoke convolutional layer that was +designed to extract information from the inputs at different spatial scales. The outputs from the convolutional layer were then +provided to a fully connected layer, which was connected to the output layer. Full details of the NN design are provided in the +methods and the supplementary information. This architecture – rather unusually – provided a link between the physical effects +of aberrations on the imaging process and the mechanisms within the NN, specifically through the weights at the output of the +first fully connected layer. +NN training was performed using a diverse set of synthesised training data. These images were calculated using an +appropriate model of the microscope imaging process in the presence of aberrations. Images were synthesised by convolutions +of specimen structures with a PSF, incorporating various likely experimental uncertainties and noise sources. The specimens +consisted of a range of artificial and realistic objects. Full details are provided in the methods. +This versatile concept could accommodate different aberration biasing strategies. Conventional modal sensorless AO +methods typically required a minimum of 2N +1 biased images to estimate N aberration modes21. However, the MLAO method +has the ability to extract more information out of the images, such that aberrations could be estimated with as few as two +images, although more biased images could provide better-conditioned information. In general, we defined methods that used +M differently biased images to estimate N Zernike modes for aberration correction. The input layer of the NN was adjusted +to accommodate the M image inputs for each method. Out of the many possibilities, we chose to illustrate the performance +using two biasing schemes: one using a single bias mode (astigmatism, Noll index50 i = 5) and one using all N modes that +were being corrected. In the first case, we used either two or four images (M = 2 or 4) each with different astigmatism bias +amplitude. We refer to these methods as ast2 MLAO or ast4 MLAO. Astigmatism was chosen as the most effective bias mode +(see supplementary document, section 6). In the second case, biased images were obtained for all modes being estimated +(M = 2N or 4N); this type is referred to in this paper as 2N MLAO or 4N MLAO. For a complete list of the settings for each +demonstration, please refer to Table S2 in the supplemental document. +Results +In order to show its broad application, the MLAO method was demonstrated in three different forms of microscopy: 2-P and 3-P +scanning microscopy and widefield 3-D structured illumination microscopy (SIM). This enabled testing in different applications +to examine its performance coping with different realistic imaging scenarios. +The MLAO methods were compared to two widely used conventional modal based sensorless AO methods (labelled as +2N+1 conv and 3N conv). The 2N+1 conv method used two biased images per modulation mode and an additional zero +biased image to determine phase correction consisting N modes simultaneously. The 3N conv method used three images per +modulation mode (two biased and one unbiased images) and determined the coefficients of the modes sequentially. For both +methods, the bias size was chosen to be ±1 rad for each mode. A suitable metric was selected to quantify the image quality. +For each mode, the coefficients were optimised by maximising the quality metric of the corresponding images using a parabolic +fitting algorithm. When used in 2-P and 3-P demonstrations, the total fluorescence intensity metric was optimised. For the +widefield 3-D SIM microscope, a Fourier based metric was optimised51. For the details of the two conventional methods, please +refer to21,36. +Different functions were defined as optimisation metrics for the conventional AO methods, and also to assist quantifiable +comparisons of image quality improvement for the MLAO methods. These were defined as an intensity based metric yI, a +Fourier based metric yF, and a sharpness metric yS. Details are provided in the methods section. +Two-photon microscopy +A range of method validations were performed on a 2-P microscope that incorporated a SLM as the adaptive correction +element, including imaging bead ensembles and extended specimen structures. The experimental set-up of the 2-P system was +included in Figure S8 (a) in the supplemental document. In order to obtain controlled and quantitative comparisons between +different AO methods, the SLM was used to both introduce and correct aberrations. This enabled statistical analysis of MLAO +4/16 + +performance with known input aberrations. System aberrations were first corrected using a beads sample before carrying out +further experiments. +We performed a statistical analysis to assess how MLAO algorithms (ast2 MLAO and 2N MLAO) performed in various +experimental conditions compared to conventional algorithms (2N+1 conv and 3N conv). Experiments were conducted on +fixed beads samples (Figure 2 (a, b)), and Bovine Pulmonary Artery Endothelial (BPAE) cells (FluoCellsTM Prepared Slide +#1) (Figure 2 (c - f)). Dependent on the experiment, either N = 5 or N = 9 Zernike modes were estimated (see Table S2 in +Supplemental document for details). +Statistical performance analysis +Figure 2 (a) and (b) showed statistical comparisons of the different correction methods. Figure 2 (a) displayed the residual +aberrations gathered from twenty experiments, each consisting of one correction cycle from random initial aberrations including +five Zernike modes. If the remaining aberration is below the pre-correction value, then the method provides effective aberration +correction. A wide shaded area indicated inconsistent and less reliable correction. The results show that when correcting small +aberrations with root mean square (RMS) = 0.63 to 1.19 rad, 2N MLAO performed similarly to 2N+1 conv. Between RMS = +1.19 to 1.92 rad, 2N MLAO corrected more accurately (lower mean aberration) and also more reliably (smaller error range). For +large aberrations above RMS = 2.12 rad, 2N+1 conv completely failed, whereas the MLAO methods still improved aberration +correction. ast2 MLAO had poor performance at small aberrations (RMS = 0.63 to 0.84 rad) but provided reasonable correction +for large aberrations (RMS = 1.92 to 2.12 rad). However, it is important to note that ast2 MLAO required only two images for +each correction cycle, far fewer that the ten and eleven images required respectively for 2N MLAO and 2N+1 conv. +Figure 2 (b) displayed the mean value of metric yI from ten experiments against the number of images acquired during +multiple iterations of the different correction methods. The corrected aberrations consisted of nine Zernike modes. It was shown +that ast2 MLAO corrects the fastest initially when the input aberration is large but converges to a moderate signal level, which +indicates only partial correction of the aberration. 2N MLAO corrects more quickly and to a higher level than the conventional +algorithms. The narrower error bars for both MLAO algorithms at the end of the correction process indicate that they are more +reliable than the two conventional methods. +Correction on extended specimen structures +Figure 2 (c)-(f) showed experimental results when imaging microtubules of BPAE cells. Specimen regions were chosen to +illustrate performance on different structures: (c) contained mainly aligned fine fibrous structures; (d) contained some large +scale structures (bottom right); (e) contained fine and sparse features. For (f) we intentionally reduced illumination laser power +and increased detector gain to simulate an imaging scenario with very low signal to noise ratio (SNR). The images showed +structured noise at the background, which could pose a challenge to estimation performance. A large randomly generated +aberration (RMS = 2.12 to 2.23 rad) consisting of five (c and f) or nine (d and e) Zernike modes was used as the input aberration. +In (c), (d) and (e), ast2 MLAO corrected the fastest initially when the aberration was large but converged to a moderate level +of correction. 2N MLAO corrected faster in general than the conventional methods and converged to a higher level of correction. +In (f) when SNR was poor and structured noise was present, ast2 MLAO failed to correct while 2N MLAO continued to perform +consistently. +Three-photon intravital microscopy +Three-photon microscopy of neural tissue imaging is a particular challenge for sensorless AO, due to the inherently low +fluorescence signal levels. While this could be alleviated by averaging over time, problems are created due to specimen motion. +Further challenges are posed for functional imaging, due to the time dependence of emission from ion or voltage sensitive dyes. +The demonstrations here show the robustness of the new MLAO methods in experimental scenarios where the conventional +methods were not effective. Importantly, the MLAO methods were able to perform effective correction based on a small number +of image frames without averaging. +The experimental set-up of the 3-P system is shown in Figure S8 (b) in the supplemental document. The microscope used +an electromagnetic DM for aberration biasing and correction. Two MLAO methods, ast4 MLAO and 4N MLAO, were used +to correct aberrations by using single frame images as inputs. In each case, more input frames were chosen than in the 2-P +demonstrations, in order to cope with the lower SNR. The NNs were trained to estimate N = 7 Zernike modes. Two types of +mice were used to perform live brain imaging of green fluorescent protein (GFP) labelled cells (Figure 3 (a)) and functional +imaging in GCaMP-expressing neurons (Figure 3 (b)). In Figure 3 (a), results were collected at 660µm depth and power at +sample was 32 mW. In Figure 3 (b), imaging was at 250µm depth and power at sample was 19 mW. Further 3-P results were +included in the section 8 of supplemental document. For the details of the sample preparation, please refer to section 9B in +supplemental document. +Figure 3 (a) shows plots of the metrics yI and yF as proxies for correction quality when imaging GFP labelled cells. Both +ast4 MLAO and 4N MLAO networks successfully improved the imaging quality. Similar to the ast2 MLAO results in the 2-P +5/16 + +A +B C +D +A +5μm +B +C +D +No. of images +A +B +E +D +F +C +A +5μm +B +C +D +E +F +B +D +C +E +F +No. of images +ast2 MLAO +2N MLAO +A +5μm +B +C +D +E +F +A +B +C +E +F +D +A +5μm +B +C +D +E +F +(c) +(d) +(f) +No. of images +No. of images +A +2.1 rad +1.6 rad +1.9 rad +1.2 rad +0.8 rad +0.6 rad +(a) +(b) +A +B +C +E +F +D +D +E +F +C +B +A +5μm +No. of images +yI +A +yI +yI +yI +yI +(e) +N = 5 +N = 9 +N = 5 +N = 9 +N = 9 +N = 5 +0.6 0.8 +1.2 +1.6 +1.9 2.1 +Applied aberration RMS (rad) +1 +2 +3 +4 +Remaining aberration RMS (rad) +Pre correction +2N+1 conv +ast2 MLAO +2N MLAO +5μm +0.0rad +Fine structure +Coarse structure +Sparse structure +Low SNR with +strructured noise +Figure 2. Comparative performance of MLAO methods in a 2-P microscope. (a) Residual aberration after one correction +cycle for three methods. Points show the mean and the shaded area indicates the standard deviations (SDs) of aberration +distributions. The images show an example field of view (FOV) when different amounts of a random aberration were +introduced. (b)-(f) show the intensity metric (yI) as a proxy for correction quality, against the number of images used for +multiple iterations of correction. Random aberrations consisting of N Zernike modes, as shown in the figure, were introduced +and corrected. In (b), an ensemble of ten random aberrations were corrected, imaging over the same FOV. Error bars on the plot +showed the SD of the fluorescence intensity before and after correction. (c)-(f) show specific corrections imaging microtubules +of BPAE cells, illustrating performance for different specimen structures and imaging conditions. The images were acquired +before and after correction through the different methods (as marked on the metric plots). Insets on the images show residual +wavefronts after correction for each image. The grayscale colorbars show phase in radians. +6/16 + +3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures2元 +0 +-22 +T3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures4.5元 +0 +-4.5元3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures1.5元 +0 +-1.5元3N conv +2N+1 conv +Fluorescence intensity +ast network +2N network +0 +10 +20 +30 +number of sample exposures6 +4 +2 +0 +-2 +-4 +-66 +4 +2 +0 +-2 +-4 +-66 +4 +2 +0 +-2 +-4 +-66 +4 +2 +0 +-2 +-4 +-66 +4 +2 +0 +-2 +-4 +-66 +2 +0 +-2 +-4 +-62元 +0 +-22 +T2元 +0 +-22 +T(b) GCaMP at 250 μm +(a) static GFP at 660 μm +i +iv +4N MLAO it:1 +20μm +ii +Pre MLAO +iii +ast4 MLAO it:5 +yI +yF +0 +4 +8 12 16 20 +28 +ast4 MLAO +4N MLAO +v +0 +4 +8 12 16 20 +28 +No. of images +ast4 +it:5 +4N +it:1 +vi +Post MLAO it:5 +iii +1 +1 +Post MLAO +it:5 +Pre +MLAO +0 +50 +100 +time / s +4 +3 +2 +it:1 +Pre MLAO +ii +20μm +A +B +C +D +E +F +G +H +ast4 MLAO +bias mode i = 5 +1 +3 +2 +4 +-1 +rad +-0.5 +rad ++1 +rad ++0.5 +rad +1 +3 +2 +4 +ast4 MLAO +4N MLAO +±0.5,±1 rad +bias mode i = 5 +6 +7 +8 +9 +10 11 ++0.5 +rad ++1 +rad +-0.5 +rad +-1 +rad +bias mode i= +i +0 +4 +8 +12 +16 +20 +yI +ast4 MLAO +iv +0 +4 +8 +12 +16 +20 +No. of images +yS +v +it:1 +it:2 +it:3 +it:4 +it:5 +A +B +0 +50 +100 +time / s +C +D +0 +50 +100 +time / s +E +F +0 +50 +100 +time / s +G +H +0 +50 +100 +time / s +vi +Figure 3. Aberration correction in three-photon microscopy of live mouse brains: (a) GFP-labelled cells at depth 660µm and +(b) functional activity of GCaMP-labelled cells at 250µm. Wavefronts inserted to the figures showed the phase modulations +applied by the DM at the relevant step; the common scale is indicated by the colorbar next to (a) and (b) ii. +(a) i shows example single-frame images used in correction with the corresponding bias modes as insets. 1-4 were the image +inputs to ast4 MLAO. For 4N MLAO, six more bias modes and thus 24 more images were also used in each iteration. (a) ii-iv +show images averaged from 20 frames after motion correction. The rectangular boxes highlight regions of interest for +comparison. (a) v and vi show the intensity metric (yI) and the Fourier metric (yF), respectively, calculated from single image +frames, against the number of images acquired for five iterations ast4 MLAO one iteration of 4N MLAO. +(b) i 1-4 shows example single-frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes as +insets. White squares highlight two cells for comparison to show the fluorescence fluctuations over time neural activity. (b) ii +and iii show respectively before and after ast4 MLAO correction through five iterations (it:1 to 5), 200 frame averages after +motion correction. In iii, time traces shown to the left were taken from the marked line (1). (b) iv and v show the intensity +metric (yI) and the sharpness metric (yS), respectively, calculated from single image frames, against the number of images +acquired for five iterations ast4 MLAO. (b) vi shows the calcium activity of 8 cells (A-H marked on ii). +7/16 + +T +1 +0 +2 +-T +1 +22 +1 +0 +-1 +-2 +-30 +50 +100 +time / s2 +1 +0 +-1 +-2 +-32 +1 +0 +-1 +-2 +-32 +1 +0 +-1 +-2 +-32 +1 +0 +-1 +-2 +-3T +1 +0 +2 +-T +1 +2demonstrations, ast4 MLAO corrected more quickly at first, but converged to a lower correction level. In contrast, 4N MLAO +preformed better overall correction, but required more images. Panels ii-iv show averaged images in which processes previously +hidden below the noise level are revealed through MLAO correction (as highlighted in the white rectangles). The example +biased images shown in Figure 3 (a) i provide an indication of the low raw-data SNR that the MLAO method can successfully +use. +Figure 3 (b) shows results from imaging calcium activity in a live mouse. The ast4 MLAO method successfully improved +image quality despite the low SNR and fluorescence fluctuations of the sample. From both time traces of line 1 and cells A-H, it +could be clearly seen that after corrections, signals were increase. The 4N MLAO method failed to correct in this experimental +scenario (results not shown). We will discuss the likely hypotheses for this in the discussion section. +The fluctuating fluorescence levels due to neural activity mean that conventional metrics would not be effective in sensorless +AO optimisation processes. This is illustrated in Figure 3 (b) iv and v, where it can be seen that no single metric can accurately +reflect the image quality during the process of ast4 MLAO correction. These observations illustrate the advantages of MLAO +methods, as their optimisation process did not rely on any single scalar metric. +Widefield 3-D structured illumination microscopy +The architecture of the NN was conceived so that it would be translatable to different forms of microscopy. In order to illustrate +this versatility, and to complement to the previously shown 2-P and 3-P laser scanning systems, we applied MLAO to a widefield +method. The 3D SIM microscope included multiple lasers and fluorescence detection channels and an electromagnetic DM as +the correction element. Structured illumination patterns were introduced using a focal plane SLM. The detailed experimental +set-up was included in Figure S6 (c) in the supplemental document. +Without AO, 3D SIM reconstruction suffers artefacts caused by aberrations. Since typical specimens contain 3D structures, +the lack of optical sectioning in widefield imaging means that the aberration correction process can be affected by out of focus +light. As total intensity metrics are not suitable for conventional AO algorithms in widefield imaging, Fourier based sharpness +metrics have often been used. However, such metrics depend on the frequency components of the specimen structure39. In +particular, emission from out of focus planes can also affect the sensitivity and accuracy of correction. However, the NN based +MLAO methods were designed and trained to mitigate against the effects of the sample structures and out of focus light. +Figure 4 shows results from two NN-based methods ast2 MLAO and 2N MLAO compared to the conventional algorithm 3N +conv, which used the yS metric. Sensorless AO was implemented using widefield images as the input (Figure 4 (a, b)). The +correction settings thus obtained by the 2N MLAO method were then applied to super-resolution 3D SIM operation (Figure 4 (c, +d)). N = 8 Zernike modes were involved in the aberration determination. The specimen was a multiple labelled Drosophila +larval neuromuscular junction (NMJ). For the details of the sample preparation, please refer to section 7B in supplemental +document. +Figure 4 (b) showed that ast2 MLAO corrected most quickly; 2N MLAO corrected to a similar level but required more +sample exposures; 3N MLAO was less effective. Figure 4 (a) showed the effectiveness of correction on raw and deconvolved +widefield images. Part iii showed the changes in image spectrum after correction. The dashed line shows a threshold where +signal falls below the noise level. It can be seen that both (C) ast2 MLAO and (D) 2N MLAO increased high frequency content +compared to (A) before AO correction and (B) after 3N conv corrections. Figure 4 (c) and (d) showed the images after 3D +SIM reconstruction. It can be clearly seen that when by-passing AO (i), there was strong artefacts due to aberrations. After +correcting using five iterations of 2N MLAO, artefacts were suppressed and z-resolution was improved (see sections through +line 1 and 2 in Figure 4 (d)) +Discussion +The power and simplicity of the MLAO method arise mainly from a combination of three aspects: the pre-processing of image +data, the bespoke NN architecture, and the definition of the training data set. All of these aspects are informed by physical +and mathematical principles of image formation. This forms a contrast with many other data-driven deep learning approaches, +where complex NNs are trained using vast amount of acquired data. +The calculation of the pseudo-PSF from pair of biased images (as shown in Figure 1 (c) and elaborated in the Methods) +acts to remove most of the effects of unknown specimen structure from the input data. The information contained within the +pseudo-PSF encodes indirectly how aberrations affect the imaging PSF (see Figure S2 in the supplemental document for more +details). There is a spatial correspondence between a pixel in the pseudo-PSF and the PSF itself. Hence, spatial correlations +across the pseudo-PSF relate to spatial effects of aberrations on the images. +The set of pseudo-PSFs forms the input to the convolutional layers of the NN. The masks in each convolutional layer probe, +in effect, different scales across the pseudo-PSF. Hence, one can attribute a correspondence between the output of these layers +and the effects aberrations have over different physical scales in the image. Such phenomena are heuristically demonstrated in +8/16 + +5μm +1 +2 +1 +2 +2 +2 +1 +1 +z +z=6μm +z +z +(d) +i +ii +A +B +D +C +5μm +(b) +(c) +i +ii +Pre +Post-MLAO +Pre +Post-MLAO +A i +B i +C i +D i +Widefield Deconvolution +Image +spectrum +10μm +ii +iii +ii +iii +ii +iii +ii +iii +(a) +a.u. +yS +Figure 4. Aberration correction in a widefield structured illumination microscope. (a) Widefield images acquired A i before +and B-D i after correction through different methods (as marked on the metric plot (b)). The second column ii shows +corresponding deconvolved widefield images. The third column iii shows corresponding image spectra; dashed lines show the +threshold where signal falls below the noise level. +(b) The sharpness metric yS against the number of images, for two iterations of 3N conv, ten iterations of ast4 MLAO and three +iterations of 2N MLAO. +(c, d) 3-D projections of 3-D reconstructed SIM image stack of (c) 10µm and (d) 6µm when (i) by-passing AO and (ii) after +five iterations of 2N MLAO correction; square inserts show zoomed in region for comparison. x-y and y-z sections are shown +through lines 1 and 2. +Insets to (a,c and d) show wavefronts corrected by the DM for each image acquisition; phase is shown on the adjacent scale bar. +9/16 + +3N conv +-ast2 MLAO +2N MLAO +2 +14 +24 +32 +48 +no. of imagesT +0 +-T2元 +0 +-2 +T2元 +0 +-2 +T102 +-2 +10 +一Layer +1 +2 +3 +4 +5 +astX MLAO +0.23 +0.19 +0.17 +0.18 +0.23 +XN MLAO +0.39 +0.14 +0.15 +0.13 +0.20 +Table 1. The RMS of the weight distributions extracted from different convolutional layers of the two classes of trained CNNs, +astX MLAO and XN MLAO. The values shown are calculated from the ensemble of corresponding layers from all CNNs of the +given class. +section 3 of the supplementary information. By extracting relevant weight connections from inside the NN, we can observe +embedded physical interpretations of how the machine learned to process aberration information contained in images. +To illustrate this, we extracted from the trained NN the weights between the layer embedding physical interpretations and +the next fully connected layer (marked by red arrows in Figure 1 (c)). Going down the convolutional layers, the scale of probed +features increases from a single pixel, through small scale features, up to large scale features (as explained in section 3 of the +supplemental document). The RMS values of the weights from each convolutional layer are shown in Table 1, where the data +are shown for the ensembles of the two classes of MLAO networks used in this paper, astX MLAO and XN MLAO (where X =2 +or 4). A full breakdown is provided in the Figure S4 of the supplementary document. +The largest weight variation was in the first layer in the XN MLAO NN, which indicates that this algorithm extracts more +information from the single pixel detail than from larger scale correlations. In contrast, astX MLAO assigns weights more +evenly across all layers. As explained in the supplementary document, the single pixel extraction from the pseudo-PSF is +related to the Strehl ratio of the PSF and the intensity information of the images in non-linear systems. Hence, it is expected +that the XN MLAO NN, which uses as similar set of bias aberrations to the conventional method, would learn as part of its +operation similar behaviour to the conventional algorithm. The same phenomena can also explain why in 3-P GCaMP imaging +of neural activity astX MLAO was less affected by the fluorescence fluctuations than XN MLAO, as astX MLAO relies less on +overall fluorescence intensity changes. Conversely, astX MLAO generally performed worse than XN MLAO in 2-P imaging +when structured noise present, as astX MLAO used fewer images and hence had access to less detectable intensity variations +than XN MLAO. The fact that astX MLAO had access to less well-conditioned image information may also explain why in +general it was able to correct aberrations to a lower final level than XN MLAO. +Conclusion +The MLAO methods achieved the aims explained at the outset. They provided more efficient aberration correction with fewer +images over a larger range, reducing time required and specimen exposure. The training procedure, which was based on +synthesised data, ensured that the AO correction was robust to uncertainty in microscope properties, the presence of noise, and +variations in specimen structure. The concept was translatable across different microscope modalities, simply requiring training +using a revised imaging model. +The new methods used NN architectures that are orders of magnitude simpler, in terms of trainable parameters, than in +previous similar work (see supplementary information, section 5). This vast simplification was achieved through pre-processing +of data to remove most of the effects of unknown specimen structure. The physics-informed design of the NN also meant that – +unusually for most NN applications – the learned weights inside the network provided indications of the physical information +used by the network. This provides constructive feedback that can inform future AO system designs and the basis for extension +of the MLAO concept to more demanding tasks in microscopy and other imaging applications. +Methods +Image pre-processing +Image data were pre-processed before being used by the NN, in order to remove effects of the unknown specimen structure. The +resulting “pseudo-PSFs” were better conditioned for the extraction of aberration information, independently of the specimen. +The image formation can be modelled as a convolution between specimen fluorescence distribution and an intensity PSF. The +AO introduced pre-chosen bias aberrations, so that multiple images with different PSFs could be acquired over the same FOV. +Mathematically, this process can be expressed as +I1 = O∗ f1 +δ1 +I2 = O∗ f2 +δ2 +(1) +where I1 and I2 were the images acquired with two different PSFs f1 and f2 for the same unknown specimen structure O. δ1 +and δ2 represent combined background and noise in each image. In order to remove (or at least reduce) the effects of specimen +10/16 + +structures, we defined the pseudo-PSF as +pseudo-PSF = F −1 +�F(I1) +F(I2) +� += F −1 +�F(O∗ f1 +δ1) +F(O∗ f2 +δ2) +� += F −1 +�F(O)×F( f1)+F(δ1) +F(O)×F( f2)+F(δ2) +� +where F was the 2D Fourier transform and F −1 was its inverse (see Figure 1 (c)). The term “pseudo-PSF” was chosen as the +function was defined in the same variable space as a PSF, although it is not used directly in any imaging process. A similar +computational process was shown elsewhere for different applications using defocussed images52. Assuming the noise is small +enough to be neglected +pseudo-PSF = F −1 +�F(I1) +F(I2) +� +≈ F −1 +�F( f1) +F( f2) +� +(2) +There is an implicit assumption here that there are no zeroes in the object spectrum F(O) or the optical transfer function F(f2). +In practice, it was found that a small non-zero value of F(δ2) mitigated against any problems caused by this. Furthermore, +although structured noise was present in the pseudo-PSFs (see e.g. Figure S1 in the supplemental document), it was found that +this did not detrimentally affect data extraction through the subsequent NN. As a further mitigation, we calculated pairs of +pseudo-PSFs from pairs of biased input images by swapping the order from ( f1, f2) for the first pseudo-PSF to ( f2, f1) for the +second. +Example pseudo-PSFs are shown in Figure S1 and S2 in the Supplemental document. As most information was contained +within the central region, to ensure more efficient computation, we cropped the central region (32×32 pixels) of the pseudo- +PSFs to be used as the input to the NN. Dependent upon the MLAO algorithm, the input to the NN would consist of a single pair +of cropped pseudo-PSFs, or multiple pairs corresponding to the multiple pairs of bias aberrations applied in different modes. +Neural network training +To estimate phase aberrations from pseudo-PSFs, a convolutional based neural network was designed incorporating physical +understanding of the imaging process and was trained through supervised learning. Synthetic data were used for training and +the trained networks were then tested on real AO microscopes. For each imaging modality (i.e. 2-P, 3-P and widefield), a +separate training dataset was generated, with the imaging model and parameters adjusted for different applications. +Neural network architecture +A convolutional neural network was designed to determine the aberrations from pseudo-PSFs, while embedding physical +understanding of image formation. The conceptual structure is shown in Figure 1 (c); more specific details of the architecture +and learning process are provided in Section S1 of the supplementary document. This CNN architecture allowed convolutional +masks to – in effect – probe different spatial scales within the pseudo-PSF images and, hence, to learn from the effects +aberrations had at different spatial scales in microscope images. The outputs from these convolutional layers acted as inputs to +a single concatenated fully connected layer (FCL). This was followed by another FCL then the output layer, whose outputs +corresponded to the Zernike mode coefficients estimated for aberration correction. This shallow architecture with the order of +104 trainable parameters was effective due to the pre-processing of data that meant the input information was better conditioned +to this estimation task than raw images. +The weight connections between the concatenated FCL immediately following the CNN layer and the subsequent FCL +(marked in red arrows in Figure 1 (c)) depended upon the significance of the information learnt from the different scales +embedded in the CNN layers. Analysis of these weights could therefore provide insight into the pseudo-PSF information that +was used by the ML process. +Synthetic data generation +Due to the impracticality of acquiring sufficient high-quality data experimentally, a large dataset of simulated image data was +constructed. The simulations were designed to resemble images collected from different microscopes when imaging a range of +samples. +We started with a collection of image stacks (containing around a total of 350 images) obtained from high-resolution 3D +microscopy of various specimens labelled with nuclear, cytoplasmic membrane and/or single-molecule markers. The images +were down-sampled to 8-bit (128×128) and separated into their individual channels. This formed a pool of realistic sample +structures which were later used to generate synthetic images. To further augment the varieties of sample structures, random +rotations were applied and synthetic shapes including dots, rings, circular shapes, curved and straight lines of varying sizes +were randomly introduced. +11/16 + +The simulated training dataset was generated by convolving the sample structures with synthetic PSFs, f (see Eq. 1). f was +modelled as a pixel array through +f = +���F +� +Pe j(Ψ+Φ+Ξ)���� +l +(3) +where F represented the 2D discrete Fourier transform. P was the circular pupil function, defined such that pixels in the region +outside the pupil had value zero. The ratio between the radius of the pupil in pixels and the size in pixels of the overall array +was adjusted to match sampling rates for different microscopes. In practical scanning optical microscopes, the sampling rates +can be easily adjusted, although perhaps not arbitrarily. Hence, for experimental flexibility, the ratio for the simulated training +dataset was tuned to be within the range of 1.0× to 1.2× the base sampling rate. The base sampling rate was defined as using +two pixels to sample the full width half maximum (FWHM) of the PSF of the system when aberration free. For the widefield +system, the ratio was tuned to simulate the projection of the camera pixel sampling rate at the specimen. Figure S5 in the +supplemental document shows how tolerable a trained network was when tested on data collected at different pixel sampling. +P also incorporated the illumination profile for different practical imaging systems, such as when using truncated Gaussian +illumination at the pupil in the 3-P microscope. The exponent l varied with imaging modes: when simulating a 3-P, a 2-P and a +widefield microscope, l was set to 6, 4 and 2 respectively. +The total aberration was expressed as a sum of chosen Zernike polynomial modes Ψ+Φ+Ξ = ∑i aiZi. Ψ was the sum +of the randomly generated specimen aberrations, which included all modes that the AO system was designed to correct. Φ +represented the additional bias aberrations. Ξ included additional non-correctable higher order Zernike modes. The coefficients +of the correctable modes were randomly generated for each data set. Representing the set of coefficients {ai} as a vector a, the +random coefficients followed a modified uniform n-sphere distribution53 where both the direction and the two-norm of a were +uniformly distributed. The maximum two-norm (size) of a were chosen differently for different imaging applications. This +distribution allowed a denser population close to zero aberration, which was intuitively beneficial to train a stable NN. We +also added random small errors to the correctable coefficients so that the labels were slightly inaccurate. This was to simulate +situations when the AO would be incapable of introducing perfect Zernike modes. The spurious high order non-correctable +Zernike modes were included to further resemble realistic scenarios in a practical microscope. +Poisson, Gaussian, pink and structured noise of varying noise level were also introduced to the generated images after the +convolution to allow the training dataset to simulate more closely real microscope images. +Note that the scalar Fourier approximation of Eq. 3 was chosen for simplicity, although more accurate, vectorial, high +numerical aperture (NA) objective lens models could have been applied54–57. Although the chosen model would deviate from +high NA and vectorial effects, the main phenomena under consideration here – namely the effects of phase aberrations on PSFs +and images – are adequately modelled by scalar theory. +Image quality metrics +Different image quality metrics were defined for use as the basis for optimisation in conventional sensorless AO methods and as +proxies to quantify the level of aberration correction. yI is an intensity based metric and can be used in non-linear imaging +systems. It is defined as +yI = +� � +I(x)d2x +yF is a Fourier based metric and provides an alternative aspect to the intensity metric. It is defined as +yF = +� � +0.1fmax<| f|<0.6fmax +|F[I(x)]|d2 f +where F[I(x)] is the 2D Fourier transform of image I(x) from x domain to f domain; fmax is the maximum frequency limit of +the imaging system. The range 0.1fmax < |f| < 0.6 fmax was selected such that most PSF related frequency information was +included in the range. +yS is a sharpness metric that can be used for optimisation in widefield systems, where the other metrics are not practical, or +applications with fluorescence fluctuations. It is defined as +yS = +� � +nfmax<| f| m > n > 0. This metric is defined as the ratio of higher to lower spatial frequency content, which is dependent upon +aberration content, but independent of changes in overall brightness. +12/16 + +Microscope implementations +Three microscopes were used to demonstrate and examine the MLAO method. The microscope implementations are briefly +described here and fully elaborated in the supplementary document section 9A. +In the home built 2-P system, a Newport-Spectra-Physics DeepSee femtosecond laser was used as the illumination with +wavelength set at 850nm. Light was modulated by a Hamamatsu spatial light modulator before passing through a water +immersion objective lens with NA equals to 1.15 and reaching the sample plane. +A commercial Scientifica microscope system was used as the basis for our 3-P demonstration. In the 3-P system, a +Ti:Sapphire laser passed through a pair of compressors and operated at 1300nm. Light was modulated by a Mirao 52E +deformable mirror before reaching a water dipping objective lens with NA equals to 0.8. +In the home built widefield 3D SIM system, two continuous wave lasers with wavelengths equal to 488 and 561nm were +used as the illumination. Light was modulated by a ALPAO 69 deformable mirror before reaching a water dipping objective +lens with NA of 1.1. +Image acquisition and processing +For 3-P imaging of live specimens, where motion was present, averaging was performed after inter-frame motion correction +using TurboReg58. Time traces were taken from 200 raw frames captured at 4 Hz consecutively for each of the pre- and +post-MLAO corrections. +For the widefield/SIM results, widefield images were processed where indicated using the Fiji iterative deconvolution 3-D +plugin59. A PSF for deconvolution was first generated using the Fiji plugin Diffraction PSF 3-D with settings the same as the +widefield microscope. For the deconvolution, the following settings were applied: Wiener filter gamma equals to 0; both x-y +and z direction low pass filter pixels equal to 1; maximum number of iterations equals to 100; and the iteration terminates when +mean delta is smaller than 0.01%. +The thresholds shown on the widefield image spectra were calculated by identifying the largest frequency in all x-y +directions with image spectrum components higher than noise level. The noise level was identified by averaging the components +of the highest spectral frequency, i.e. at the four corners of the image spectrum. 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J. 94, 4957–4970, DOI: https://doi.org/10.1529/biophysj.107.120345 (2008). +Acknowledgements +This work was supported by grants from the European Research Council (to MJB: AdOMiS, No. 695140, to AMP: No. 852765), +Wellcome Trust (to MJB: 203285/C/16/Z, to ID and MJB: 107457/Z/15/Z, to AMP: 204651/Z/16/Z, to HA: 222807/Z/21/Z), +Engineering and Physical Sciences Research Council (to MJB: EP/W024047/1), +Author contributions +QH and MJB conceived the overall physics-informed approach including data pre-processing and bespoke NN architecture. +MH, QH and MJB developed NN architectures and the training approach. QH, MH, MW, JA and DS developed the software +packages. JW, QH, AMP set up the microscopes for the experimental demonstrations. QH performed the two-photon +experiments, supervised by MJB. HA, JW and QH performed the three-photon experiments, supervised by AMP and MJB. JW, +MW, DS, QH and RMP performed the widefield/SIM experiments, for which DG, TC and RMP prepared specimens, supervised +by ID and MJB. QH performed data analysis. QH and MJB wrote the manuscript. All authors reviewed the manuscript. +Additional information +All experimental procedures involving animals were conducted in accordance with the UK animals in Scientific Procedures Act +(1986). +16/16 + +Universal adaptive optics for +microscopy through embedded neural +network control: supplemental +document +1. MLAO PROCESS AND CNN ARCHITECTURE +The MLAO aberration estimation process consists of two parts: image pre-processing to compute +pseudo-PSFs from images and a CNN-based machine learning process for mode coefficient +determination. A stack of M images over the same field of view, each with a different pre- +determined bias phase modulation, was used to calculate pseudo-PSFs according to the procedure +in the methods section. It was observed and understood that most of the information was +contained within the central region of the calculated pseudo-PSFs. 1 A central patch of 32 × 32 +pixels was then cropped and used as the inputs to the CNN. Cropped pseudo-PSFs were processed +by a sequence of convolutional layers (CL) with trainable 3 × 3 kernels, each followed by a local +2 × 2 max-pooling and thus the x and y sizes were reduced by half but the stack size was increased +twice going down each CL. For the input pseudo-PSFs and each of the CL outputs, a global +max-pooling was applied and concatenated into a fully connected layer (FCL). This concatenated +FCL was connected to the next FCL containing 32 neurons, which in turn was connected to the +output layer, which produced the coefficients of the N chosen Zernike modes. The activation +functions were chosen to be tanh and linear (only for the last layer connection FCL 32 and the +output). The regularizer used was L1L2, the initializer was glorot-uniform and the optimizer +was AdamW. The CNN architecture was built and the network training was conducted using +TensorFlow.[1] As elaborated in the results section of the manuscript, M and N may be varied to +suit different applications. +The weights in the connection between the concatenated FCL and FCL32 (enclosed by a grey +dashed square) were extracted and analysed to understand the physical significance of structures +in the pseudo-PSFs in influencing the learning of the CNN. Further analysis of such weights is +provided in Discussion of the main paper and section 4. +1The process of calculating pseudo-PSFs can be interpreted as a deconvolution between two PSFs. Depending on the +sampling size of the imaging system, most details of a deformed PSF typically occupy a central region of a few pixels. Most +features of the pseudo-PSFs were thus captured within the central region. +arXiv:2301.02647v1 [eess.IV] 6 Jan 2023 + +Image stack over the +same field of view +(128×128×M) +Cropped +pseudo-PSFs +(32×32×M) +CL +16×16×8 +CL +8×8×16 +CL +4×4×32 +CL +2×2×64 +M +8 +16 +32 +64 +FCL +32 +Concatenate +FCL +Output +N +Pseudo-PSF computation +Convolution ++ +local maxpooling +Global maxpooling +Fully connected layers +Calculated +pseudo-PSFs +(128×128×M) +CNN +Fig. S1. A schematic illustration of the MLAO process and CNN architecture (enclosed by a +black dashed square) designed for phase determination applications. CL: convolutional layer +followed by local max-pooling; FCL: fully connected layer; M: number of input images and +computed pseudo-PSFs; N: number of estimated output Zernike modes. +2. ZERNIKE POLYNOMIALS AND EXAMPLE PSEUDO-PSFS +A total of ten Zernike polynomials were used for aberration estimation and correction presented +in the paper. A list of the polynomials, sequenced using Noll’s indices, were included in Figure +S2 (a). +Figure S2 (b) included some examples of pseudo-PSFs. It can be observed that when aberration +size increases, the maximum pixel value of the Pseudo-PSF decreases; a global max-pooling of +the pseudo-PSF extracts information related to the Strehl ratio of the PSFs. Pseudo-PSFs also have +shapes that are related to the aberrated PSF shapes. +3. PHYSICAL INFORMATION EMBEDDED IN THE CNN ARCHITECTURE +As mentioned in the main paper, the bespoke CNN architecture embedded information about +the physical effects of aberrations on images within the trainable parameters. To illustrate these +phenomena, we designed six input patterns and two filters to calculate how values obtained +after global max-poolings from different convolutional layers were related to the features of the +patterns. Normally, the filters would be learned as part of the training process, but for illustrative +purposes, we have defined them manually here. +As shown in Figure S3, patterns 1 to 3 had the same general shape but varying sizes. They were +all convolved with the same filter 1. Pattern 1 had the largest feature and the values obtained were +almost constant throughout layers 1 to 5 (see Figure S3 (b)). Patterns 2 and 3 had smaller features +and the extracted values reduced when moving further down the layers, where the embedded +physical scales were more closely related to large scale features. Patterns 4 to 6 had the same +general shape with four peaks positioned at the corners of a square. They were all convolved +with filter 2, which shared a similar general shape. Pattern 4 had the smallest feature size and +resulted a largest value in layer 2. Patterns 5 and 6 had larger feature sizes and resulted in largest +values in layers 3 and 4, respectively. This trend confirms the expectation that layers later in the +CNN probe larger scales in the input images. Note that all the patterns were designed in such a +way that the maximum pixel reading (and thus the value max-pooled from layer 1) equalled to 1. +2 + +i = 5 +astigmatism +8 +coma +9 +trefoil +10 +trefoil +11 +primary +spherical +12 +secondary +astigmatism +13 +secondary +astigmatism +22 +secondary +spherical +6 +astigmatism +7 +coma +(a) +(b) +0 rad +0.5 rad +i = 7 +±1 rad +i = 7 +1.5 rad +i = 7 +2.5 rad +i = 7 +Aberration +Bias +Pseudo-PSF +Aberration +Bias +Pseudo-PSF +0 rad +0.5 rad +i = 5 +±1 rad +i = 5 +0.8 rad +i = 5 +1.5 rad +i = 5 +0 +1 a.u. +0 +π rad +Fig. S2. (a) Zernike polynomials Noll’s index 5-13, 22. This is a whole list of the polynomi- +als used for aberration determinations in the paper. (b) Examples of pseudo-PSFs. The first +column is the input aberration and the second column is the bias mode used in pseudo-PSFs +generation. +3 + +∗ filter1 +∗ filter2 +Local +max-pooling +Convolution +Layer +1 +Layer +2 +Layer +3 +Layer +4 +Layer +5 +Pattern +1 +Pattern +4 +Pattern +5 +Pattern +6 +(a) +(b) +Pattern +2 +Pattern +3 +Layer 1-5 +Normalised max-pooling value +Pattern 1 +2 +3 +4 +5 +6 +Single +pixel +feature +Small +scale +feature +Large +scale +feature +0 +1 a.u. +3×3 +3×3 +Fig. S3. Demonstrations of the link between feature sizes and convolutional layers. (a) Pattern +1 to 6 each underwent a series of convolutions followed by a 2 × 2 local max-pooling. Pattern +1 to 3 were convolved with filter 1 and pattern 4 to 6 were convolved with filter 2. For each +layer, a global max-pooling were carried out to extract the maximum reading of each layer. The +physical interpretations of the extracted values of the different layers were related to Strehl +ratio (layer 1) and shapes with features ranging from small scales (layer 2) to large scales (layer +5). The extracted readings was normalised with the readings of their respective previous layer +and displayed in (b). The horizontal axis of each plot in (b) indicates from which layer the +normalised maximum reading (indicated by the vertical axis) was extracted from. +4 + +ast2 2-P +0 +0.4 +2N 2-P +0 +0.4 +ast4 3-P +0 +0.4 +4N 3-P +0 +0.4 +ast2 widefield +0 +0.4 +2N widefield +0 +0.4 +Layer 1-5 +RMS of weights +Fig. S4. Analysis of the weight distributions across convolutional layers in the CNNs trained +for different biasing schemes and microscopes. +4. WEIGHT ANALYSIS OF DIFFERENT TRAINED NEURAL NETWORKS +Figure S4 shows the root-mean-square (RMS) values of the weights at the output of each section +of the concatenated FCL following the convolutional layers of the CNN. These weights encode +information about physical phenomena in the pseudo-PSF that is related to the spatial effects +of aberrations on images. Higher numbered layers correspond to larger scale features. Similar +distributions are seen for all of the ast CNNs class and all of the 2/4N class. Most notably, it can +be seen that the 2/4N networks all carry heavier weights in layer 1, which is most similar to the +Strehl ratio variations of the PSFs. +5. TRAINABLE NEURAL NETWORK PARAMETERS +The bespoke NN and data pre-processing steps were designed with knowledge of the physical +basis of image formation. This permitted signficant reduction in NN complexity compared +to previous methods for aberration estimation. This architecture not only allowed improved +performances, providing insights on internal workings, but also had a structure size orders of +magnitude smaller than common NNs used in similar applications (see the comparison in Table +S1). This will be beneficial for future applications as NN with fewer trainable parameters would +generally require less training data and a shorter training time. Furthermore, the simplified design +means that there is greater potential for extending the method to more challenging applications. +5 + +Neural network method +Number of trainable parameters +ResNet[2] +>0.27M +Inception V3/ GoogLeNet[3, 4] +23.6M +Xception[5, 6] +22.8M +Deep Image Prior[7] +2M +PHASENET[8, 9] +1M +MLAO in this paper +0.028M to 0.032M +Table S1. A list of NNs used in image processing and phase determination with their number +of trainable parameters. Inception V3[3], Xception[5] and PHASENET[8] have been directly +demonstrated for phase determination. ResNet is a common basic NN architecture that has +been used in many different image processing and phase determination architectures[8]. A 20 +layer ResNet is the smallest architecture proposed in the ResNet paper[2] that has ∼0.27M +trainable parameters. Deep Image Prior employs a U-Net architecture that is a commonly +used in many biomedical image processing applications. Deep phase decoder[10], a network +designed for wavefront and image reconstruction, was also inspired and adapted from Deep +Image Prior. +6. CHOICE OF BIAS MODE +The simplest MLAO implementation uses a pair of biased images as the input. The nature of +the bias aberrations is a design choice. In order to investigate this, we tested individual Zernike +modes as the bias and trained different MLAO networks with identical architecture to correct the +same randomly generated aberrations. The loss function of the different NNs during training +was shown in Fig. S5 (a). Results from correcting 20 randomly generated aberrations were shown +in Fig. S5 (b). +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +1k +2k +3k +4k +5k +6k +5 +8 +7 +6 +11 +i= +Training epochs +RMS loss function +0 +0.5 +1 +1.5 +2 +2.5 +Aberration RMS / rad +pre +correction +4 +5 +6 +7 +8 +11 +Bias mode i +(a) +(b) +188nm/px +5μm aberration +free +aberration +1.88 rad +Fig. S5. Testing Zernike modes as choice of bias aberration. (a) A plot of the root mean square +(RMS) loss function against the number of epochs when training NNs of the same architec- +ture from the same dataset but using different bias modes. (b) Statistical results of testing the +trained NNs to correct the same sets of random aberrations over 2-P microscope images of +beads. Twenty randomly generated aberrations consisting five Zernike modes and RMS value +smaller than 2.2 radians were introduced for correction (dark gray bar). The remaining aberra- +tions after correction by different networks were averaged and shown in the figure; standard +deviations of the remaining aberrations are represented as the error bar. Insets showed an +example of the FOV when no aberration was introduced and an example when 1.88 rad of +aberration was introduced into the system. +The two networks using oblique and vertical astigmatism (index i =5 and 6) converged to +similar loss function during training (Fig. S5 (a)). The same two networks also gave similar +6 + +averaged remaining aberrations during experimental aberration correction on a bead sample +(Fig. S5 (b)). The two networks using vertical and horizontal coma (index 7 and 8) also showed +mutually similar values. This was expected as these pairs of modes (5 and 6; 7 and 8) differ only +by rotation, which should not have an effect on how effective the networks determine aberrations. +From these results, the NNs using astigmatism as the bias modes converged to the smallest +loss function during training. This possibly suggested that the astigmatism modes, on average, +allowed the network to learn more from the training data. It was also observed from the ex- +perimental results where, in general, the NN obtained the smallest remaining aberrations. We +therefore chose to use astigmatism as the modulation modes for the two-bias NN methods in the +experiments conducted in this paper. +7. TOLERANCE TO SAMPLING RATE +As described in the paper, the networks for scanning microscopy were trained on simulated +dataset with pixel sampling within the range of 1.0× to 1.2× of the base sampling rate (see the +method section in the main paper for more details). However in many practical cases, there can +be uncertainty in pixel sampling for a system or constraints on the sampling rates that may be +used. We hence tested the tolerance of our networks to pixel sampling rates outside the range of +the training dataset (see Fig. S6). +0 +1 +2 +3 +4 +pre correction +2N+1 conv +ast2 MLAO +2N MLAO +219nm/px +5μm +188nm/px +5μm +156nm/px +5μm +5μm +125nm/px +Aberration RMS / rad +Image sampling rate +Fig. S6. Testing of robustness to pixel sampling. Statistical results of remaining aberrations +before (red plot) and after correction using 2N+1 conv, ast2 MLAO and 2N MLAO methods. +The results were averaged from 20 randomly generated aberrations and the SDs were shown +as the error bars. The same algorithms were used to correct the same aberrations over images +collected at different pixel sampling as shown by the horizontal axis. Insets show examples of +the images collected at different sampling rates. +In this case, 188nm per pixel was close to the sampling of the generated dataset on which the +two NNs were trained. When images were sampled at a smaller or larger rate, ast2 MLAO and +2N MLAO were still able to correct aberrations, but were slightly less effective. +8. FURTHER THREE-PHOTON MICROSCOPE DEMONSTRATIONS +Figure S7 showed the performance of the ast4 MLAO algorithm, for imaging neuronal activity +at a depth of 670 µm in a mouse brain. Despite the very low SNR of the image data, the image +quality and cell activity data were considerably improved. +9. DETAILS OF THE EXPERIMENTAL METHODOLOGY +Three optical systems, a 2-P, 3-P and widefield microscope, were used for demonstrations on +different samples. Networks with different parameter settings are also adjusted for different +applications. +7 + +vi + +A +B +C +D +E +F +G +H +Post MLAO it:3 +Pre MLAO +Post MLAO it:3 +1 +2 +Post MLAO +it:3 +Pre +MLAO +Post +MLAO +it:3 +iii +1 +2 +Pre +MLAO +0 +50 +100 +0 +50 +100 +time / s +time / s +yS +yI +Images +Images +it:1 +it:2 +it:3 +iv +v +it:2 +it:1 +20μm +A +B +C +D +E +F +G +H +GCaMP at 670 μm +vii +Pre MLAO +i +-1 rad +-0.5 rad ++0.5 rad ++1 rad +ast4 MLAO +1 +3 +2 +4 +Bias mode i=5 +ii +Fig. S7. Three-photon microscopy imaging GCaMP neuronal activities at depth 670µm. Power +at sample was 44 mW. Wavefronts inserted to the figures showed the phase modulations ap- +plied by the DM at the relevant step; the common scale is indicated by the colorbar above v. i +and iii show respectively before and after ast4 MLAO correction through three iterations (it:1 +to 3), 200 frame averages after motion correction. In iii, time traces shown to the right and +bottom were taken from the marked lines (1) and (2) respectively. ii 1-4 shows example single- +frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes +as insets. iv and v show the intensity metric (yI) and the sharpness metric (yS), respectively, +calculated from single image frames, against the number of images acquired for three iterations +ast4 MLAO. vi shows the Calcium activity of 8 cells (A-H marked on i). vii shows a histogram +of the 200 frames collected pre MLAO (blue), post MLAO (red) and the differences between pre +and post MLAO (yellow). +8 + +A +0 +50 +100 +time / s0 +S +time / s +50 +1002 +1 +0 +-1 +-2 +-3based metric (y) +Sharpness +*ast4 MLAO +0 +4 +8 +12 +number of +sample exposuresFluorescence +intensity (y,) +*一ast4 MLAO +0 +4 +8 +12 +number of +sample exposures2 +1 +0 +-1 +-2 +-32 +1 +0 +-1 +-2 +-3×105 +Pre MLAO +Post MLAO it:5 +Difference between +post and pre MLAO +0T +1 +0 +2 +-T +1 +2×105 +PreMLAO +Post MLAO it:3 +Difference between +post and pre MLAO0 +50 +100 +time / sA. Experimental setups +c +Widefield +3-D SIM +microscope +f200 +SLM +DM +f175 +Obj:W-D +f60 +BX +f175 +f125 +f200 +FS +AP +f175 +f75 +f400 +f50 +C1 +C2 +M +M +M +PL +M +M +DF +DF +M +PR +/2 +f175 +f200 +f200 +EF +EF +SF +EF +LS488 +LS561 +M +ST +M +/2 +FS laser +HWP +PBS +Dump +compressor +M +f50 +f200 +M +Galvo scanners +f206 +f30 +DF +PZ +EF +PMT +Obj:W-D +b +Three-photon +microscope +f500 +f75 +auto- +correlator +DM +FS laser +HWP +PBS +Dump +f50 +f150 +M +M +Galvo x +FM +f75 +f75 +Galvo y +f75 +f120 +M +M +f150 +f75 +DF +EF +PMT +PZ +f150 +f100 +SLM +M +f200 +f200 +FS +Obj:W-I +a +Two-photon +microscope +DF +DF +FM +FM +M +Fig. S8. Configuration of the (a) 2-P (b) 3-P (c) widefield 3-D SIM microscope. (Caption contin- +ued on the next page.) +9 + +Femtosecond (FS) Laser; Continuous-wave lasers with wavelenths 488nm and 561nm (LS488 and +LS561); half wave plate (HWP); polarisation beam splitter (PBS); laser beam dump (Dump); lens +with focal length = x mm (fx); broadband dielectric mirror (M); flip mirror (FM); Hamamatsu +spatial light modulator (SLM); Mirao 52E deformable mirror (DM) in the 3-P system; ALPAO +69 deformable mirror (DM) in the widefield 3-D SIM system; aperture (AP); spatial filter (SF); +field stopper (FS); X galvanometer (Galvo x); Y galvanometer (Galvo y); beam expansion (BX); +half waveplate (λ/2); linear polariser (PL); polarisation rotator (PR); Olympus 40× numerical +aperture (NA) 1.15 water immersion objective lens (Obj:W-I) used in the 2-P system; Nikon 16× +NA 0.8 water dipping objective lens (Obj:W-D) used in the 3-P system; Olympus 60× NA 1.1 +water dipping objective lens (Obj:W-D) in the widefield 3-D SIM system; Z-piezo translation stage +(PZ); X-Y-Z translational sample mounting stage (ST); Dichroic filter (DF) allow emission signal +from fluorophores to be reflected through emission filter (EF) into a photo-multiplier tube (PMT) +in a multi-photon system; cameras (C1 and C2) +B. Sample preparation +The 3-P results were collected from imaging male (Lhx6-eGFP)BP221Gsat; Gt(ROSA)26Sortm32(CAG- +COP4*H134R/EYFP)Hze mice (static imaging) and female and male Tg(tetO-GCaMP6s)2Niell +mice (calcium imaging). Mice were between 8-12 weeks of age when surgery was performed. The +scalp was removed bilaterally from the midline to the temporalis muscles, and a metal headplate +with a 5 mm circular imaging well was fixed to the skull with dental cement (Super-Bond C&B, +Sun-Medical). A 4–5 mm circular craniotomy was performed during which any bleeding was +washed away with sterile external solution or staunched with Sugi-sponges (Sugi, Kettenbach). +Cranial windows composed of 4 or 5 mm circular glass coverslips were press-fit into the cran- +iotomy, sealed to the skull by a thin layer of cyanoacrylate (VetBond) and fixed in place by dental +cement. +The widefield 3-D SIM results were collected from imaging NMJ of Drosophila larvae. For +the immunofluorescence sample with one coloured channel, it was prepared as previously [11]. +Crawling 3rd instar larvae of wildtype Oregon-R Drosophila melanogaster were dissected on a +Sylgard-coated Petri Dish in HL3 buffer with 0.3mM Ca2+ to prepare larval fillet [12]. Then, the +larval fillet samples were fixed in Paraformaldehyde 4% in PBS containing 0.3% (v/v) Triton +X-100 (PBSTX) for 30 minutes. The brains were removed post-fixation, and the fillet samples were +transferred to a Microcentrifuge tube containing PBSTX for 45 minutes of permeabilisation. The +samples were stained with HRP conjugated to Alexa Fluor 488 and DAPI for 1 hour at room +temperature (21C◦). After the washes, the samples were mounted in Vectashield. +For the 3-D SIM results collected on the Drosophila larvae sample with two coloured channels, +it was prepared by following the protocol presented in [11]. 3rd instar Drosophila melanogaster +larvae (Brp-GFP strain) were dissected in HL3 buffer with 0.3mM Ca2+ to prepare a so-called +larval fillet, and the larval brains were removed. After this, larvae were stained for 15 minutes +with HRP conjugated to Alexa Fluor 568 to visualise the neurons, washed with HL3 buffer with +0.3mM Ca2+ and imaged in HL3 buffer without Ca2+ to prevent the larvae from moving. +C. Network parameters +Table S2 showed the network settings used in different imaging applications. +10 + +Results in +Method label +M +N +Bias +Bias +Corrected +modes, i +depths +modes, i +Fig. 2 (a, c, f) +ast2 MLAO +2 +5 +5 +±1 rad +5–8, 11 +Fig. S3 +Fig. 2 (a, c, f) +2N MLAO +10 +5 +5–8, 11 +±1 rad +5–8, 11 +Fig. 2 (b, d, e) +ast2 MLAO +2 +9 +5 +±1 rad +5–13 +Fig. 2 (b, d, e) +2N MLAO +18 +9 +5–13 +±1 rad +5–13 +Fig. 3 (a, b) +ast4 MLAO +4 +7 +5 +±0.5 +5–11 +Fig. S4 +±1 rad +Fig. 3 (a) +4N MLAO +28 +7 +5–11 +±0.5 +5–11 +±1 rad +Fig. 4 +ast2 MLAO +2 +8 +5 +±1 rad +5–11, 22 +Fig. 4 +2N MLAO +2 +8 +5–11, 22 +±1 rad +5–11, 22 +Table S2. A list of MLAO parameters chosen for different imaging applications. The Zernike +modes were sequenced using Noll’s indices. +REFERENCES +1. +M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, +J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Joze- +fowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, +M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Va- +sudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, +“TensorFlow: Large-scale machine learning on heterogeneous systems,” (2015). Software +available from tensorflow.org. +2. +K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in +Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016). +3. +T. Andersen, M. Owner-Petersen, and A. Enmark, “Neural networks for image-based wave- +front sensing for astronomy,” Opt. Lett. 44, 4618–4621 (2019). +4. +C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and +A. Rabinovich, “Going deeper with convolutions,” in 2015 IEEE Conference on Computer Vision +and Pattern Recognition (CVPR), (2015), pp. 1–9. +5. +P. A. Khorin, A. P. Dzyuba, P. G. Serafimovich, and S. N. Khonina, “Neural networks +application to determine the types and magnitude of aberrations from the pattern of the +point spread function out of the focal plane,” J. Physics: Conf. Ser. 2086, 012148 (2021). +6. +F. Chollet, “Xception: Deep learning with depthwise separable convolutions,” (2016). +7. +D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Deep image prior,” Int. J. Comput. Vis. 128, +1867–1888 (2020). +8. +D. Saha, U. Schmidt, Q. Zhang, A. Barbotin, Q. Hu, N. Ji, M. J. Booth, M. Weigert, and E. W. +Myers, “Practical sensorless aberration estimation for 3D microscopy with deep learning,” +Opt. Express 28, 29044–29053 (2020). +9. +D. Saha and U. Schmidt, “Phasenet,” https://github.com/mpicbg-csbd/phasenet (2020). +10. +E. Bostan, R. Heckel, M. Chen, M. Kellman, and L. Waller, “Deep phase decoder: self- +calibrating phase microscopy with an untrained deep neural network,” Optica 7, 559–562 +(2020). +11. +J. R. Brent, K. M. Werner, and B. D. McCabe, “Drosophila larval nmj dissection.” J Vis Exp +(2009). +12. +R. M. Parton, A. M. Vallés, I. M. Dobbie, and I. Davis, “Drosophila Larval Fillet Preparation +and Imaging of Neurons,” Cold Spring Harb. Protoc. 2010, pdb.prot5405 (2010). +11 + diff --git a/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/load_file.txt b/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78973c5d329b117e1e4b64ceb01b9c9705a6e9ff --- /dev/null +++ b/ANE0T4oBgHgl3EQfxgKB/content/tmp_files/load_file.txt @@ -0,0 +1,1718 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf,len=1717 +page_content='Universal adaptive optics for microscopy through embedded neural network control Qi Hu1, Martin Hailstone2, Jingyu Wang1, Matthew Wincott1, Danail Stoychev2, Huriye Atilgan3, Dalia Gala2, Tai Chaiamarit2, Richard M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Parton2, Jacopo Antonello1, Adam M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Packer3, Ilan Davis2, and Martin J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Booth1,* 1Department of Engineering Science, University of Oxford 2Department of Biochemistry, University of Oxford 3Department of Physiology, Anatomy, and Genetics, University of Oxford martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='booth@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='uk ABSTRACT The resolution and contrast of microscope imaging is often affected by aberrations introduced by imperfect optical systems and inhomogeneous refractive structures in specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Adaptive optics (AO) compensates these aberrations and restores diffraction limited performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A wide range of AO solutions have been introduced, often tailored to a specific microscope type or application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Until now, a universal AO solution – one that can be readily transferred between microscope modalities – has not been deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We propose versatile and fast aberration correction using a physics-based machine learning (ML) assisted wavefront-sensorless AO control method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Unlike previous ML methods, we used a bespoke neural network (NN) architecture, designed using physical understanding of image formation, that was embedded in the control loop of the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The approach means that not only is the resulting NN orders of magnitude simpler than previous NN methods, but the concept is translatable across microscope modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We demonstrated the method on a two-photon, a three-photon and a widefield three-dimensional (3D) structured illumination microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Results showed that the method outperformed commonly-used modal-based sensorless AO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We also showed that our ML-based method was robust in a range of challenging imaging conditions, such as extended 3D sample structures, specimen motion, low signal to noise ratio and activity-induced fluorescence fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Moreover, as the bespoke architecture encapsulated physical understanding of the imaging process, the internal NN configuration was no-longer a “black box”, but provided physical insights on internal workings, which could influence future designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Introduction The imaging quality of high-resolution optical microscopes is often detrimentally affected by aberrations which result in compromised scientific information in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' These aberrations can arise from imperfections in the optical design of the microscope, but are most commonly due to inhomogeneous refractive index structures within the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Adaptive optics (AO) has been built into many microscopes, restoring image quality through aberration correction by reconfigurable elements, such as deformable mirrors (DMs) or liquid crystal spatial light modulators (LC-SLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1–6 Applications of AO-enabled microscopes have ranged from deep tissue imaging in multiphoton microscopy through to the ultra-high resolution required for optical nanoscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This range of applications has led to a wide variety of AO solutions that have invariably been tailored to a specific microscope modality or application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' There are two main classes AO operation: in one case, a wavefront sensor measures aberrations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' in the other case, aberrations are inferred from images – so called “wavefront sensorless AO”, or “sensorless AO” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For operations with a wavefront sensor, phase aberrations are measured directly by wavefront sensors such as a Shack-Hartmann sensor7,8 or an interferometer9–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Such operations are direct and fast but also have intrinsic disadvantages such as requiring a complex optical design and suffering from non-common path errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Furthermore, such wavefront sensors often have limitations and are less versatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For example, an interferometer requires a coherent source and all such methods suffer from problems due to out-of-focus light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' On the other hand, sensorless AO methods normally function with a simpler optical design and thus are more easily adaptable for a wide range of imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, sensorless AO methods are based on iterative deductions of phase aberrations and thus tend to be more time consuming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' this is coupled with repeated and prolonged sample exposures, which inevitably lead to photo-damage or motion related errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' There have been many developments in AO technology, and in particular sensorless AO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Conventionally, sensorless AO operates based on the principle that the optimal image quality corresponds to the best aberration correction12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A suitably defined metric, such as the total signal intensity14–27 or a spatial frequency based sharpness metric28–33, is used to quantify the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='02647v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='IV] 6 Jan 2023 image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Phase is modulated by the AO while this quality metric reading is measured and optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' There have been discussions on how the phase should be modulated12,24,34,35 and how the optimisation algorithm should be designed21,36–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, as mentioned before, such “conventional” sensorless AO methods depend on iterative optimisation of a scalar metric, where all image information is condensed into a single number, and the optimisation process is usually through mode by mode adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Such methods were thus not the most efficient approach to solving this multi-dimensional optimisation problem and the effective range of correction was limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' While a higher dimensional metric was considered to extract more information from images39, the optimisation of such a vector metric was not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' While the utility of each of these conventional sensorless AO methods has been demonstrated separately, each method had been defined for a particular microscope type and application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Until now, no such AO solution has been introduced that can be universally transferred between microscope modalities and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We propose in this article a new approach to sensorless AO that addresses the limitations of previous methods and provides a route to a universal AO solution that is applicable to any form of microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This solution is constructed around a physics- based machine learning (ML) framework that incorporates novel neural network (NN) architectures with carefully crafted training procedures, in addition to data pre-processing that is informed by knowledge of the image formation process of the microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The resulting NN is embedded into the control of the microscope, improving the efficiency and range of sensorless AO estimation beyond that possible with conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This approach delivers versatile aberration measurement and correction that can be adapted to the application, such as the correction of different types of aberration, over an increased range of aberration size, across different microscope modalities and specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In recent years, machine learning (ML) has been trialled in AO for its great computational capability to extract and process information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, many of these approaches required access to point spread functions (PSFs) or experimentally acquired bead images40–46 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' these requirements limited the translatability of these methods to a wider range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Reinforcement learning was applied to correct for phase aberrations when imaging non point-like objects47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' however, the method still involved iterative corrections and was not advantageous in terms of its correction efficiency, accuracy and correction working range compared to conventional sensorless AO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Untrained neural networks (NN) were used to determine wavefront phase and were demonstrated on non point-like objects48,49;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' however, such methods were reported to normally require a few minutes of network convergence, which limits their potential in live imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Our new approach differs considerably from previous ML assisted aberration estimation, as previous methods mostly employed standard deep NN architectures that used raw images as the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Our method builds upon physical knowledge of the imaging process and is designed around the abilities of the AO to introduce aberration biases, which improve the information content of the NN input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This approach means that the resulting NN is orders of magnitude simpler, in terms of trainable parameters, than previous NN methods (See Table S1 in supplemental document).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Furthermore, our method is readily translatable across microscope modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As NN training is carried out on a synthetic data set, adaptation for a different modality simply requires regeneration of the image data using a new imaging model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The NN architecture and training process are otherwise similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' To illustrate the versatility of this concept, we have demonstrated the method on three different types of fluorescence microscopes with different forms of AO corrector: a two-photon (2-P) microscope using a SLM, a three-photon (3-P) intravital microscope using a DM, and a widefield three dimensional (3-D) SIM microscope using a DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In all cases, we showed that the new method outperformed commonly used conventional sensorless AO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The results further showed that the ML-based method was robust in a range of challenging imaging conditions, such as specimen motion, low signal to noise ratio, and fluorescence fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Moreover, as the bespoke architecture encapsulated into its design physical understanding of the imaging process, there was a link between the weights in the trained NN and physical properties of the imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This means that the internal NN configuration needs no-longer to be considered as a “black box”, but can be used to provide physical insights on internal workings and how information about aberrations is encoded into images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Concept and implementation The overall MLAO concept is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The experimental application follows closely the concept of modal sensorless AO, whereby a sequence of images are taken, each with a different bias aberration applied using the adaptive element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The set of images are then used as the input to the ML-enabled estimator, which replaces the previous conventional method of optimisation of an image quality metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The estimated correction aberration is then applied to the adaptive element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' If necessary, the process can be iterated for refined correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The significant advantage of the new method is the way in which the estimator can more efficiently use image information to determine the aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The concept has been designed in order to achieve particular capabilities that extend beyond those of conventional sensorless AO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The new method should ideally achieve more efficient aberration estimation from fewer images, to reduce time and exposure of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It should operate over a larger range of aberration amplitudes, compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='particular estimator should be robust to variations between similar microscopes and the concept should be translatable across ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2/16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Microscope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Adaptive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Corrected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Efficient MLAO estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Corrected image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Aberrated image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Iterative correction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Versatile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Aberration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='correction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Maximum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='reading ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Small ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Large ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='bespoke CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FCL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='F -1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image pre-processing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='maxpooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Global ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='maxpooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Fully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Pseudo-PSF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image variations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Synthesised images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Image = Object ∗ PSF + Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='microscope imaging model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='PSF2-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' PSF3-P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' PSFWF Fluorescence fluctuations Spatial sampling size Sample motion Sample structure/ sparsity Aberration and brightness Detector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' photon and structured noise 3-D structures/ background Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The MLAO concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) Overview of the AO correction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A minimum of two bias aberrations were introduced by the adaptive element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' corresponding images of the same field were captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The images were passed to the MLAO estimator, which determined the Zernike coefficients for correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The correction speed was limited only by the speed of image acquisition, not by computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) Training data generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A range of image variations were included in the synthetic data set for NN training to cope with variations in real experimental scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The data was a combination of artificial and real microscope images, chosen to model a wide range of realistic specimen structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Images were created through convolution of specimen structures with an appropriate PSF, generated for the specific microscope modality, incorporating aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (c) Image pre-processing and NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Images were pre-processed to compute pseudo-PSFs, which were predominantly independent of specimen structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' F and F −1 represent the forward and inverse Fourier transform, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A central cropped region of the pseudo-PSF images was used as the inputs to a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The CNN was designed and trained specifically for aberration determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The output from the overall network was the correction coefficients for the Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The NN architecture was such that the convolutional layer outputs could be correlated with spatial scales of the aberration effects on the pseudo-PSFs and hence the imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hence, the distribution of weights in the network had physical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3/16 3 2 1 0 1 2 3different microscope types and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' From a practical perspective, it is also important that training can be performed on synthetic data, as it would be impractical to obtain the vast data set necessary for training from experimentally obtained images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' An essential step towards efficient use of image data is the image pre-processing before they are presented to the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Rather than taking raw image data as the inputs, the NN receives pre-processed data calculated from pairs of biased images, which we term a “pseudo-PSF”, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 1 and explained in the methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This pseudo-PSF contains information about the input aberration and is mostly independent of the unknown specimen structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' By removing the specimen information at this stage, we can reduce the demands on the subsequent NN, hence vastly simplifying the architecture required to retrieve the aberration information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As most of the useful information related to aberrations was contained within the central pixels of the pseudo-PSF, a region of 32×32 pixels was extracted as the input to the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The first section of the NN was a bespoke convolutional layer that was designed to extract information from the inputs at different spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The outputs from the convolutional layer were then provided to a fully connected layer, which was connected to the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Full details of the NN design are provided in the methods and the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This architecture – rather unusually – provided a link between the physical effects of aberrations on the imaging process and the mechanisms within the NN, specifically through the weights at the output of the first fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' NN training was performed using a diverse set of synthesised training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' These images were calculated using an appropriate model of the microscope imaging process in the presence of aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Images were synthesised by convolutions of specimen structures with a PSF, incorporating various likely experimental uncertainties and noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The specimens consisted of a range of artificial and realistic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Full details are provided in the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This versatile concept could accommodate different aberration biasing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Conventional modal sensorless AO methods typically required a minimum of 2N +1 biased images to estimate N aberration modes21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, the MLAO method has the ability to extract more information out of the images, such that aberrations could be estimated with as few as two images, although more biased images could provide better-conditioned information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In general, we defined methods that used M differently biased images to estimate N Zernike modes for aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The input layer of the NN was adjusted to accommodate the M image inputs for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Out of the many possibilities, we chose to illustrate the performance using two biasing schemes: one using a single bias mode (astigmatism, Noll index50 i = 5) and one using all N modes that were being corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In the first case, we used either two or four images (M = 2 or 4) each with different astigmatism bias amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We refer to these methods as ast2 MLAO or ast4 MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Astigmatism was chosen as the most effective bias mode (see supplementary document, section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In the second case, biased images were obtained for all modes being estimated (M = 2N or 4N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' this type is referred to in this paper as 2N MLAO or 4N MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For a complete list of the settings for each demonstration, please refer to Table S2 in the supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Results In order to show its broad application, the MLAO method was demonstrated in three different forms of microscopy: 2-P and 3-P scanning microscopy and widefield 3-D structured illumination microscopy (SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This enabled testing in different applications to examine its performance coping with different realistic imaging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The MLAO methods were compared to two widely used conventional modal based sensorless AO methods (labelled as 2N+1 conv and 3N conv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The 2N+1 conv method used two biased images per modulation mode and an additional zero biased image to determine phase correction consisting N modes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The 3N conv method used three images per modulation mode (two biased and one unbiased images) and determined the coefficients of the modes sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For both methods, the bias size was chosen to be ±1 rad for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A suitable metric was selected to quantify the image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For each mode, the coefficients were optimised by maximising the quality metric of the corresponding images using a parabolic fitting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' When used in 2-P and 3-P demonstrations, the total fluorescence intensity metric was optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the widefield 3-D SIM microscope, a Fourier based metric was optimised51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the details of the two conventional methods, please refer to21,36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Different functions were defined as optimisation metrics for the conventional AO methods, and also to assist quantifiable comparisons of image quality improvement for the MLAO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' These were defined as an intensity based metric yI, a Fourier based metric yF, and a sharpness metric yS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Details are provided in the methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Two-photon microscopy A range of method validations were performed on a 2-P microscope that incorporated a SLM as the adaptive correction element, including imaging bead ensembles and extended specimen structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The experimental set-up of the 2-P system was included in Figure S8 (a) in the supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In order to obtain controlled and quantitative comparisons between different AO methods, the SLM was used to both introduce and correct aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This enabled statistical analysis of MLAO 4/16 performance with known input aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' System aberrations were first corrected using a beads sample before carrying out further experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We performed a statistical analysis to assess how MLAO algorithms (ast2 MLAO and 2N MLAO) performed in various experimental conditions compared to conventional algorithms (2N+1 conv and 3N conv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Experiments were conducted on fixed beads samples (Figure 2 (a, b)), and Bovine Pulmonary Artery Endothelial (BPAE) cells (FluoCellsTM Prepared Slide #1) (Figure 2 (c - f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Dependent on the experiment, either N = 5 or N = 9 Zernike modes were estimated (see Table S2 in Supplemental document for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Statistical performance analysis Figure 2 (a) and (b) showed statistical comparisons of the different correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 2 (a) displayed the residual aberrations gathered from twenty experiments, each consisting of one correction cycle from random initial aberrations including five Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' If the remaining aberration is below the pre-correction value, then the method provides effective aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A wide shaded area indicated inconsistent and less reliable correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The results show that when correcting small aberrations with root mean square (RMS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='63 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='19 rad, 2N MLAO performed similarly to 2N+1 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Between RMS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='19 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='92 rad, 2N MLAO corrected more accurately (lower mean aberration) and also more reliably (smaller error range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For large aberrations above RMS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='12 rad, 2N+1 conv completely failed, whereas the MLAO methods still improved aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' ast2 MLAO had poor performance at small aberrations (RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='63 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='84 rad) but provided reasonable correction for large aberrations (RMS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='92 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='12 rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, it is important to note that ast2 MLAO required only two images for each correction cycle, far fewer that the ten and eleven images required respectively for 2N MLAO and 2N+1 conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 2 (b) displayed the mean value of metric yI from ten experiments against the number of images acquired during multiple iterations of the different correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The corrected aberrations consisted of nine Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It was shown that ast2 MLAO corrects the fastest initially when the input aberration is large but converges to a moderate signal level, which indicates only partial correction of the aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2N MLAO corrects more quickly and to a higher level than the conventional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The narrower error bars for both MLAO algorithms at the end of the correction process indicate that they are more reliable than the two conventional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Correction on extended specimen structures Figure 2 (c)-(f) showed experimental results when imaging microtubules of BPAE cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Specimen regions were chosen to illustrate performance on different structures: (c) contained mainly aligned fine fibrous structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (d) contained some large scale structures (bottom right);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (e) contained fine and sparse features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For (f) we intentionally reduced illumination laser power and increased detector gain to simulate an imaging scenario with very low signal to noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The images showed structured noise at the background, which could pose a challenge to estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A large randomly generated aberration (RMS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='12 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='23 rad) consisting of five (c and f) or nine (d and e) Zernike modes was used as the input aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In (c), (d) and (e), ast2 MLAO corrected the fastest initially when the aberration was large but converged to a moderate level of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2N MLAO corrected faster in general than the conventional methods and converged to a higher level of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In (f) when SNR was poor and structured noise was present, ast2 MLAO failed to correct while 2N MLAO continued to perform consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Three-photon intravital microscopy Three-photon microscopy of neural tissue imaging is a particular challenge for sensorless AO, due to the inherently low fluorescence signal levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' While this could be alleviated by averaging over time, problems are created due to specimen motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Further challenges are posed for functional imaging, due to the time dependence of emission from ion or voltage sensitive dyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The demonstrations here show the robustness of the new MLAO methods in experimental scenarios where the conventional methods were not effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Importantly, the MLAO methods were able to perform effective correction based on a small number of image frames without averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The experimental set-up of the 3-P system is shown in Figure S8 (b) in the supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The microscope used an electromagnetic DM for aberration biasing and correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Two MLAO methods, ast4 MLAO and 4N MLAO, were used to correct aberrations by using single frame images as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In each case, more input frames were chosen than in the 2-P demonstrations, in order to cope with the lower SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The NNs were trained to estimate N = 7 Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Two types of mice were used to perform live brain imaging of green fluorescent protein (GFP) labelled cells (Figure 3 (a)) and functional imaging in GCaMP-expressing neurons (Figure 3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In Figure 3 (a), results were collected at 660µm depth and power at sample was 32 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In Figure 3 (b), imaging was at 250µm depth and power at sample was 19 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Further 3-P results were included in the section 8 of supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the details of the sample preparation, please refer to section 9B in supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 3 (a) shows plots of the metrics yI and yF as proxies for correction quality when imaging GFP labelled cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Both ast4 MLAO and 4N MLAO networks successfully improved the imaging quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Similar to the ast2 MLAO results in the 2-P 5/16 A B C D A 5μm B C D No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images A B E D F C A 5μm B C D E F B D C E F No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images ast2 MLAO 2N MLAO A 5μm B C D E F A B C E F D A 5μm B C D E F (c) (d) (f) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images A 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1 rad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 rad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='9 rad 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 rad (a) (b) A B C E F D D E F C B A 5μm No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images yI A yI yI yI yI (e) N = 5 N = 9 N = 5 N = 9 N = 9 N = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1 Applied aberration RMS (rad) 1 2 3 4 Remaining aberration RMS (rad) Pre correction 2N+1 conv ast2 MLAO 2N MLAO 5μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='0rad Fine structure Coarse structure Sparse structure Low SNR with strructured noise Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Comparative performance of MLAO methods in a 2-P microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) Residual aberration after one correction cycle for three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Points show the mean and the shaded area indicates the standard deviations (SDs) of aberration distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The images show an example field of view (FOV) when different amounts of a random aberration were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b)-(f) show the intensity metric (yI) as a proxy for correction quality, against the number of images used for multiple iterations of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Random aberrations consisting of N Zernike modes, as shown in the figure, were introduced and corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In (b), an ensemble of ten random aberrations were corrected, imaging over the same FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Error bars on the plot showed the SD of the fluorescence intensity before and after correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (c)-(f) show specific corrections imaging microtubules of BPAE cells, illustrating performance for different specimen structures and imaging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The images were acquired before and after correction through the different methods (as marked on the metric plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Insets on the images show residual wavefronts after correction for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The grayscale colorbars show phase in radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 6/16 3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures2元 0 22 T3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5元 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5元3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5元 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5元3N conv 2N+1 conv Fluorescence intensity ast network 2N network 0 10 20 30 number of sample exposures6 4 2 0 2 4 66 4 2 0 2 4 66 4 2 0 2 4 66 4 2 0 2 4 66 4 2 0 2 4 66 2 0 2 4 62元 0 22 T2元 0 22 T(b) GCaMP at 250 μm (a) static GFP at 660 μm i iv 4N MLAO it:1 20μm ii Pre MLAO iii ast4 MLAO it:5 yI yF 0 4 8 12 16 20 28 ast4 MLAO 4N MLAO v 0 4 8 12 16 20 28 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images ast4 it:5 4N it:1 vi Post MLAO it:5 iii 1 1 Post MLAO it:5 Pre MLAO 0 50 100 time / s 4 3 2 it:1 Pre MLAO ii 20μm A B C D E F G H ast4 MLAO bias mode i = 5 1 3 2 4 1 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad +1 rad +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad 1 3 2 4 ast4 MLAO 4N MLAO ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5,±1 rad bias mode i = 5 6 7 8 9 10 11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad +1 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad 1 rad bias mode i= i 0 4 8 12 16 20 yI ast4 MLAO iv 0 4 8 12 16 20 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of images yS v it:1 it:2 it:3 it:4 it:5 A B 0 50 100 time / s C D 0 50 100 time / s E F 0 50 100 time / s G H 0 50 100 time / s vi Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Aberration correction in three-photon microscopy of live mouse brains: (a) GFP-labelled cells at depth 660µm and (b) functional activity of GCaMP-labelled cells at 250µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Wavefronts inserted to the figures showed the phase modulations applied by the DM at the relevant step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' the common scale is indicated by the colorbar next to (a) and (b) ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) i shows example single-frame images used in correction with the corresponding bias modes as insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 1-4 were the image inputs to ast4 MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For 4N MLAO, six more bias modes and thus 24 more images were also used in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) ii-iv show images averaged from 20 frames after motion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The rectangular boxes highlight regions of interest for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) v and vi show the intensity metric (yI) and the Fourier metric (yF), respectively, calculated from single image frames, against the number of images acquired for five iterations ast4 MLAO one iteration of 4N MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) i 1-4 shows example single-frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes as insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' White squares highlight two cells for comparison to show the fluorescence fluctuations over time neural activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) ii and iii show respectively before and after ast4 MLAO correction through five iterations (it:1 to 5), 200 frame averages after motion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In iii, time traces shown to the left were taken from the marked line (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) iv and v show the intensity metric (yI) and the sharpness metric (yS), respectively, calculated from single image frames, against the number of images acquired for five iterations ast4 MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) vi shows the calcium activity of 8 cells (A-H marked on ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 7/16 T 1 0 2 T 1 22 1 0 1 2 30 50 100 time / s2 1 0 1 2 32 1 0 1 2 32 1 0 1 2 32 1 0 1 2 3T 1 0 2 T 1 2demonstrations, ast4 MLAO corrected more quickly at first, but converged to a lower correction level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In contrast, 4N MLAO preformed better overall correction, but required more images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Panels ii-iv show averaged images in which processes previously hidden below the noise level are revealed through MLAO correction (as highlighted in the white rectangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The example biased images shown in Figure 3 (a) i provide an indication of the low raw-data SNR that the MLAO method can successfully use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 3 (b) shows results from imaging calcium activity in a live mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The ast4 MLAO method successfully improved image quality despite the low SNR and fluorescence fluctuations of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' From both time traces of line 1 and cells A-H, it could be clearly seen that after corrections, signals were increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The 4N MLAO method failed to correct in this experimental scenario (results not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We will discuss the likely hypotheses for this in the discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The fluctuating fluorescence levels due to neural activity mean that conventional metrics would not be effective in sensorless AO optimisation processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This is illustrated in Figure 3 (b) iv and v, where it can be seen that no single metric can accurately reflect the image quality during the process of ast4 MLAO correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' These observations illustrate the advantages of MLAO methods, as their optimisation process did not rely on any single scalar metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Widefield 3-D structured illumination microscopy The architecture of the NN was conceived so that it would be translatable to different forms of microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In order to illustrate this versatility, and to complement to the previously shown 2-P and 3-P laser scanning systems, we applied MLAO to a widefield method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The 3D SIM microscope included multiple lasers and fluorescence detection channels and an electromagnetic DM as the correction element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Structured illumination patterns were introduced using a focal plane SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The detailed experimental set-up was included in Figure S6 (c) in the supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Without AO, 3D SIM reconstruction suffers artefacts caused by aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Since typical specimens contain 3D structures, the lack of optical sectioning in widefield imaging means that the aberration correction process can be affected by out of focus light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As total intensity metrics are not suitable for conventional AO algorithms in widefield imaging, Fourier based sharpness metrics have often been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, such metrics depend on the frequency components of the specimen structure39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In particular, emission from out of focus planes can also affect the sensitivity and accuracy of correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However, the NN based MLAO methods were designed and trained to mitigate against the effects of the sample structures and out of focus light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 4 shows results from two NN-based methods ast2 MLAO and 2N MLAO compared to the conventional algorithm 3N conv, which used the yS metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Sensorless AO was implemented using widefield images as the input (Figure 4 (a, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The correction settings thus obtained by the 2N MLAO method were then applied to super-resolution 3D SIM operation (Figure 4 (c, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' N = 8 Zernike modes were involved in the aberration determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The specimen was a multiple labelled Drosophila larval neuromuscular junction (NMJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the details of the sample preparation, please refer to section 7B in supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 4 (b) showed that ast2 MLAO corrected most quickly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2N MLAO corrected to a similar level but required more sample exposures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3N MLAO was less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 4 (a) showed the effectiveness of correction on raw and deconvolved widefield images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Part iii showed the changes in image spectrum after correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The dashed line shows a threshold where signal falls below the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It can be seen that both (C) ast2 MLAO and (D) 2N MLAO increased high frequency content compared to (A) before AO correction and (B) after 3N conv corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure 4 (c) and (d) showed the images after 3D SIM reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It can be clearly seen that when by-passing AO (i), there was strong artefacts due to aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' After correcting using five iterations of 2N MLAO, artefacts were suppressed and z-resolution was improved (see sections through line 1 and 2 in Figure 4 (d)) Discussion The power and simplicity of the MLAO method arise mainly from a combination of three aspects: the pre-processing of image data, the bespoke NN architecture, and the definition of the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' All of these aspects are informed by physical and mathematical principles of image formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This forms a contrast with many other data-driven deep learning approaches, where complex NNs are trained using vast amount of acquired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The calculation of the pseudo-PSF from pair of biased images (as shown in Figure 1 (c) and elaborated in the Methods) acts to remove most of the effects of unknown specimen structure from the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The information contained within the pseudo-PSF encodes indirectly how aberrations affect the imaging PSF (see Figure S2 in the supplemental document for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' There is a spatial correspondence between a pixel in the pseudo-PSF and the PSF itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hence, spatial correlations across the pseudo-PSF relate to spatial effects of aberrations on the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The set of pseudo-PSFs forms the input to the convolutional layers of the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The masks in each convolutional layer probe, in effect, different scales across the pseudo-PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hence, one can attribute a correspondence between the output of these layers and the effects aberrations have over different physical scales in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Such phenomena are heuristically demonstrated in 8/16 5μm 1 2 1 2 2 2 1 1 z z=6μm z z (d) i ii A B D C 5μm (b) (c) i ii Pre Post-MLAO Pre Post-MLAO A i B i C i D i Widefield Deconvolution Image spectrum 10μm ii iii ii iii ii iii ii iii (a) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' yS Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Aberration correction in a widefield structured illumination microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) Widefield images acquired A i before and B-D i after correction through different methods (as marked on the metric plot (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The second column ii shows corresponding deconvolved widefield images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The third column iii shows corresponding image spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' dashed lines show the threshold where signal falls below the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) The sharpness metric yS against the number of images, for two iterations of 3N conv, ten iterations of ast4 MLAO and three iterations of 2N MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (c, d) 3-D projections of 3-D reconstructed SIM image stack of (c) 10µm and (d) 6µm when (i) by-passing AO and (ii) after five iterations of 2N MLAO correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' square inserts show zoomed in region for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' x-y and y-z sections are shown through lines 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Insets to (a,c and d) show wavefronts corrected by the DM for each image acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' phase is shown on the adjacent scale bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 9/16 3N conv ast2 MLAO 2N MLAO 2 14 24 32 48 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' of imagesT 0 T2元 0 2 T2元 0 2 T102 2 10 一Layer 1 2 3 4 5 astX MLAO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='23 XN MLAO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='20 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The RMS of the weight distributions extracted from different convolutional layers of the two classes of trained CNNs, astX MLAO and XN MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The values shown are calculated from the ensemble of corresponding layers from all CNNs of the given class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' section 3 of the supplementary information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' By extracting relevant weight connections from inside the NN, we can observe embedded physical interpretations of how the machine learned to process aberration information contained in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' To illustrate this, we extracted from the trained NN the weights between the layer embedding physical interpretations and the next fully connected layer (marked by red arrows in Figure 1 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Going down the convolutional layers, the scale of probed features increases from a single pixel, through small scale features, up to large scale features (as explained in section 3 of the supplemental document).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The RMS values of the weights from each convolutional layer are shown in Table 1, where the data are shown for the ensembles of the two classes of MLAO networks used in this paper, astX MLAO and XN MLAO (where X =2 or 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A full breakdown is provided in the Figure S4 of the supplementary document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The largest weight variation was in the first layer in the XN MLAO NN, which indicates that this algorithm extracts more information from the single pixel detail than from larger scale correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In contrast, astX MLAO assigns weights more evenly across all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As explained in the supplementary document, the single pixel extraction from the pseudo-PSF is related to the Strehl ratio of the PSF and the intensity information of the images in non-linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hence, it is expected that the XN MLAO NN, which uses as similar set of bias aberrations to the conventional method, would learn as part of its operation similar behaviour to the conventional algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The same phenomena can also explain why in 3-P GCaMP imaging of neural activity astX MLAO was less affected by the fluorescence fluctuations than XN MLAO, as astX MLAO relies less on overall fluorescence intensity changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Conversely, astX MLAO generally performed worse than XN MLAO in 2-P imaging when structured noise present, as astX MLAO used fewer images and hence had access to less detectable intensity variations than XN MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The fact that astX MLAO had access to less well-conditioned image information may also explain why in general it was able to correct aberrations to a lower final level than XN MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Conclusion The MLAO methods achieved the aims explained at the outset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' They provided more efficient aberration correction with fewer images over a larger range, reducing time required and specimen exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The training procedure, which was based on synthesised data, ensured that the AO correction was robust to uncertainty in microscope properties, the presence of noise, and variations in specimen structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The concept was translatable across different microscope modalities, simply requiring training using a revised imaging model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The new methods used NN architectures that are orders of magnitude simpler, in terms of trainable parameters, than in previous similar work (see supplementary information, section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This vast simplification was achieved through pre-processing of data to remove most of the effects of unknown specimen structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The physics-informed design of the NN also meant that – unusually for most NN applications – the learned weights inside the network provided indications of the physical information used by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This provides constructive feedback that can inform future AO system designs and the basis for extension of the MLAO concept to more demanding tasks in microscopy and other imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Methods Image pre-processing Image data were pre-processed before being used by the NN, in order to remove effects of the unknown specimen structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The resulting “pseudo-PSFs” were better conditioned for the extraction of aberration information, independently of the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The image formation can be modelled as a convolution between specimen fluorescence distribution and an intensity PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The AO introduced pre-chosen bias aberrations, so that multiple images with different PSFs could be acquired over the same FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Mathematically, this process can be expressed as I1 = O∗ f1 +δ1 I2 = O∗ f2 +δ2 (1) where I1 and I2 were the images acquired with two different PSFs f1 and f2 for the same unknown specimen structure O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' δ1 and δ2 represent combined background and noise in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In order to remove (or at least reduce) the effects of specimen 10/16 structures, we defined the pseudo-PSF as pseudo-PSF = F −1 �F(I1) F(I2) � = F −1 �F(O∗ f1 +δ1) F(O∗ f2 +δ2) � = F −1 �F(O)×F( f1)+F(δ1) F(O)×F( f2)+F(δ2) � where F was the 2D Fourier transform and F −1 was its inverse (see Figure 1 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The term “pseudo-PSF” was chosen as the function was defined in the same variable space as a PSF, although it is not used directly in any imaging process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A similar computational process was shown elsewhere for different applications using defocussed images52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Assuming the noise is small enough to be neglected pseudo-PSF = F −1 �F(I1) F(I2) � ≈ F −1 �F( f1) F( f2) � (2) There is an implicit assumption here that there are no zeroes in the object spectrum F(O) or the optical transfer function F(f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In practice, it was found that a small non-zero value of F(δ2) mitigated against any problems caused by this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Furthermore, although structured noise was present in the pseudo-PSFs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure S1 in the supplemental document), it was found that this did not detrimentally affect data extraction through the subsequent NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As a further mitigation, we calculated pairs of pseudo-PSFs from pairs of biased input images by swapping the order from ( f1, f2) for the first pseudo-PSF to ( f2, f1) for the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Example pseudo-PSFs are shown in Figure S1 and S2 in the Supplemental document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As most information was contained within the central region, to ensure more efficient computation, we cropped the central region (32×32 pixels) of the pseudo- PSFs to be used as the input to the NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Dependent upon the MLAO algorithm, the input to the NN would consist of a single pair of cropped pseudo-PSFs, or multiple pairs corresponding to the multiple pairs of bias aberrations applied in different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Neural network training To estimate phase aberrations from pseudo-PSFs, a convolutional based neural network was designed incorporating physical understanding of the imaging process and was trained through supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Synthetic data were used for training and the trained networks were then tested on real AO microscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For each imaging modality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2-P, 3-P and widefield), a separate training dataset was generated, with the imaging model and parameters adjusted for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Neural network architecture A convolutional neural network was designed to determine the aberrations from pseudo-PSFs, while embedding physical understanding of image formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The conceptual structure is shown in Figure 1 (c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' more specific details of the architecture and learning process are provided in Section S1 of the supplementary document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This CNN architecture allowed convolutional masks to – in effect – probe different spatial scales within the pseudo-PSF images and, hence, to learn from the effects aberrations had at different spatial scales in microscope images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The outputs from these convolutional layers acted as inputs to a single concatenated fully connected layer (FCL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This was followed by another FCL then the output layer, whose outputs corresponded to the Zernike mode coefficients estimated for aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This shallow architecture with the order of 104 trainable parameters was effective due to the pre-processing of data that meant the input information was better conditioned to this estimation task than raw images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The weight connections between the concatenated FCL immediately following the CNN layer and the subsequent FCL (marked in red arrows in Figure 1 (c)) depended upon the significance of the information learnt from the different scales embedded in the CNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Analysis of these weights could therefore provide insight into the pseudo-PSF information that was used by the ML process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Synthetic data generation Due to the impracticality of acquiring sufficient high-quality data experimentally, a large dataset of simulated image data was constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The simulations were designed to resemble images collected from different microscopes when imaging a range of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We started with a collection of image stacks (containing around a total of 350 images) obtained from high-resolution 3D microscopy of various specimens labelled with nuclear, cytoplasmic membrane and/or single-molecule markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The images were down-sampled to 8-bit (128×128) and separated into their individual channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This formed a pool of realistic sample structures which were later used to generate synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' To further augment the varieties of sample structures, random rotations were applied and synthetic shapes including dots, rings, circular shapes, curved and straight lines of varying sizes were randomly introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 11/16 The simulated training dataset was generated by convolving the sample structures with synthetic PSFs, f (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' f was modelled as a pixel array through f = ���F � Pe j(Ψ+Φ+Ξ)���� l (3) where F represented the 2D discrete Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' P was the circular pupil function, defined such that pixels in the region outside the pupil had value zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The ratio between the radius of the pupil in pixels and the size in pixels of the overall array was adjusted to match sampling rates for different microscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In practical scanning optical microscopes, the sampling rates can be easily adjusted, although perhaps not arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hence, for experimental flexibility, the ratio for the simulated training dataset was tuned to be within the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='0× to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2× the base sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The base sampling rate was defined as using two pixels to sample the full width half maximum (FWHM) of the PSF of the system when aberration free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the widefield system, the ratio was tuned to simulate the projection of the camera pixel sampling rate at the specimen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure S5 in the supplemental document shows how tolerable a trained network was when tested on data collected at different pixel sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' P also incorporated the illumination profile for different practical imaging systems, such as when using truncated Gaussian illumination at the pupil in the 3-P microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The exponent l varied with imaging modes: when simulating a 3-P, a 2-P and a widefield microscope, l was set to 6, 4 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The total aberration was expressed as a sum of chosen Zernike polynomial modes Ψ+Φ+Ξ = ∑i aiZi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Ψ was the sum of the randomly generated specimen aberrations, which included all modes that the AO system was designed to correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Φ represented the additional bias aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Ξ included additional non-correctable higher order Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The coefficients of the correctable modes were randomly generated for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Representing the set of coefficients {ai} as a vector a, the random coefficients followed a modified uniform n-sphere distribution53 where both the direction and the two-norm of a were uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The maximum two-norm (size) of a were chosen differently for different imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This distribution allowed a denser population close to zero aberration, which was intuitively beneficial to train a stable NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We also added random small errors to the correctable coefficients so that the labels were slightly inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This was to simulate situations when the AO would be incapable of introducing perfect Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The spurious high order non-correctable Zernike modes were included to further resemble realistic scenarios in a practical microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Poisson, Gaussian, pink and structured noise of varying noise level were also introduced to the generated images after the convolution to allow the training dataset to simulate more closely real microscope images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Note that the scalar Fourier approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3 was chosen for simplicity, although more accurate, vectorial, high numerical aperture (NA) objective lens models could have been applied54–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Although the chosen model would deviate from high NA and vectorial effects, the main phenomena under consideration here – namely the effects of phase aberrations on PSFs and images – are adequately modelled by scalar theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Image quality metrics Different image quality metrics were defined for use as the basis for optimisation in conventional sensorless AO methods and as proxies to quantify the level of aberration correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' yI is an intensity based metric and can be used in non-linear imaging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It is defined as yI = � � I(x)d2x yF is a Fourier based metric and provides an alternative aspect to the intensity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It is defined as yF = � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1fmax<| f|<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6fmax |F[I(x)]|d2 f where F[I(x)] is the 2D Fourier transform of image I(x) from x domain to f domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' fmax is the maximum frequency limit of the imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1fmax < |f| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 fmax was selected such that most PSF related frequency information was included in the range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' yS is a sharpness metric that can be used for optimisation in widefield systems, where the other metrics are not practical, or applications with fluorescence fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It is defined as yS = � � nfmax<| f| m > n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This metric is defined as the ratio of higher to lower spatial frequency content, which is dependent upon aberration content, but independent of changes in overall brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 12/16 Microscope implementations Three microscopes were used to demonstrate and examine the MLAO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The microscope implementations are briefly described here and fully elaborated in the supplementary document section 9A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In the home built 2-P system, a Newport-Spectra-Physics DeepSee femtosecond laser was used as the illumination with wavelength set at 850nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Light was modulated by a Hamamatsu spatial light modulator before passing through a water immersion objective lens with NA equals to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='15 and reaching the sample plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A commercial Scientifica microscope system was used as the basis for our 3-P demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In the 3-P system, a Ti:Sapphire laser passed through a pair of compressors and operated at 1300nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Light was modulated by a Mirao 52E deformable mirror before reaching a water dipping objective lens with NA equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In the home built widefield 3D SIM system, two continuous wave lasers with wavelengths equal to 488 and 561nm were used as the illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Light was modulated by a ALPAO 69 deformable mirror before reaching a water dipping objective lens with NA of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Image acquisition and processing For 3-P imaging of live specimens, where motion was present, averaging was performed after inter-frame motion correction using TurboReg58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Time traces were taken from 200 raw frames captured at 4 Hz consecutively for each of the pre- and post-MLAO corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the widefield/SIM results, widefield images were processed where indicated using the Fiji iterative deconvolution 3-D plugin59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A PSF for deconvolution was first generated using the Fiji plugin Diffraction PSF 3-D with settings the same as the widefield microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the deconvolution, the following settings were applied: Wiener filter gamma equals to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' both x-y and z direction low pass filter pixels equal to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' maximum number of iterations equals to 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' and the iteration terminates when mean delta is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='01%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The thresholds shown on the widefield image spectra were calculated by identifying the largest frequency in all x-y directions with image spectrum components higher than noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The noise level was identified by averaging the components of the highest spectral frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' at the four corners of the image spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Starting from the lowest frequency, each angular and radial fragment was averaged and compared to the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The largest component which was still above the noise level was traced on the image spectra by the dashed line and identified as the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Each 3D-SIM frame were extracted from a set of 15 image frames using the SoftWorx package (Applied Precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='60 The projected images were obtained by summing frames at different z depths into an extended focus xy image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Booth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Adaptive optics in microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Transactions Royal Soc.' metadata={'source': 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2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Booth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=', Andrade, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=', Burke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=', Patton, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' & Zurauskas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} 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+page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='01004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Marsh, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=', Burns, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' & Girkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Practical implementation of adaptive optics in multiphoton microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Express 11, 1123–1130, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1364/OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='11.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Jesacher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Adaptive harmonic generation microscopy of mammalian embryos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 34, 3154–3156, DOI: 10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Facomprez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=', Beaurepaire, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' & D´ebarre, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Accuracy of correction in modal sensorless adaptive optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Opt.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 ‘Accessible adaptive optics and super-resolution microscopy to enable improved imaging’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' thesis, University of Oxford (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Xin, Q.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 0 in Aeroacoustics Conferences (American Institute of Aeronautics and Astronautics, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Gustafsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='120345 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Acknowledgements This work was supported by grants from the European Research Council (to MJB: AdOMiS, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 695140, to AMP: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 852765), Wellcome Trust (to MJB: 203285/C/16/Z, to ID and MJB: 107457/Z/15/Z, to AMP: 204651/Z/16/Z, to HA: 222807/Z/21/Z), Engineering and Physical Sciences Research Council (to MJB: EP/W024047/1), Author contributions QH and MJB conceived the overall physics-informed approach including data pre-processing and bespoke NN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' MH, QH and MJB developed NN architectures and the training approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' QH, MH, MW, JA and DS developed the software packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' JW, QH, AMP set up the microscopes for the experimental demonstrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' QH performed the two-photon experiments, supervised by MJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' HA, JW and QH performed the three-photon experiments, supervised by AMP and MJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' JW, MW, DS, QH and RMP performed the widefield/SIM experiments, for which DG, TC and RMP prepared specimens, supervised by ID and MJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' QH performed data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' QH and MJB wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' All authors reviewed the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Additional information All experimental procedures involving animals were conducted in accordance with the UK animals in Scientific Procedures Act (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 16/16 Universal adaptive optics for microscopy through embedded neural network control: supplemental document 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' MLAO PROCESS AND CNN ARCHITECTURE The MLAO aberration estimation process consists of two parts: image pre-processing to compute pseudo-PSFs from images and a CNN-based machine learning process for mode coefficient determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A stack of M images over the same field of view, each with a different pre- determined bias phase modulation, was used to calculate pseudo-PSFs according to the procedure in the methods section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It was observed and understood that most of the information was contained within the central region of the calculated pseudo-PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 1 A central patch of 32 × 32 pixels was then cropped and used as the inputs to the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Cropped pseudo-PSFs were processed by a sequence of convolutional layers (CL) with trainable 3 × 3 kernels, each followed by a local 2 × 2 max-pooling and thus the x and y sizes were reduced by half but the stack size was increased twice going down each CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the input pseudo-PSFs and each of the CL outputs, a global max-pooling was applied and concatenated into a fully connected layer (FCL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This concatenated FCL was connected to the next FCL containing 32 neurons, which in turn was connected to the output layer, which produced the coefficients of the N chosen Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The activation functions were chosen to be tanh and linear (only for the last layer connection FCL 32 and the output).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The regularizer used was L1L2, the initializer was glorot-uniform and the optimizer was AdamW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The CNN architecture was built and the network training was conducted using TensorFlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' [1] As elaborated in the results section of the manuscript, M and N may be varied to suit different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The weights in the connection between the concatenated FCL and FCL32 (enclosed by a grey dashed square) were extracted and analysed to understand the physical significance of structures in the pseudo-PSFs in influencing the learning of the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Further analysis of such weights is provided in Discussion of the main paper and section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 1The process of calculating pseudo-PSFs can be interpreted as a deconvolution between two PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Depending on the sampling size of the imaging system, most details of a deformed PSF typically occupy a central region of a few pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Most features of the pseudo-PSFs were thus captured within the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='02647v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='IV] 6 Jan 2023 Image stack over the same field of view (128×128×M) Cropped pseudo-PSFs (32×32×M) CL 16×16×8 CL 8×8×16 CL 4×4×32 CL 2×2×64 M 8 16 32 64 FCL 32 Concatenate FCL Output N Pseudo-PSF computation Convolution + local maxpooling Global maxpooling Fully connected layers Calculated pseudo-PSFs (128×128×M) CNN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A schematic illustration of the MLAO process and CNN architecture (enclosed by a black dashed square) designed for phase determination applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' CL: convolutional layer followed by local max-pooling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' FCL: fully connected layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' M: number of input images and computed pseudo-PSFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' N: number of estimated output Zernike modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' ZERNIKE POLYNOMIALS AND EXAMPLE PSEUDO-PSFS A total of ten Zernike polynomials were used for aberration estimation and correction presented in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A list of the polynomials, sequenced using Noll’s indices, were included in Figure S2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Figure S2 (b) included some examples of pseudo-PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It can be observed that when aberration size increases, the maximum pixel value of the Pseudo-PSF decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' a global max-pooling of the pseudo-PSF extracts information related to the Strehl ratio of the PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Pseudo-PSFs also have shapes that are related to the aberrated PSF shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' PHYSICAL INFORMATION EMBEDDED IN THE CNN ARCHITECTURE As mentioned in the main paper, the bespoke CNN architecture embedded information about the physical effects of aberrations on images within the trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' To illustrate these phenomena, we designed six input patterns and two filters to calculate how values obtained after global max-poolings from different convolutional layers were related to the features of the patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Normally, the filters would be learned as part of the training process, but for illustrative purposes, we have defined them manually here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' As shown in Figure S3, patterns 1 to 3 had the same general shape but varying sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' They were all convolved with the same filter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Pattern 1 had the largest feature and the values obtained were almost constant throughout layers 1 to 5 (see Figure S3 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Patterns 2 and 3 had smaller features and the extracted values reduced when moving further down the layers, where the embedded physical scales were more closely related to large scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Patterns 4 to 6 had the same general shape with four peaks positioned at the corners of a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' They were all convolved with filter 2, which shared a similar general shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Pattern 4 had the smallest feature size and resulted a largest value in layer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Patterns 5 and 6 had larger feature sizes and resulted in largest values in layers 3 and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This trend confirms the expectation that layers later in the CNN probe larger scales in the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Note that all the patterns were designed in such a way that the maximum pixel reading (and thus the value max-pooled from layer 1) equalled to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2 i = 5 astigmatism 8 coma 9 trefoil 10 trefoil 11 primary spherical 12 secondary astigmatism 13 secondary astigmatism 22 secondary spherical 6 astigmatism 7 coma (a) (b) 0 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad i = 7 ±1 rad i = 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad i = 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad i = 7 Aberration Bias Pseudo-PSF Aberration Bias Pseudo-PSF 0 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad i = 5 ±1 rad i = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8 rad i = 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad i = 5 0 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 0 π rad Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) Zernike polynomials Noll’s index 5-13, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This is a whole list of the polynomi- als used for aberration determinations in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) Examples of pseudo-PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The first column is the input aberration and the second column is the bias mode used in pseudo-PSFs generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3 ∗ filter1 ∗ filter2 Local max-pooling Convolution Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Pattern 1 Pattern 4 Pattern 5 Pattern 6 (a) (b) Pattern 2 Pattern 3 Layer 1-5 Normalised max-pooling value Pattern 1 2 3 4 5 6 Single pixel feature Small scale feature Large scale feature 0 1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3×3 3×3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Demonstrations of the link between feature sizes and convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) Pattern 1 to 6 each underwent a series of convolutions followed by a 2 × 2 local max-pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Pattern 1 to 3 were convolved with filter 1 and pattern 4 to 6 were convolved with filter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For each layer, a global max-pooling were carried out to extract the maximum reading of each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The physical interpretations of the extracted values of the different layers were related to Strehl ratio (layer 1) and shapes with features ranging from small scales (layer 2) to large scales (layer 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The extracted readings was normalised with the readings of their respective previous layer and displayed in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The horizontal axis of each plot in (b) indicates from which layer the normalised maximum reading (indicated by the vertical axis) was extracted from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 4 ast2 2-P 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 2N 2-P 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 ast4 3-P 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 4N 3-P 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 ast2 widefield 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 2N widefield 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 Layer 1-5 RMS of weights Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Analysis of the weight distributions across convolutional layers in the CNNs trained for different biasing schemes and microscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' WEIGHT ANALYSIS OF DIFFERENT TRAINED NEURAL NETWORKS Figure S4 shows the root-mean-square (RMS) values of the weights at the output of each section of the concatenated FCL following the convolutional layers of the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' These weights encode information about physical phenomena in the pseudo-PSF that is related to the spatial effects of aberrations on images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Higher numbered layers correspond to larger scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Similar distributions are seen for all of the ast CNNs class and all of the 2/4N class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Most notably, it can be seen that the 2/4N networks all carry heavier weights in layer 1, which is most similar to the Strehl ratio variations of the PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' TRAINABLE NEURAL NETWORK PARAMETERS The bespoke NN and data pre-processing steps were designed with knowledge of the physical basis of image formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This permitted signficant reduction in NN complexity compared to previous methods for aberration estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This architecture not only allowed improved performances, providing insights on internal workings, but also had a structure size orders of magnitude smaller than common NNs used in similar applications (see the comparison in Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This will be beneficial for future applications as NN with fewer trainable parameters would generally require less training data and a shorter training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Furthermore, the simplified design means that there is greater potential for extending the method to more challenging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 5 Neural network method Number of trainable parameters ResNet[2] >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='27M Inception V3/ GoogLeNet[3, 4] 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6M Xception[5, 6] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8M Deep Image Prior[7] 2M PHASENET[8, 9] 1M MLAO in this paper 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='028M to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='032M Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A list of NNs used in image processing and phase determination with their number of trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Inception V3[3], Xception[5] and PHASENET[8] have been directly demonstrated for phase determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' ResNet is a common basic NN architecture that has been used in many different image processing and phase determination architectures[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A 20 layer ResNet is the smallest architecture proposed in the ResNet paper[2] that has ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='27M trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Deep Image Prior employs a U-Net architecture that is a commonly used in many biomedical image processing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Deep phase decoder[10], a network designed for wavefront and image reconstruction, was also inspired and adapted from Deep Image Prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' CHOICE OF BIAS MODE The simplest MLAO implementation uses a pair of biased images as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The nature of the bias aberrations is a design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In order to investigate this, we tested individual Zernike modes as the bias and trained different MLAO networks with identical architecture to correct the same randomly generated aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The loss function of the different NNs during training was shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S5 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Results from correcting 20 randomly generated aberrations were shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S5 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 0 1k 2k 3k 4k 5k 6k 5 8 7 6 11 i= Training epochs RMS loss function 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 Aberration RMS / rad pre correction 4 5 6 7 8 11 Bias mode i (a) (b) 188nm/px 5μm aberration free aberration 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='88 rad Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Testing Zernike modes as choice of bias aberration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (a) A plot of the root mean square (RMS) loss function against the number of epochs when training NNs of the same architec- ture from the same dataset but using different bias modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (b) Statistical results of testing the trained NNs to correct the same sets of random aberrations over 2-P microscope images of beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Twenty randomly generated aberrations consisting five Zernike modes and RMS value smaller than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2 radians were introduced for correction (dark gray bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The remaining aberra- tions after correction by different networks were averaged and shown in the figure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' standard deviations of the remaining aberrations are represented as the error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Insets showed an example of the FOV when no aberration was introduced and an example when 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='88 rad of aberration was introduced into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The two networks using oblique and vertical astigmatism (index i =5 and 6) converged to similar loss function during training (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S5 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The same two networks also gave similar 6 averaged remaining aberrations during experimental aberration correction on a bead sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S5 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The two networks using vertical and horizontal coma (index 7 and 8) also showed mutually similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This was expected as these pairs of modes (5 and 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 7 and 8) differ only by rotation, which should not have an effect on how effective the networks determine aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' From these results, the NNs using astigmatism as the bias modes converged to the smallest loss function during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' This possibly suggested that the astigmatism modes, on average, allowed the network to learn more from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' It was also observed from the ex- perimental results where, in general, the NN obtained the smallest remaining aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We therefore chose to use astigmatism as the modulation modes for the two-bias NN methods in the experiments conducted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' TOLERANCE TO SAMPLING RATE As described in the paper, the networks for scanning microscopy were trained on simulated dataset with pixel sampling within the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='0× to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='2× of the base sampling rate (see the method section in the main paper for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' However in many practical cases, there can be uncertainty in pixel sampling for a system or constraints on the sampling rates that may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' We hence tested the tolerance of our networks to pixel sampling rates outside the range of the training dataset (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 0 1 2 3 4 pre correction 2N+1 conv ast2 MLAO 2N MLAO 219nm/px 5μm 188nm/px 5μm 156nm/px 5μm 5μm 125nm/px Aberration RMS / rad Image sampling rate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Testing of robustness to pixel sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Statistical results of remaining aberrations before (red plot) and after correction using 2N+1 conv, ast2 MLAO and 2N MLAO methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The results were averaged from 20 randomly generated aberrations and the SDs were shown as the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The same algorithms were used to correct the same aberrations over images collected at different pixel sampling as shown by the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Insets show examples of the images collected at different sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In this case, 188nm per pixel was close to the sampling of the generated dataset on which the two NNs were trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' When images were sampled at a smaller or larger rate, ast2 MLAO and 2N MLAO were still able to correct aberrations, but were slightly less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' FURTHER THREE-PHOTON MICROSCOPE DEMONSTRATIONS Figure S7 showed the performance of the ast4 MLAO algorithm, for imaging neuronal activity at a depth of 670 µm in a mouse brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Despite the very low SNR of the image data, the image quality and cell activity data were considerably improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' DETAILS OF THE EXPERIMENTAL METHODOLOGY Three optical systems, a 2-P, 3-P and widefield microscope, were used for demonstrations on different samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Networks with different parameter settings are also adjusted for different applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 7 vi A B C D E F G H Post MLAO it:3 Pre MLAO Post MLAO it:3 1 2 Post MLAO it:3 Pre MLAO Post MLAO it:3 iii 1 2 Pre MLAO 0 50 100 0 50 100 time / s time / s yS yI Images Images it:1 it:2 it:3 iv v it:2 it:1 20μm A B C D E F G H GCaMP at 670 μm vii Pre MLAO i 1 rad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 rad +1 rad ast4 MLAO 1 3 2 4 Bias mode i=5 ii Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Three-photon microscopy imaging GCaMP neuronal activities at depth 670µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Power at sample was 44 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Wavefronts inserted to the figures showed the phase modulations ap- plied by the DM at the relevant step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' the common scale is indicated by the colorbar above v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' i and iii show respectively before and after ast4 MLAO correction through three iterations (it:1 to 3), 200 frame averages after motion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' In iii, time traces shown to the right and bottom were taken from the marked lines (1) and (2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' ii 1-4 shows example single- frame images used as inputs to the ast4 MLAO correction with the corresponding bias modes as insets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' iv and v show the intensity metric (yI) and the sharpness metric (yS), respectively, calculated from single image frames, against the number of images acquired for three iterations ast4 MLAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' vi shows the Calcium activity of 8 cells (A-H marked on i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FS laser ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='HWP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='PBS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Dump ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Galvo x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Galvo y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='DF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='EF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='PMT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='PZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='SLM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='f200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Obj:W-I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Two-photon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='microscope ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='DF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='DF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='FM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Configuration of the (a) 2-P (b) 3-P (c) widefield 3-D SIM microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' (Caption contin- ued on the next page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=') 9 Femtosecond (FS) Laser;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Continuous-wave lasers with wavelenths 488nm and 561nm (LS488 and LS561);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' half wave plate (HWP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' polarisation beam splitter (PBS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' laser beam dump (Dump);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' lens with focal length = x mm (fx);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' broadband dielectric mirror (M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' flip mirror (FM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Hamamatsu spatial light modulator (SLM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Mirao 52E deformable mirror (DM) in the 3-P system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' ALPAO 69 deformable mirror (DM) in the widefield 3-D SIM system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' aperture (AP);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' spatial filter (SF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' field stopper (FS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' X galvanometer (Galvo x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Y galvanometer (Galvo y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' beam expansion (BX);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' half waveplate (λ/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' linear polariser (PL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' polarisation rotator (PR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Olympus 40× numerical aperture (NA) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='15 water immersion objective lens (Obj:W-I) used in the 2-P system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Nikon 16× NA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='8 water dipping objective lens (Obj:W-D) used in the 3-P system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Olympus 60× NA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='1 water dipping objective lens (Obj:W-D) in the widefield 3-D SIM system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Z-piezo translation stage (PZ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' X-Y-Z translational sample mounting stage (ST);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Dichroic filter (DF) allow emission signal from fluorophores to be reflected through emission filter (EF) into a photo-multiplier tube (PMT) in a multi-photon system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' cameras (C1 and C2) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Sample preparation The 3-P results were collected from imaging male (Lhx6-eGFP)BP221Gsat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Gt(ROSA)26Sortm32(CAG- COP4*H134R/EYFP)Hze mice (static imaging) and female and male Tg(tetO-GCaMP6s)2Niell mice (calcium imaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Mice were between 8-12 weeks of age when surgery was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The scalp was removed bilaterally from the midline to the temporalis muscles, and a metal headplate with a 5 mm circular imaging well was fixed to the skull with dental cement (Super-Bond C&B, Sun-Medical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A 4–5 mm circular craniotomy was performed during which any bleeding was washed away with sterile external solution or staunched with Sugi-sponges (Sugi, Kettenbach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Cranial windows composed of 4 or 5 mm circular glass coverslips were press-fit into the cran- iotomy, sealed to the skull by a thin layer of cyanoacrylate (VetBond) and fixed in place by dental cement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The widefield 3-D SIM results were collected from imaging NMJ of Drosophila larvae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the immunofluorescence sample with one coloured channel, it was prepared as previously [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Crawling 3rd instar larvae of wildtype Oregon-R Drosophila melanogaster were dissected on a Sylgard-coated Petri Dish in HL3 buffer with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='3mM Ca2+ to prepare larval fillet [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Then, the larval fillet samples were fixed in Paraformaldehyde 4% in PBS containing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='3% (v/v) Triton X-100 (PBSTX) for 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The brains were removed post-fixation, and the fillet samples were transferred to a Microcentrifuge tube containing PBSTX for 45 minutes of permeabilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The samples were stained with HRP conjugated to Alexa Fluor 488 and DAPI for 1 hour at room temperature (21C◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' After the washes, the samples were mounted in Vectashield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' For the 3-D SIM results collected on the Drosophila larvae sample with two coloured channels, it was prepared by following the protocol presented in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3rd instar Drosophila melanogaster larvae (Brp-GFP strain) were dissected in HL3 buffer with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='3mM Ca2+ to prepare a so-called larval fillet, and the larval brains were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' After this, larvae were stained for 15 minutes with HRP conjugated to Alexa Fluor 568 to visualise the neurons, washed with HL3 buffer with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='3mM Ca2+ and imaged in HL3 buffer without Ca2+ to prevent the larvae from moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Network parameters Table S2 showed the network settings used in different imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 10 Results in Method label M N Bias Bias Corrected modes, i depths modes, i Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2 (a, c, f) ast2 MLAO 2 5 5 ±1 rad 5–8, 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2 (a, c, f) 2N MLAO 10 5 5–8, 11 ±1 rad 5–8, 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2 (b, d, e) ast2 MLAO 2 9 5 ±1 rad 5–13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2 (b, d, e) 2N MLAO 18 9 5–13 ±1 rad 5–13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3 (a, b) ast4 MLAO 4 7 5 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 5–11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' S4 ±1 rad Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 3 (a) 4N MLAO 28 7 5–11 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='5 5–11 ±1 rad Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 4 ast2 MLAO 2 8 5 ±1 rad 5–11, 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 4 2N MLAO 2 8 5–11, 22 ±1 rad 5–11, 22 Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' A list of MLAO parameters chosen for different imaging applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' The Zernike modes were sequenced using Noll’s indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} 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Preparation and Imaging of Neurons,” Cold Spring Harb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' Protoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 2010, pdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content='prot5405 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE0T4oBgHgl3EQfxgKB/content/2301.02647v1.pdf'} diff --git a/D9FQT4oBgHgl3EQfQDZ4/content/tmp_files/2301.13281v1.pdf.txt b/D9FQT4oBgHgl3EQfQDZ4/content/tmp_files/2301.13281v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..81eff27927f9e878de912cc4064cf96c29c2aca0 --- /dev/null +++ b/D9FQT4oBgHgl3EQfQDZ4/content/tmp_files/2301.13281v1.pdf.txt @@ -0,0 +1,3019 @@ +MNRAS 000, 1–19 (2023) +Preprint 1 February 2023 +Compiled using MNRAS LATEX style file v3.0 +Short Timescale Evolution of the Polarized Radio Jet during V404 Cygni’s +2015 Outburst +A. K. Hughes,1★ G. R. Sivakoff,1 C. E. Macpherson,2 J. C. A. Miller-Jones,2 A. J. Tetarenko,3,4†, +D. Altamirano,5 G. E. Anderson,2 T. M. Belloni,6 S. Heinz,7 P. G. Jonker8,9, E. G. Körding9, +D. Maitra10, S. B. Markoff11,12, S. Migliari13,14, K. P. Mooley15,16, M. P. Rupen17, D. M. Russell18, +T. D. Russell19, C. L. Sarazin20, R. Soria21,22,23, and V. Tudose24 +Affiliations are listed at the end of the paper +Accepted 2023 January 25. Received 2023 January 25; in original form 2022 October 4 +ABSTRACT +We present a high time resolution, multi-frequency linear polarization analysis of Very Large Array (VLA) radio observations +during some of the brightest radio flaring (∼1 Jy) activity of the 2015 outburst of V404 Cygni. The VLA simultaneously +captured the radio evolution in two bands (each with two 1 GHz base-bands), recorded at 5/7 GHz and 21/26 GHz, allowing for +a broadband polarimetric analysis. Given the source’s high flux densities, we were able to measure polarization on timescales of +∼13 minutes, constituting one of the highest temporal resolution radio polarimetric studies of a black hole X-ray binary (BHXB) +outburst to date. Across all base-bands, we detect variable, weakly linearly polarized emission (<1%) with a single, bright peak +in the time-resolved polarization fraction, consistent with an origin in an evolving, dynamic jet component. We applied two +independent polarimetric methods to extract the intrinsic electric vector position angles and rotation measures from the 5 and +7 GHz base-band data and detected a variable intrinsic polarization angle, indicative of a rapidly evolving local environment or +a complex magnetic field geometry. Comparisons to the simultaneous, spatially-resolved observations taken with the Very Long +Baseline Array at 15.6 GHz, do not show a significant connection between the jet ejections and the polarization state. +Key words: black hole physics — ISM: jets and outflows — polarization — radio continuum: stars — stars: individual (V404 +Cygni, GS 2023+338) — X-rays: binaries +1 INTRODUCTION +A black hole X-ray binary (BHXB) is an interacting binary sys- +tem composed of a stellar-mass black hole accreting material from +a companion star. Standard features of BHXBs are jets and winds, +making them ideal candidates for the study of accretion-fed outflows. +The majority of these systems spend most of their lifetimes in quies- +cence, accreting small amounts of matter, at low X-ray luminosities +(𝐿𝑋 ≲ 1032 erg s−1). Most of the known systems sporadically enter +into bright (𝐿𝑋 > 1035 erg s−1), transient outbursts that last weeks to +years (e.g., Tetarenko et al. 2016), allowing for real-time observations +of the evolving accretion flow (best measured at X-ray frequencies; +e.g., Belloni et al. 1999; Tomsick et al. 2004; Kylafis et al. 2012; +Plant et al. 2014) and relativistic jets (best measured at radio through +infrared frequencies; e.g., Corbel & Fender 2002; van der Horst et al. +2013; Russell et al. 2015; Tetarenko et al. 2015a). +During an outburst, the morphological evolution of the jet closely +correlates with the X-ray properties (i.e., accretion states; e.g., Mc- +Clintock & Remillard 2006; Belloni 2010; Fender 2010). In the +hard accretion state, an optically thin X-ray corona dominates the +X-ray emission and the jet adopts a steady, compact structure. In +★ E-mail: hughes1@ualberta.ca +† NASA Einstein Fellow +some systems, the compact jet is observed to persist into quies- +cence (e.g., Gallo et al. 2006; Plotkin et al. 2016, we note that, +for many BHXBs, jets in quiescence will have flux densities be- +low the detection capabilities of most facilities). Compact jet spectra +are described by optically-thick, partially self-absorbed synchrotron +emission with an inverted or flat spectral index (𝛼 ≳ 0; with flux den- +sity 𝐹𝜈 ∝ 𝜈𝛼) up to a break frequency (typically in the sub-mm or +infrared regime) where the spectra become optically thin (𝛼 ∼ −0.6) +to higher frequency emission (Russell et al. 2013). The flat/inverted +spectral index is thought to result from the superposition of mul- +tiple spatially-unresolved synchrotron components originating from +different positions along the jet axis (Blandford & Königl 1979). +Conversely, in the soft accretion state, thermal emission from the +accretion disk dominates the X-ray spectrum, and the radio emission +from the compact jet decreases or is fully quenched (e.g., Russell et al. +2011, 2020). During the hard-to-soft transition, one or more blobs +of discrete jet ejecta are typically launched, and these ejecta have +been spatially resolved in several sources (e.g., Mirabel & Rodríguez +1994; Hjellming & Rupen 1995; Hannikainen et al. 2001; Rushton +et al. 2017; Miller-Jones et al. 2019). +The ejection events are attributed to brief periods of highly efficient +plasma production at the base of the jet, creating (often adiabatically) +expanding plasma knots threaded with complex magnetic fields (e.g., +the van der Laan —vdL— model, van der Laan 1966; Hjellming +© 2023 The Authors +arXiv:2301.13281v1 [astro-ph.HE] 30 Jan 2023 + +2 +A. K. Hughes et al. +& Johnston 1988; Hjellming & Han 1995, and references therein). +Each ejection has an emission spectrum characterized by a single +self-absorbed synchrotron source with a temporally evolving elec- +tron/lepton population. As the ejection propagates and expands, the +self-absorption turnover transitions to lower frequencies, extending +deep into the radio regime and resulting in an observing bandwidth +that is optically thin in its entirety (e.g., Curran et al. 2014, 2015; +Williams et al. 2020). In the radio, these ejections are observed as +multi-frequency flares with well-defined rise and decay phases that +last minutes to days. Due to the expansion-driven, evolving optical +depth, the lower frequency components are broadened in time and +temporally delayed with respect to the higher frequency counterparts +(Mirabel et al. 1998). Flaring events from Active Galactic Nuclei +(AGN, the large-scale analogous of BHXBs) have also been mod- +elled based on the adiabatic expansion of jet plasma (Yusef-Zadeh +et al. 2008; Falcke et al. 2009; Maitra et al. 2009; Ball et al. 2021; +Michail et al. 2021). +Alternative flaring models can also be applied to BHXB observa- +tions, such as the “shock-in-jet” picture that is typically associated +with AGN (e.g., Marscher & Gear 1985; Spada et al. 2001). Within +this framework, each flare is the result of shocks within a (quasi-) +steady jet accelerating particles and temporarily enhancing emission +intensities (e.g., Fender et al. 2004; Türler et al. 2004; Türler 2011; +Malzac 2013). +Most radio jets from BHXBs are described by their photometric, +spectral, and (when available) spatial properties. However, a much +smaller fraction of studies explore the linear polarization that results +from a synchrotron dominated emission spectrum. For optically thin +and optically thick synchrotron emission, the maximum expected lin- +ear polarization fraction is 𝑓𝜆 = (3𝑝 + 3)/(3𝑝 + 7) × 100 % ≈ 70 % +and 𝑓𝜆 = 3/(6𝑝 + 13) × 100 % ≈ 10 %, respectively (assuming a +uniform magnetic field and adopting a typical value for the electron +energy distribution index, 𝑝 = 2.2; Ginzburg & Syrovatskii 1969; +Longair 2011). Complex or evolving magnetic fields, disadvanta- +geous lines of sight, Faraday depolarization, and the superposition of +multiple components are a few mechanisms that can depolarize the +observed radio emission. BHXBs with polarimetric radio analyses +typically have linear polarization fractions ≲ 10% with a rare few +reaching ∼ 50% (e.g., Han & Hjellming 1992; Hannikainen et al. +2000; Fender 2003; Brocksopp et al. 2007, 2013; Curran et al. 2014, +2015). The polarization fraction measures how “ordered” the local +magnetic field is (or appears to be), while the direction of the ob- +served electric vector position angle (EVPA) is a measure of the local +absorption conditions, jet position angle, magnetic field orientation, +as well as Faraday rotation between the emission and the observer. +After measuring and removing the effect of the Faraday rotation, the +derived intrinsic EVPA can provide an indirect measure of the jet +orientation (e.g., Curran et al. 2014; Russell et al. 2015). In cases +where polarized emission is combined with spatially resolved, total +intensity observations, polarimetry can directly probe the underlying +magnetic field strength and orientation (e.g., Stirling et al. 2004). +Despite the established observational relationship between the X- +ray and radio properties (i.e., the accretion flow and relativistic jet), +the physical mechanisms responsible for the launching and evolution +of jets are yet to be fully understood. Most theories recognize that +the local magnetic fields (and their disk/black-hole interactions) play +an essential role in extracting energy from the black hole/accretion +disk (e.g., Blandford & Znajek 1977; Blandford & Payne 1982) and +the initial launching and collimation of relativistic jets (e.g., Vla- +hakis & Königl 2004; Komissarov et al. 2007; Mignone et al. 2010). +These highly energetic processes can leave imprints on the evolving +magnetic fields, making time-resolved radio polarimetry, particu- +larly around BHXB ejection events, a valuable (yet underutilized) +tool. Outbursts from BHXBs occur at a moderate frequency (e.g., +several times a year; Tetarenko et al. 2016), with a rare subset (e.g., +once per decade) achieving X-ray luminosities near (or exceeding) +the Eddington luminosity, and Jansky level radio flux densities (e.g., +V404 Cygni’s 1989 outburst reached 1.6 Jy at 4.9 GHz; Oosterbroek +et al. 1996; Han & Hjellming 1992). During highly luminous out- +bursts, we can study accretion and accretion-rooted phenomena with +extraordinary levels of detail, capturing, in real-time, jet ejections at +flux densities that allow for a refined spectral and temporal resolution +for both total intensity and polarimetric observations. On 2015 June +15, the BHXB V404 Cygni (henceforth V404 Cyg) began one of +these rare outbursts. +1.1 V404 Cygni +First discovered in 1989, V404 Cyg (also known as GS 2023+338) +is a low-mass transient BHXB that has undergone four recorded +outbursts. Of the four outbursts, two were caught in real-time; +the initial discovery with the Ginga satellite (Makino 1989) and +the most recent outburst discovered by the Burst Alert Telescope +aboard the Neil Gehrels Swift Observatory (Barthelmy et al. 2015). +Searches through historical photo plates identified that there were +additional outbursts in 1938 and 1956 (Richter 1989). Observa- +tions of the main-sequence companion star revealed an orbital +period of 6.4714 ± 0.0001 days and a binary mass function of +𝑓 = 6.08 ± 0.06 𝑀⊙ = 𝑀3 +BH sin3 𝑖/(𝑀BH + 𝑀donor)2, where 𝑀BH +and 𝑀donor are the masses of the black hole and donor, respectively, +and 𝑖 is the orbital inclination angle (Casares & Charles 1994). The +K spectral type of the companion star, coupled with near-infrared +spectroscopy (and modeling of the H-band ellipsoidal modulations), +infer a BH mass of 9.0+0.2 +−0.6 𝑀⊙ with a best fit orbital inclination angle +of 67+3 +−1 +◦ (Khargharia et al. 2010). The modelled orbital inclination +angle assumes that the optical light curve of the companion star has +≲7% contamination from accretion disk (or jet) emission. However, +V404 Cyg has exhibited optical variability in quiescence (e.g., Zu- +rita et al. 2003; Bernardini et al. 2016) and, as a result, may have a +larger contamination fraction, larger inclination angle, and smaller +black hole mass. On the other hand, narrow emission lines suggest +that V404 Cyg has a low inclination angle 𝑖 < 40◦ (Casares et al. +1993) and a higher black hole mass. This uncertainty suggests the +mass of the black hole is not yet accurate at the 2–7% precision level +quoted above. High angular resolution radio parallax measurements +determined a source distance of 2.39 ± 0.14 kpc (Miller-Jones et al. +2009), making it one of the closest known BHXBs and a superb +laboratory for the study of accretion physics. +During the 1989 outburst, Han & Hjellming (1992) monitored the +radio emission of V404 Cyg between 1989 May 30 and 1991 May 31. +The monitoring began when the radio light curves were dominated +by the tail of a rapidly decaying (decay timescales of ∼5 days) “major +synchrotron bubble event". At later times, the radio light curves were +dominated by a slowly-decaying, nonthermal, optically thick source +(e.g., a compact jet) that lasted hundreds of days. Linear polarization +was detected for the first 50 days of observations, except for the first +observation on 1989 May 30 which did not include adequate polar- +ization calibration. During the decay of the synchrotron bubble, the +polarization fraction was a few tenths of a percent, before increas- +ing to a few percent during the period when the slowly-decaying +component dominated the radio emission. +On 2015 June 15, V404 Cyg entered its fourth recorded outburst, +and a follow-up campaign showed bright multi-wavelength flaring ac- +MNRAS 000, 1–19 (2023) + +V404 Cyg’s rapidly evolving polarized jet +3 +tivity in radio through X-ray wavelengths (e.g., Mooley et al. 2015; +Motta et al. 2015a,b; Tetarenko et al. 2015b,c; Gandhi et al. 2016; +Maitra et al. 2017), and rapid (∼ 15 s) transitions between accretion +states (see, Kajava et al. 2020, and references therein). With radio- +through-optical flux densities reaching ∼ Jy levels, V404 Cyg became +the brightest BHXB outburst observed in the last decade, character- +istic of a high, near-Eddington accretion rate, and its close proximity. +The source remained in outburst until the end of June, from which it +began decaying, eventually reaching quiescence in mid-August (with +the source having a brief period of renewed activity in 2015 Decem- +ber through 2016 January; Plotkin et al. 2017; Muñoz-Darias et al. +2017). +The MASTER Global Robotic Net detected three linear polariza- +tion “events” using their optical telescope network (Lipunov et al. +2016, 2019). In both events, the source exhibited a significant in- +crease in the linear polarization fraction, following a (total intensity) +flare, followed by a rapid decrease in the linear polarization frac- +tion during the rise of another flare. The authors favoured a model +where decreased X-ray irradiation of the secondary also decreased +its optical brightness. In turn, this makes it easier to detect the po- +larized non-thermal emission from the jet. The authors favoured this +model after having discarded the potential that the jet orientation +varied on timescales of tens of minutes; however, a rapid, variable jet +orientation was later confirmed (see below and Miller-Jones et al. +2019). +Shahbaz et al. (2016), detected another linear polarization flare +using observations with the Nordic Optical Telescope. This flare +occurred during a steady rise of optical flux, and preceded some of +the brightest optical flaring of the entire outburst. Moreover, the flare +preceded the start of a bright radio flare. These authors proposed +that the increase in linear polarization could result from multiple +ejecta collisions establishing a dominant magnetic field direction +perpendicular to the jet axis, and may be the signature for the birth +of the ejection that produced the subsequent radio flare. +During this same outburst, modeling of radio-through-sub-mm +observations (Tetarenko et al. 2017) and Very Long Baseline Ar- +ray (VLBA) observations (Miller-Jones et al. 2019) uncovered short +time-scale flaring of the jet. Here, the jet ejecta account for most, +if not all, of the observed flaring. However, we note that detailed +modelling of the X-ray emission suggested that the source may have +been continuously accreting at an Eddington accretion rate during +the brightest phase of the 2015 outburst (Muñoz-Darias et al. 2017). +As a result, the jet ejecta in V404 Cyg are not clearly associated with +the same discrete-jet launching process that occurs during hard-to- +soft state transitions in BHXBs, which occur around lower accretion +rates (Fender et al. 2004). The VLBA observations directly resolved +several of these ejection events on top of a continuous (but vari- +able) emission from an empirically defined unresolved radio core. +The position angle (PA) of the ejecta varied rapidly with time, a +phenomenon that was attributed to the Lense-Thirring precession of +the accretion disk (Miller-Jones et al. 2019). Although the authors +constrained the period to less than 2.6 hr hours, the rapidly varying +PAs suggested that the true period was substantially shorter. +In this paper, we add to the detailed radio analysis of the 2015 +outburst detailed in Tetarenko et al. (2017) and Miller-Jones et al. +(2019). Here, our primary focus is the extraction and analysis of +V404 Cyg’s (radio) polarization properties — derived from National +Science Foundation’s Karl G. Jansky Very Large Array (VLA) ob- +servations — during some of the outburst’s brightest flaring activity +on 2015 June 22. The remainder of this paper is structured as follows; +in Section 2, we introduce our observation and analysis procedure, +while in Sections 3 and 4, we present and discuss our results. Finally, +we summarize our findings in Section 5. +2 OBSERVATIONS AND ANALYSIS +2.1 VLA Data Reduction +The details of the primary VLA observations were first discussed +in Tetarenko et al. (2017). V404 Cyg was observed with the VLA +(Project Code: 15A-504) on 2015 June 22 with scans on source +between 10:37:24 and 14:38:39 UTC in the 4–8 GHz and 18–26 GHz +bands. All observations were made with an 8-bit sampler, comprised +of two base-bands, with eight spectral windows of sixty-four 2 MHz +channels each, giving a total (unflagged) bandwidth of 1.024 GHz +per base-band. Henceforth, we will refer to each base-band by its +characteristic frequency values of (∼) 5, 7, 21, and 26 GHz. +The array was in its most extended A configuration, and was split +into two sub-arrays of 14 (sub-array 1) and 13 (sub-array 2) an- +tennas. Sub-array 1 observed the sequence (5/7 GHz)-(21/26 GHz)- +(5/7 GHz), while sub-array 2 observed the sequence (21/26 GHz)- +(5/7 GHz)-(21/26 GHz). Both sub-arrays cycled between V404 Cyg, +observed for 88 s per cycle flanked by 32 s observations of a nearby +gain calibrator (J2025+3343). A second epoch was observed during +the source’s return to quiescence, taken on 2015 July 2, with scans +on source from 10:31:08 to 14:01:32 UTC. The observing bands +and sub-array schemes remained consistent with the primary June +22 observations (Tetarenko et al. 2019). We also analyzed 5 epochs +taken between July 11 and August 5, during the source’s return to +quiescence (Project Code: SG0196; Plotkin et al. 2016), although we +were unable to detect any polarized signal in these latter observations +(see Section 3.2). +We applied standard flagging and calibration to the Stokes 𝐼 (i.e., +total continuum flux density) data using the Common Astronomy +Software Application package (casa v5.6; McMullin et al. 2007). We +used 3C48 (0137+331) as a flux and absolute (linear) polarization an- +gle calibrator, J2025+3343 as a complex gain (aka phase) calibrator, +and J2355+4950 as an unpolarized leakage calibrator for both sub- +arrays. Due to V404 Cyg being weakly polarized, we grouped our po- +larization calibration solutions on 16 MHz (8 channel) intervals. For +our Stokes 𝐼 flux calibration model, we used the default casa model +repository (Perley & Butler 2017). However, the standard calibration +routine for Stokes 𝑄 and 𝑈 (i.e. linearly polarized flux densities) +assumes that the polarization calibrators are point sources. Since we +observed with the VLA in its most extended configuration, 3C48 was +resolved. The “degree” of resolution ranges from a slightly extended +Gaussian at 5/7 GHz to multiple distinct components at 21/26 GHz. +As a result, we constructed a spatially resolved model image for +each observing band. Our model contained information on all four +Stokes parameters, assuming no circular polarization, and adopted +the spatial distribution of the Stokes 𝐼 repository models. A detailed +description of our polarized model image can be found in Appendix +B. We note that the spatial distribution of the flux densities may +differ between Stokes 𝐼, and Stokes 𝑄/𝑈 (i.e. linearly polarized flux +densities), and, as a result, our measured polarizations are suscepti- +ble to systematic calibration errors; in particular, for the 21/26 GHz +basebands, where our calibrator is significantly resolved. +2.2 Imaging +Since V404 Cygni was expected (and found) to be unresolved re- +gardless of the chosen visibility weighting, we applied a natural +MNRAS 000, 1–19 (2023) + +4 +A. K. Hughes et al. +𝑢𝑣-weighting scheme to all of our images, maximizing sensitivity. +Moreover, since we also knew that the Stokes 𝐼 flux density was +rapidly changing, we generated our analysis images in Stokes 𝐼, 𝑄, +and 𝑈, on short timescales of 12 or 14 min (6 or 7 scans; i.e., ∼8 +or 9.5 min on source)1. These timescales, some of the shortest ever +used in a radio polarimetric analysis of a BHXB, balance cadence +and polarized sensitivity. +We used the wsclean package (Offringa et al. 2014) to make all of +our polarimetric images. We imaged each base-band independently, +as well as Stokes 𝐼 separately from 𝑄 and 𝑈. In each base-band +for every time-bin we had wsclean output a set of images across a +user-set number of channels, as well as a single “multi-frequency- +synthesis” (MFS) image that stacks all the individual channels. We +measured the linear polarization intensities (𝑃 = +√︁ +𝑄2 + 𝑈2) for each +base-band/time-bin pair from the MFS images. +Any observed EVPA at an arbitrary observing wavelength 𝜆, is +related to the intrinsic EVPA, 𝜒0, through the linear relationship, +𝜒(𝜆) = 𝜒0 + RM · 𝜆2. The slope (i.e., the rotation measure, RM) +quantifies the wavelength-dependent Faraday rotation of an EVPA +due to linearly polarized light propagating through a magneto-ionic +plasma. Since our observables are 𝜒 and 𝜆, the largest detectable +rotation measure is inversely proportional to the 𝜆2 channel spacing. +The linearly-spaced frequency channels result in a 𝜆2 channel density +that increases with increasing central frequency. To avoid potential +biasing of results by the higher frequency observations, we scaled +the imaging frequency bins used for rotation measure analysis to +maintain a (roughly) constant 𝜆2 channel spacing; this resulted in a +frequency-space channelization of 16 MHz for the 5 GHz baseband, +and 64 MHz for the 7 GHz baseband. Due to their large temporal +delays (≳ 30 min) with respect to the 5/7 GHz base-bands, we chose +to omit the 21/26 GHz base-bands from the rotation measure anal- +ysis. The omission will minimize the overlap of optically thick and +optically thin emission, as well as any overlap of emission from dif- +ferent jet components (see Appendix A for a more comprehensive +motivation behind the omission). As a result, we did not scale the +frequency binning any broader than 64 MHz. +The larger Stokes 𝐼 flux densities allowed us to image the total +flux density light curves on much shorter timescales (∼10 s) than is +required for accurate polarimetry. For each spectral window, we pro- +duced a high time-resolution light curve, using the publicly available2 +imaging scripts detailed in Tetarenko et al. (2017). These images crit- +ically allow us to compare the simultaneous Stokes 𝐼 flux density, +and linear polarization evolution. +We observed an elevated rms noise in each image when compared +to the predicted values (see Table 1 for a summary). These effects +are most significant in the Stokes 𝐼 images and appear to worsen +at higher central frequencies and when larger frequency ranges are +used to create a single image. Therefore, we implemented a phase +self-calibration routine to explore if the elevated Stokes 𝐼 noise is +biasing the polarimetric results. Our self-calibration routine was bro- +ken into three steps that refined the phase calibration solutions on +progressively shorter timescales: first, half the length of a source +scan, 44 s; then a quarter, 22 s; and ending with solutions on the in- +tegration timescale, 2 s. We excluded amplitude self-calibration due +to the known Stokes 𝐼 variability within our imaging intervals. Al- +though the phase self-calibration improved the Stokes 𝐼 rms noise, +1 We made the time bins a variable integer number of scans to avoid com- +bining scans from different sub-arrays. We note that the ∼20% difference in +on-source time has a negligible effect on the analysis. +2 https://github.com/Astroua/AstroCompute_Scripts +we were unable to reach the theoretical limit expected from ther- +mal noise. This result is not unexpected: (i) we are averaging over +variable emission (spectrally and temporally) during our imaging +routines; (ii) the reduction in baseline coverage due to the division +into sub-arrays coupled with the bright emission is expected to limit +the dynamic range; (iii) completely automated self-calibration, like +we employ, can have difficulties achieving high dynamic ranges; (iv) +we only image a ≈51′′ × 51′′ field-of-view, and there can be some +added noise across the entire image due to our nearby phase calibrator +(approximately 16.6′from V404 Cyg) — our primary beams range +from 1.6–8.9′, leading to noise that would be stronger in our lower +frequency basebands. Since the self-calibration and its reduction of +the Stokes 𝐼 rms had a negligible effect on both the noise of the +Stokes 𝑄 and 𝑈 images and our measured polarimetric parameters, +we are confident that the elevated noise is not a significant issue. For +the remainder of this analysis, we have adopted our self-calibrated +results. +2.3 Flux Density Extraction +We measured the Stokes 𝐼 flux densities and linear polarization inten- +sities (from the MFS images) from an image plane analysis using the +casa task imfit. We fit an elliptical Gaussian component in a small +sub-region around the source, fitting for the position, flux density, +and shape of the component. Due to the source’s weakly polarized +emission, at fine spectral resolutions (e.g., the 16 MHz channeliza- +tion), the Stokes 𝑄 and 𝑈 flux densities are similar in magnitude +to (or weaker than) the local peaks in the rms noise (see Appendix +D). Often our attempts to freely fit the Stokes 𝑄 and 𝑈 images us- +ing imfit did not converge or converged on artificial noise signals. +Therefore, we decided to fix the shape of the component, and only fit +for the flux density in the region (i.e., we performed forced aperture +photometry). We set the component shape to be the synthesized beam +of each image, and used the position of the 𝑃 peak (for each time bin +and base-band) as the position of our aperture. We extracted the rms +of each image using a large annular region centred on the source. To +check for bias by a non-zero background we subtracted the mean flux +density in the rms region from the flux density of the source. The +background subtraction had a negligible effect on our results. +The fine (spectral) resolution images uncovered anomalous chan- +nels (∼1–2 per time bin) that were missed during flagging and cali- +bration, or corrupted during imaging. We apply a 𝜎-clipping routine +to remove these channels from the Stokes 𝐼 spectrum of each time +bin. After constructing a model spectrum by passing our Stokes 𝐼 +data through a narrow Gaussian filter with 𝜎 = 2.5 data points, +corresponding to 5 and 20 MHz at 5 and 7 GHz, respectively, any +flux density point that was > 3 residual standard deviations from the +model spectrum was flagged. We continued the routine until the frac- +tional difference in residual standard deviations between the current +and previous iteration was ≤ 0.001%. The channels removed from +the total intensity spectra were recorded and subsequently removed +from the 𝑄 and 𝑈 spectra. No further data manipulation was applied. +2.4 Polarization Properties +We derived all polarization properties from the flux densities ex- +tracted during image plane analysis. The polarization intensity im- +ages, 𝑃𝜆 = +√︃ +𝑄2 +𝜆 + 𝑈2 +𝜆, for an image with a central wavelength 𝜆, +were created from the Stokes 𝑄 and 𝑈 images using the native casa +task immath. Since 𝑃𝜆 is positive definite, we debiased each po- +larization intensity using the correction from George et al. (2012); +MNRAS 000, 1–19 (2023) + +V404 Cyg’s rapidly evolving polarized jet +5 +Table 1. Table of imaging properties. The highlighted frequency parameters for each base-band are the central frequency of the lowest (𝜈𝑖) and highest (𝜈 𝑓 ) +channels, in addition to the imaging bandwidth (Δ𝜈) assuming a typical ∼15% loss during flagging and calibration. Δ𝑡, is the average time on source. The +theoretical rms noise (𝜎rms) and the median rms noise for each Stokes parameter (𝜎𝐼 , 𝜎𝑄, 𝜎𝑈) are also highlighted. The high time-resolution images +(Δ𝜈 ∼ 110 MHz) were excluded from the self-calibration procedure due to the number of images (∼45000). The theoretical noise estimates were calculated +using the VLA exposure calculator; obs.vla.nrao.edu/ect/. +Base-band +𝜈𝑖 (MHz) +𝜈 𝑓 (MHz) +Δ𝜈 (MHz) +Δ𝑡 (s) +𝜎rms (mJy) +𝜎𝐼 (mJy) +𝜎𝑄 (mJy) +𝜎𝑈 (mJy) +5 GHz +4738 +5762 +850 +520 +0.03 +0.2 +0.05 +0.05 +16 +520 +0.2 +0.4 +0.3 +0.3 +110 +10 +0.6 +2 +— +— +7 GHz +6938 +7962 +850 +520 +0.03 +0.3 +0.06 +0.06 +64 +520 +0.1 +0.4 +0.14 +0.14 +110 +10 +0.6 +3 +— +— +21 GHz +20288 +21312 +850 +520 +0.09 +1 +0.14 +0.15 +110 +10 +1.5 +10 +— +— +26 GHz +25388 +26412 +850 +520 +0.08 +1.7 +0.3 +0.2 +110 +10 +1.5 +12 +— +— +𝑃𝜆,0 = +√︃ +𝑃2 +𝜆 − 2.3𝜎2 +𝑄𝑈. To remain consistent with the RM synthe- +sis routine (Section 2.4.1), we have chosen 𝜎𝑄𝑈 ≡ 1 +2 (𝜎𝑄 + 𝜎𝑈) +to parameterize the noise in 𝑃𝜆, noting that 𝜎𝑄 ≈ 𝜎𝑈 for all of +our images. The polarization fraction adopts its standard definition, +𝑓𝜆 ≡ 𝑃𝜆,0/𝐼𝜆, and we approximated its error using Gaussian error +propagation. We recognize that the MFS images will experience a +degree of bandwidth depolarization due to averaging over an intra- +band Faraday rotation. However, at our detected rotation measures +(|RM| ∼ 100 rad m−2), even at the lowest frequencies, the amount of +depolarization is insignificant; Δ 𝑓𝜆/ 𝑓𝜆 ≲ 1%. +To extract the intrinsic EVPA and rotation measure from each +time bin, we applied two independent methods: rotation measure +synthesis and a custom Markov-Chain Monte Carlo (MCMC) routine. +Meaningful RM synthesis results requires a band-averaged, polarized +S/N of 𝑃𝜆,0/𝜎𝑄𝑈 ≳ 7 (e.g., Brentjens & de Bruyn 2005; Macquart +et al. 2012). To ensure the significance of each detection, we enforce +the 𝑃𝜆,0/𝜎𝑄𝑈 > 7 restriction on the 5/7 GHz base-bands separately. +Our aggressive restriction was motivated by the susceptibility of +weakly polarized data to spurious effects from imperfect leakage +calibration. As a result, we limited the intrinsic EVPA and rotation +measure analysis to the 13 time bins between 11:15 and 13:53 UTC. +Data tables including our polarimetric measurements can be found +in Appendix E. +2.4.1 Rotation Measure Synthesis +Rotation measure synthesis derives the linear polarization parame- +ters of a source through its structure(s) in Faraday space; i.e., its +Faraday dispersion function (FDF; see, Burn 1966; Brentjens & de +Bruyn 2005; Macquart et al. 2012; Hales et al. 2012, for a compre- +hensive description). We generated each FDF using the rm-tools3 +repository, currently developed and maintained by the Canadian Ini- +tiative for Radio Astronomy Data Analysis (CIRADA). To mitigate +any aliasing at large rotation measures, we fixed the FDF domains at +±1.5×105 rad m−2, i.e., twice the rotation measure that corresponds +to a ∼50% drop in sensitivity at our spectral channelization. Further- +more, we fixed the bin size at 75 rad m−2, a factor of 20 (twice the +median polarized S/N) smaller than the full width at half maximum +of the rotation measure synthesis function. The package typically +quantifies the noise in each FDF (𝜎RM) using the median absolute +deviation after masking the strongest rotation measure component. +3 https://github.com/CIRADA-Tools/RM-Tools +We chose to use the rms noise as it was a factor of ∼2 larger, and thus, +increased our confidence in each detection. Any FDF component that +satisfied a > 5𝜎RM condition was recorded. During the construction +of each FDF, the observed EVPAs are de-rotated to their values at +the weighted mean of the 𝜆2 channels, with a 1/𝜎2 +𝑄𝑈 weighting. +The intrinsic EVPA is calculated from a further de-rotation using +the best-fit rotation measure; i.e., 𝜒0 = 𝜒𝑤 − RM · 𝜆2𝑤, where 𝜆2𝑤 +is the weighted average of all 𝜆2 channels and 𝜒𝑤 is the observed +polarization angle at 𝜆2𝑤. +2.4.2 MCMC +Since V404 Cyg is weakly polarized, we also employ a simple +Bayesian forward model to fit the polarization parameters directly +to the Stokes fluxes. Consistency between the two methods is an im- +portant check to mitigate the potential that our derived polarization +parameters originate from noise, as opposed to an intrinsic signal. +Our fitting functions adopt the following forms; +� +𝑄𝜆 = �𝐼𝜆 �𝑓𝜆 cos +� +2𝜒𝑤 + 2RM · (𝜆2 − 𝜆2 +𝑤) +� +; and +(1) +� +𝑈𝜆 = �𝐼𝜆 �𝑓𝜆 sin +� +2𝜒𝑤 + 2RM · (𝜆2 − 𝜆2 +𝑤) +� +. +(2) +We chose to fit for 𝜒𝑤, to remain consistent with the RM synthe- +sis routine. The superscript, 𝜆, in equations (1) and (2) denotes the +central wavelength of the spectral channel of interest. The model +parameters for Stokes 𝐼 (�𝐼𝜆) and the linear polarization fraction ( �𝑓𝜆) +were excluded from the fitting procedure, due to negligible correla- +tion with the quantities of interest (RM and 𝜒𝑤). Instead, the Stokes +𝐼 and polarization fraction models were smoothed using a Savitzky- +Golay filter (Savitzky & Golay 1964), retaining the overall structure +while removing stochastic variability, and stabilizing the fitting rou- +tine. +We assumed the sampled flux densities were independently dis- +tributed normal random variables, resulting in a log-likelihood func- +tion (L) of the following form, +log L = − +∑︁ +𝜆 +� +log +√︃ +2𝜋�𝜎2 +𝑄,𝜆 + (𝑄𝜆 − � +𝑄𝜆)2 +2�𝜎2 +𝑄,𝜆 ++ log +√︃ +2𝜋�𝜎2 +𝑈,𝜆 + (𝑈𝜆 − � +𝑈𝜆)2 +2�𝜎2 +𝑈,𝜆 +� +, +(3) +where 𝑄𝜆/𝑈𝜆 and � +𝑄𝜆/� +𝑈𝜆 are the measured and modelled flux den- +sities, respectively. We added two additional modeling parameters, +MNRAS 000, 1–19 (2023) + +6 +A. K. Hughes et al. +𝜎𝑄,sys and 𝜎𝑈,sys, that are channel independent variances to account +for missed systematic effects. The variances seen in equation (5) are +the sum of the measured rms noise variance and our systematic ad- +dition (e.g., �𝜎2 +𝑄,𝜆 ≡ 𝜎2 +𝑄,𝜆 + 𝜎2 +𝑄,sys). +We used the Markov-Chain Monte Carlo algorithm implemented +through Python’s emcee package. emcee is a pure-Python implemen- +tation of Goodman and Weare’s Affine Invariant Markov chain Monte +Carlo Ensemble Sampler (Foreman-Mackey et al. 2013; Goodman & +Weare 2010); a modified version of the classic Metropolis-Hastings +algorithm, simultaneously evolving a select number of walkers +through parameter space. The number of (sampling) walkers was +fixed at five times the number of dimensions, 20. We chose four +broad, uniform, and uninformative priors to reflect the lack of a pri- +ori information on V404 Cyg’s polarization state. The systematic +variance priors were positive definite, with maximum values chosen +to be twice the variance of the measured flux densities. The rotation +measure prior adopted the FDF domain, ±1.5 × 105 rad m−2. A uni- +form prior was unable to capture the circularity of the EVPA. As a +result, individual walkers frequently would become trapped in the +local minima created by the prior’s edges, subsequently inhibiting +convergence. To combat this, we expanded the prior to ±3𝜋/2 rad, +while maintaining the initial condition distribution for the physi- +cally meaningful range of ±𝜋/2 rad. We initialized each run with 80 +walkers, four times the number intended for sampling. Following an +initial set of “burn-in" iterations, we removed the 60 walkers with +the lowest posterior probabilities and adopted the remaining 20 as +the starting positions for sampling. After sampling, we verified that +each simulation converged by visually inspecting the walkers over a +large number of autocorrelation times. +We adopted the median of each posterior distribution as the best- +fit value of our model, and the ranges between the median and the +15th/ 85th percentiles as the 1𝜎 (−)/(+) uncertainties, noting that the +measured uncertainties are purely statistical. Once again, the intrinsic +EVPA was solved for using, 𝜒0 = 𝜒𝑤 − RM · 𝜆2𝑤, and we calculated +its error using standard Gaussian error propagation. +3 RESULTS +By splitting our ∼3.5 hr observation into sixteen ∼13 min time bins, +we have measured the temporal evolution of the linear polarization +fraction (Figure 1), rotation measure, and intrinsic EVPA (both Fig- +ure 2) during the 2015 June 22 flaring events of V404 Cyg. In this +section, we present our polarimetric results. We note that weak linear +polarization fractions should be treated with caution; in Appendix C, +we compare our results to the simultaneous evolution of the phase +calibrator ( 𝑓𝜆 ∼ 2%) to ensure that significant changes we see in +V404 Cyg arise from physical evolution, and not systematic calibra- +tion effects. +3.1 Linear Polarization Fraction +Each base-band showed a weak but variable degree of linear polar- +ization with a maximum linear polarization fraction that decreased +with decreasing frequency; i.e, maxima of ∼0.22, 0.25, 0.5, 0.75% +for the 5, 7, 21, and 26 GHz base-bands, respectively. The maximum +linear polarization fraction occurs between the peaks of the first and +second Stokes 𝐼 flare, and, like Stokes 𝐼, occurs at later times for +lower-frequency observations. There is evidence of a second (much +weaker) linear polarization fraction peak in the 21 GHz base-band, +between the second and third flare (at ∼ 12:50 UTC). This sec- +ondary peak is marginally detected in the 5/7 GHz base-bands, but +is consistent with noise at 26 GHz. Additionally, in the 21/26 GHz +base-bands, at late times the linear polarization fraction begins to +increase alongside the decay of the third flare. A similar increase is +not observed in the 5 GHz base-band, with a marginal trend seen in +7 GHz, although temporal delays would have likely shifted any peak +at these frequencies beyond our observing time. +In the 2015 July 2 observations, during V404 Cyg’s return to qui- +escence, the Stokes 𝐼 flux densities had decreased to ∼ 4 mJy across +all base-bands. As a result, we are unable to detect weakly polar- +ized emission, and the source showed no polarization with a 99% +confidence upper limit on the polarization fraction of 1.0, 0.9, 2.2, +2.4% for the 5, 7, 21, and 26 GHz base-bands, respectively. Here +we calculated upper limits following Vaillancourt (2006). Further- +more, we analyzed 5 subsequent epochs of the 5/7 GHz observations +taken between 2015 July 15 and 2015 August 5 (see, Plotkin et al. +2017, for details). Of these 5 epochs, data on 2015 August 5 had the +most constraining upper limits, 6.0, and 5.1% (for the 5 and 7 GHz +base-bands respectively), with all other epochs having upper limits +between 10−25%. While we cannot detect a weakly polarized signal +with these observations, a ∼ 5% limit is lower than some past linearly +polarized fractions detected in BHXBs (e.g., Han & Hjellming 1992; +Brocksopp et al. 2007; Curran et al. 2014, 2015). +The S/N of all linear polarization intensities, are ≳ 5 with the +strongest detections reaching 𝑃𝜆,0/𝜎𝑄𝑈 ∼ 25 (see Table E1). At +all 𝑓𝜆, imperfect leakage calibration systematically increases the ob- +served linear polarization fraction. This effect is not included in the +calculations of the S/N of linear polarization intensities or the errors +on 𝑓𝜆 that we present. While our higher 𝑓𝜆 values may be (slightly) +overestimated due to imperfect leakage corrections, the lower values +could be due to spurious signals and are actually consistent with no +linear polarization. Following Hales (2017), the predicted level of +spurious linear polarization fraction is Rayleigh distributed with a +mean given by +𝑓spur,mean ≈ +√︂ 𝜋 +4𝑁𝑎 +� +𝑓 2 +true + 𝑁𝑎 [(𝑆/𝑁)𝐼 ]−2� +, +(4) +where, 𝑁𝑎 is the number of antenna in each sub-array (𝑁𝑎=11/13 +for Sub-array 1/2, respectively), (𝑆/𝑁)𝐼 is the Stokes 𝐼 signal-to- +noise ratio of the leakage calibrator at the frequency of interest (with +a 16 MHz leakage solution bandwidth), and 𝑓true is the true linear +polarization fraction of the leakage calibrator. For 𝑓true, we adopted +the mean linear polarization fraction from the VLA polarization cal- +ibrator catalog4, corresponding to, 0.04%, and 0.17% for the 5/7 and +21/26 GHz bands respectively. We measured the Stokes 𝐼 signal from +an image plane analysis using imfit. Due to the leakage calibrator’s +large Stokes 𝐼 flux densities and our sparse 𝑢𝑣-coverage (from a sin- +gle scan, 13-element sub-array), the Stokes 𝐼 images are dynamic +range limited. As a result, we chose to use 𝜎𝑄𝑈 as the noise value +in (𝑆/𝑁)𝐼 , as opposed to 𝜎𝐼 . The Stokes 𝑄 and 𝑈 images were not +dynamic range limited, and thus would better quantify the instrumen- +tal noise that also affects the Stokes 𝐼 data. We present the spurious +linear polarization parameters in Table 2. The differences between +the two sub-arrays are the result of elevated noise in sub-array 2. +We note that the minimum linear polarization fraction we detect in +each base-band is approximately equal to the predicted values of +𝑓spur,mean. +Henceforth, we define the significance level (SL) as the proba- +bility that a detection is not the result of a purely spurious signal +4 The VLA catalog can be found here; http://www.vla.nrao.edu/ +astro/evlapolcal/index.html +MNRAS 000, 1–19 (2023) + +V404 Cyg’s rapidly evolving polarized jet +7 +Table 2. Table of spurious linear polarization properties. All symbols adopt +their definitions as defined in the text. +Sub-array +𝑁𝑎 +Base-band +(𝑆/𝑁 )𝐼 +𝑓spur,mean (%) +1 +13 +5 GHz +1702 +0.053 +7 GHz +1656 +0.054 +21 GHz +516 +0.18 +26 GHz +418 +0.22 +2 +11 +5 GHz +1331 +0.067 +7 GHz +1335 +0.066 +21 GHz +480 +0.19 +26 GHz +357 +0.25 +(See Fig. 1, horizontal-dotted lines). The significance levels for all +maxima are > 99%, confirming that we have observed an intrinsic +polarized signal in each base-band; the SLs for each time bin are +tabulated in Table E1). Only one scan per sub-array was used to +correct leakage, and leakage converts Stokes 𝐼 into 𝑃. This leads +to a single offset in fractional linear polarization in the absence of +noise. On the other hand, imperfect leakage can potentially lead to +dynamic systematic-error-induced changes in the measured EVPA +due to parallactic rotation. +3.2 Rotation Measure and EVPA +The derived rotation measures exhibit stochastic variability, with +values between −330 ≲ RM ≲ −20 rad m−2 (top panel, Fig. 2), +and show a strong agreement between the two polarimetric meth- +ods in almost all time bins (the only ≳ 1𝜎 disagreement occurs in +the final, lowest S/N time bin). The weighted means of the rotation +measure, are −100 ± 16 rad m−2 for the RM synthesis method and +−100±12 rad m−2 for the MCMC method. We applied a simple vari- +ability analysis by calculating the 𝜒2 statistic against a constant RM +model equal to the weighted mean. The 𝜒2 values of 10.6 (RM Syn- +thesis) and 17.4 (MCMC) for 12 degrees of freedom are consistent +with a constant rotation measure at probabilities of 56% and 13%, +respectively. +When we use the default RM synthesis prescription (i.e., quan- +tifying the noise in the dispersion function with an appropriately +scaled median absolute deviation rather than the rms), the detection +uncertainties reduce by a factor of ∼2, and the RM Synthesis 𝜒2 +value become significantly larger than the MCMC value (53.5/12), +consistent with a variable rotation measure. However, this choice +also increases the population of >5𝜎RM components to ≳10 for each +FDF, with rotation measure magnitudes ∼103 − 105 rad m−2. These +rotation measures are characteristic of extremely particle-rich lines +of sight (e.g., towards the Galactic centre) and, historically, have +not been observed in outbursting BHXBs. Our phase calibrator, a +source with ∼ 2% linear polarization, showed a similar population +of secondary components. We find it very unlikely that these sources +would exist while evading detection during recent Galactic rotation +measure analyses (Oppermann et al. 2012, 2015; Hutschenreuter & +Enßlin 2020). Therefore, we propose that these components are ar- +tifacts from imperfect 𝜆2 sampling (cf., the effects of poor/patchy +𝑢𝑣-coverage during synthesis imaging; Taylor et al. 1999) or a sys- +tematic effect in the modern RM synthesis routine(s). We conclude +our decision to use the FDF rms noise is more reflective of the sta- +tistical significance of each detection, and that we cannot identify +any significant rotation measure variability from V404 Cyg. As a +result, for our analyses, we have adopted a constant rotation measure +equal to the (inverse-variance) weighted mean of rotation measures +across all time bins; i.e., RM = −100 ± 16 (12) rad m−2 for the RM +Synthesis (MCMC) method. +Both the observed EVPA (𝜒𝑤; Second panel, Fig. 2) and intrinsic +EVPA (𝜒0; Third panel, Fig. 2) exhibit a clear temporal evolution, +with strong agreement between the RM synthesis and MCMC rou- +tines. Moreover, due to the stable rotation measure, this evolution +suggests an intrinsic change in the (polarized) emission environ- +ment. In the SL ≥ 90% regime, both observed and intrinsic EVPA +evolve gradually, with a ∼ 30◦ change. The intrinsic EVPA evolves +from ∼ 80◦ to ∼ 50◦ between 11:30 and 12:30 UTC. The intrinsic +EVPA then stabilized at the ∼ 50◦ for the remaining time bins. +4 DISCUSSION +In this section we describe the short timescale evolution of our polari- +metric results and compare the observed behaviours to the 1989 out- +burst. Moreover, we correlate this evolution with total intensity light +curves, high (spatial) resolution imaging, and optical polarization +detections. When analysing the connection between the polarization +flaring and the high resolution radio imaging, we limit our discussion +to the spatially-resolved VLBA components that dominate the VLBA +light curve at a given time. This implicitly assumes that any resolved +polarized flux density would track the resolved Stokes 𝐼 flux density +(i.e., 𝑃0,𝜆 ∝ 𝐼𝜆). Although we make this assumption, we cannot rule +out the possibility that the dominant VLBA components are unpolar- +ized and the sub-dominant components (with average Stokes I flux +densities ≲ 10% of the dominant counterparts) are the source of the +polarized emission. +4.1 Linear Polarization Fraction +The majority of BHXB outbursts have measured linear polariza- +tion fractions of ≳2% at 1–10 GHz (e.g., Fender 2003; Brocksopp +et al. 2007; Curran et al. 2015), with rare cases reaching appreciable +fractions of the theoretical limits (e.g., the ∼50% detections of XTE +J1752−223 and Swift J1745−26; Brocksopp et al. 2013; Curran et al. +2014). Even when considering the typical reduction compared to the +theoretical maxima, our measured maximum linear polarization frac- +tion (∼0.2% in the 5/7 GHz base-bands) for V404 Cyg is a factor of +∼10 less than a standard, weakly polarized signal during a BHXB +outburst. However, we acknowledge that past outbursts with compa- +rably weak polarization fractions may not have had sufficient S/N +for a clear detection and/or the larger (average) polarization fractions +may suffer from a publication bias where strongly polarized outbursts +are more often introduced within the literature. +Observers caught a glimpse of a comparably low polarization frac- +tion during the monitoring of the 1989 outburst of V404 Cyg. The +first day of polarization observations — 1989 June 1, during the de- +cay of the “major synchrotron bubble event” (i.e., the ejection of a +bright cloud of synchrotron emitting plasma) — recorded the lowest +polarization fraction of the entire campaign, measuring 0.4±0.1% at +central frequencies of 4.9 and 8.4 GHz. During the decay of the 1989 +outburst, the radio emission exhibited an inverted spectrum and lin- +ear polarization fraction of ∼3%, characteristic of a typical compact +jet (Han & Hjellming 1992). The polarized signal was consistently +detected for 50 days (between 1989 June 1 and 1989 July 18) before +the flux density decayed below the detection threshold. In contrast, +we did not observe an increase to a few percent polarization frac- +tion during the decay of the 2015 outburst. The 2015 July 2 epoch +places a 99% confidence interval upper limit of ∼1% on the 5/7 GHz +polarization fraction, consistent with the 2015 June 22 observations, +MNRAS 000, 1–19 (2023) + +8 +A. K. Hughes et al. +Figure 1. Temporal evolution of the linear polarization fraction; 26 GHz (top), 21 GHz (2nd from the top), 7 GHz (4th from top), 5 GHz (5th from top). The +two-point spectral indexes of the 21/26 GHz (3rd from top) and the 5/7 GHz (bottom), show the simultaneous evolution of the absorption conditions. The vertical +dashed lines across all panels highlight the time of the maximum fractional polarization in the 26 GHz base-band. The horizontal dashed lines highlight the value +of the mean spurious linear polarization fraction for each base-band; the discontinuities are the result of elevated noise in Sub-array 2. The diamond markers +correspond to SL≥ 90% , and the squares to SL< 90%. The grey curves display the simultaneous Stokes I flux density evolution for each base-band. We can see +that the linear polarization fraction exhibits a similar frequency-dependent delay as the Stokes 𝐼 light curves, and is offset from the (Stokes 𝐼) maxima. +MNRAS 000, 1–19 (2023) + +0.8 +1400 +0.7 E +26 GHz +1200 +0.6E +1000 +0.5E +Stokes +Fraction +800 +0.4 +600 +0.3 +Flux +Polarization +00F +0.2 +0.5 +1200 +21 GHz +isity +0.4 +Linear +Q0QT +mJy) +0.3 +800 +0.2 +600 +0 +0.25 +700 +7 GHz +0.20 +600 +(%) +0.15 +500 +Stokes +Fraction +400 +0.10 +OCE +Flux +tion +0.05 +Density +0.200 +E +5 GHz +600 +0.175 +0.150 +(m) +ear +500 +E +月 +0.125 +0.100 +400 +0.075 +Q.050 +OCE +0.5 +0.03 +QE-T +12:00 +12:30 +13:00 +13:30 +14:00 +Time on 2015 Tune 22(UTC)V404 Cyg’s rapidly evolving polarized jet +9 +Figure 2. Polarization properties measured from the 5 and 7 GHz base-band observations, for both the MCMC and RM synthesis routines. The vertical shaded +region corresponds to the detections with an average significance level ≥ 90% between the 5 and 7 GHz base-bands. The underlying grey curve in each panel +is the average Stokes I light curve between the 5 and 7 GHz base-bands. There is strong agreement between the two polarimetric methods. (top) The rotation +measure; the horizontal dotted line shows the weighted average of the rotation measures (∼ − 100 rad m−2). (2nd from top) The observed EVPA de-rotated to +the weighted mean of all 𝜆2 channels. (3rd from top) The intrinsic EVPA. The horizontal bars show the PAs (+90◦) of the dominant VLBA ejecta identified +in Miller-Jones et al. (2019). The length of the bar span the times between the ejection time and when an ejected component is not longer detected, and the +darker part of the bar shows when it was the brightest ejected component (excluding the compact core; see Figure 3). The vertical size of the bars is fixed at the +uncertainty in the PA. We adopt the naming conventions from the original paper. (bottom) The average linear polarization fraction between the 5 and 7 GHz +base-bands. We can see the rotation measure is constant and the EVPA exhibits a ∼ 30◦ rotation between ∼11:30 and 12:30 during the decay of the polarization +fraction maximum. +MNRAS 000, 1–19 (2023) + +600 +-100 +l) +500 +200 +RM +300 +400 +RM Svnthesis +-400 +MCMC +300 +600 +09 +500 +400 +8 +40 +kes +30 +ux +90 +D +80 +s3 +70 +S6 +60 +400 +50 +300 +40F +600 +0.20 +500 +0.15 +0.10 +400 +0.05 +300 +10:30 +11:00 +11:30 +12:00 +12:30 +13:00 +13:30 +14:00 +14:30 +Time on 2015 June 22 (UTC)10 +A. K. Hughes et al. +Figure 3. 15.6 GHz VLBA light curve using the data from Miller-Jones et al. (2019). The open black circles represent the total integrated the flux density, the +solid black circles represent the core flux density, and all other marker types represent a single spatially-resolved component. We adopt the naming convention +from the original paper. The grey shaded region corresponds to the three time bins that encompass the 7 GHz fractional polarization peak; we shifted the region +back in time by 10 minutes to account for the delay between the bands (see Appendix A). We can see that at any point in time the total integrated (VLBA) flux +density is a superposition of multiple radio-bright (and potentially polarized) components that are unresolved in our VLA observations. +but below the level seen in the compact jet during the 1989 outburst. +Furthermore, none of the epochs in Plotkin et al. (2017) showed +any polarized emission, although we note that the upper limits are +significantly larger than the maximum polarization fraction detected +during the 1989 outburst. +The Stokes 𝐼 flux density of V404 Cyg decayed significantly faster +in the most recent outburst, taking ∼30 days in 2015, as opposed +to ∼300 days in 1989. Tetarenko et al. (2018) suggested that the +more rapid decay was the result of the strong winds originating from +the accretion disk (detected by Muñoz-Darias et al. 2016) rapidly +depleting the disk and leaving less matter to fuel the jets. Other +factors may have also played a role; these could include the total +mass reservoir built during the quiescent periods prior to the two +outbursts — 33/26 years for the 1989/2015 outbursts respectively — +or differences in the total mass accreted during the bright outburst +phases. In contrast, the polarization fraction depends on the structure +of the jet(s), and is only indirectly related to the Stokes 𝐼 flux density +(i.e., ideal, optically-thick synchrotron emission at 1 Jy vs. 1 mJy +would both have a linear polarization fraction of 10%). Therefore, +the <1% upper limits on 2015 July 2 when the radio emission was +dominated by a compact jet (compared to ∼3% during similar epochs +of compact jet dominance in 1989), suggests the most recent outburst +had a less-ordered magnetic field in the jet or suffered from higher +depolarization due to independent unresolved components within the +VLA beam. +There are two clear features of the linear polarization fraction: (i) it +is continuously weak (<1%) regardless of the time bin; (ii) it evolves +in time, with maxima and minima linear polarization fractions (in +each base-band) separated by a factor of ∼5. +4.1.1 Origin of Low Linear Polarization Fraction +The short times between flares, the precession of the jet axis, and +the energetics required for such a luminous outburst are character- +istic of a complex (magnetic and geometric) environment, and are +expected to inhibit strongly polarized emission. At a spatial resolu- +tion of ∼1 AU the core emission identified by the VLBA could arise +from an unresolved population of ejecta (likely on top of a compact +jet). The jet axis precession would cause these ejecta to have vari- +able PAs, and, assuming similar internal magnetic fields, variable +EVPAs. The superposition of the unresolved (and resolved) ejecta +in our VLA observations will decrease the polarization fraction, un- +less all unresolved components have the same polarization fraction +and EVPA. The effects of multi-component superposition (i.e., when +the coherence length of the magnetic field is significantly smaller +than the angular resolution) was seen in the recent Event Horizon +Telescope (EHT) observations of M87; the lower spatial resolution +of ALMA reduced multiple components with linear polarizations of +≳20% (resolved with the Event Horizon telescope) to a net polar- +ization fraction of ∼2% for the M87 core (Event Horizon Telescope +Collaboration et al. 2021). +In the 2015 outburst of V404 Cyg, the maximum linear polariza- +tion fraction in each base-band decreases as the frequency decreases. +Here we consider the maximum of each frequency because of the +potential time delays between base-bands. A decreasing polariza- +tion fraction with decreasing frequency is a common characteristic +of Faraday depolarization. In particular, sources with strong Faraday +rotation within their emission regions can appear depolarized (in Sec- +tion 4.3 we find that the jet itself may be a strong source of Faraday +MNRAS 000, 1–19 (2023) + +c +4 +N6 +S5 +800 +N1 ++ +N8 +9S +(mJy) +N2 +x +N9 +S7 +N3 +S2 +o +Total +N4 +S3 +Stokes 1 Flux Density +009 +400 +G +Q +GGO +. +200 +. +0 +10:30 +11:00 +11:30 +12:00 +12:30 +13:00 +13:30 +14:00 +14:30 +Time on 2015 June 22(UTC)V404 Cyg’s rapidly evolving polarized jet +11 +rotation). Faraday depolarization has been well established in radio +studies of AGN (e.g., Pasetto et al. 2018) and was observed for the +candidate BHXB SS 433 (Stirling et al. 2004). While the complexity +of the spectral and temporal evolution of the 2015 outburst of V404 +Cyg makes it difficult to determine if we are in fact seeing Faraday +depolarization, here we make some simple calculations. The simplest +Faraday screen geometry (i.e., a single uniform slab of synchrotron +emitting plasma Burn 1966; Sokoloff et al. 1998) predicts a depo- +larization of Δ 𝑓𝜆/ 𝑓𝜆 ∼ 10% between the 26 and 5 GHz base-bands. +Therefore, a more complex model (see, Pasetto et al. 2018) would be +required for Faraday depolarization to explain the ∼ 70% (0.75% at +26 GHz to 0.22% at 5 GHz) depolarization we have observed (such +an analysis is beyond the scope of this paper). +We cannot ignore the possibility that during this outburst, the +magnetic fields in the jet(s) are intrinsically more disordered than +typical BHXB outbursts. The BHXB GRO J1655−40 entered a multi- +flaring highly-luminous state during its 1994 outburst, similar to the +2015 outburst of V404 Cyg (although the decay timescales of each +flare were significantly longer in GRO J1655−40). However, GRO +J1655−40 reached a maximum 4.9 and 8.4 GHz linear polariza- +tion fraction of 1–10% with linearly polarized variability as high as +Δ 𝑓𝜆 ∼ 4% on timescales less than half a day, suggesting that weakly +polarized emission is not an inherent aspect of multi-flaring outbursts +(Hjellming & Rupen 1995; Hannikainen et al. 2000). +4.1.2 Origin of Temporally Evolving Linear Polarization +A transition of the absorption conditions (e.g., from optically-thick +to optically-thin synchrotron emission) of a dominant polarized com- +ponent will result in a temporally evolving polarization fraction (e.g., +as seen in Swift J1745-26; Curran et al. 2014). During these transi- +tions, we expect the intrinsic EVPA to rotate by 90◦. For optically-thin +synchrotron emission, the EVPA and the magnetic field vector are +perpendicular (Longair 2011), and for optically thick synchrotron +emission, the EVPA tracks the direction of the magnetic field (see, +Ginzburg & Syrovatskii 1969, and references therein). The EVPA +will thus rotate as the source transitions from 𝜏 ∼ 10 to 𝜏 ∼ 0.5, +where 𝜏 is the optical depth; this takes about half the rise timescale +of a vdL plasmoid (Aller 1970). We do not observe a ∼90◦ rotation +of the intrinsic EVPA, at any time during our monitoring (see Section +4.2). Moreover, we know that the light curves are a superposition of +multiple short-lived (≲ 1.5 hr) ejecta, and a compact core, further +reducing the plausibility of a single component origin for each radio +flare. +An ensemble of polarized components with evolving optical +depths can exhibit a more complex evolution. As an investigation, +we calculated the two-point spectral indexes for the 21/26 GHz and +5/7 GHz VLA observations (see bottom panels of Fig. 1). We are un- +able to disentangle the emission from the multiple unresolved compo- +nents (seen in the VLBA), and, as a result, we are measuring the “net” +spectral index. Moreover, we are measuring a simultaneous spectral +index, which may be less appropriate for rapidly evolving ejecta. +An optically thick “net” spectral index (𝛼 > 0) requires that a sub- +population of the unresolved components are optically thick (with +the inverse being true for optically thin, 𝛼 < 0, spectral indexes). +The spectral indexes show an evolution in time, exhibiting multiple +transitions of the absorption conditions, consistent with an ensemble +of evolving components, with both optically thick and optically thin +sub-populations. Intuitively, one might expect that a negative “net” +spectral index measured would correspond to a higher contribution +of optically thin synchrotron emission, and, as a result, a higher polar- +ization fraction. The peak polarization fraction does in fact coincide +with a negative spectral index; i.e., 𝛼 ∼ −0.2 and −0.5 in the 5/7 GHz +and 21/26 GHz base-bands, respectively (Fig 1). Furthermore, the +late time rise seen in the 21/26 GHz base-bands (∼14:00–14:30), +also coincides with a negative spectral index (𝛼 ∼ −1). However, +at ∼12:45 and 13:15 in the 5/7 GHz and 21/26 GHz base-bands, we +also have 𝛼 ∼ −0.3 and −1. During these times the polarization +fraction shows a (weak) peak at 21 GHz, with marginal features at +5/7 GHz, and no evolution at 26 GHz (i.e., a “missing” polarization +peak). Therefore, we are unable to conclusively connect the spectral +index to the polarization fraction evolution. +Comparing the short time-scale temporal evolution to the +15.6 GHz VLBA light curves (reproducing data from Miller-Jones +et al. 2019 as Fig. 3 of this paper), we do not see any clear con- +nection between the resolved components and the evolution of the +polarization fraction, and cannot distinguish between a polarized +core, polarized ejecta, or a combination of the two. However, the +“missing” polarization peak coincides with a period of time when +the S5 component clearly dominates the VLBA light curve. It is pos- +sible that S5 was less polarized than the components that preceded +and followed its ejection. As a result, a complete explanation of the +polarization fraction evolution may require a combination of evolv- +ing optical depths, and intrinsic differences between the different +polarized components launched at different times. Regarding a po- +tential intrinsic evolution, Brocksopp et al. (2007) expanded upon the +shock-in-jet picture outlined in Fender et al. (2004), suggesting that +the collisions between ejecta temporarily disorder the magnetic field +lines while producing shock fronts that propagate through the ejecta, +reestablishing a dominant field direction at later times. Shahbaz et al. +(2016) proposed a similar mechanism to explain the behaviour of +the polarized optical emission during V404 Cyg’s 2015 outburst. +A flare in the optical polarization fraction that preceded a 16 GHz +radio flare, was attributed to the compression of the jet’s magnetic +field by many small shocks travelling along the jet axis. The exis- +tence of these shocks is consistent with the detection of sub-second +optical flares by Gandhi et al. (2016) during the same time period. +The polarization flare was attributed to “a major ejection event” that +followed optical flaring that began a couple of hours earlier; i.e., a +large outflow imprinted with the recently ordered magnetic field. +In both of the scenarios proposed by Brocksopp et al. (2007) and +Shahbaz et al. (2016), the ordering of the magnetic field is a result +of multiple colliding components, and, as a result, the timescales +separating collisions would have to be significantly shorter than the +precession period of the jet. This is a plausible theory if the sub- +second optical flaring is characteristic of the collision timescales. +Any such model would also need to explain the temporal offset +between the Stokes 𝐼 and polarization fraction peaks. +4.2 Intrinsic EVPA +In the first few days of the 1989 outburst, the EVPA evolved through +a ∼90◦ rotation at 4.9 and 8.4 GHz. This rotation coincided with the +transition from an optically-thin to optically-thick radio spectrum. +During the 2015 outburst the dominant feature of the intrinsic EVPA +evolution is a ∼30◦ rotation that occurs alongside the decay of the +maximum polarization (bottom two panels, Fig. 2). This rotation +occurs across 6 time bins (80 min) suggesting that a full 90◦ rotation +would take ∼4 hr, a timescale longer than the lifetimes of any of +the dominant VLBA components (see Fig. 3). Under the assumption +that the contemporaneous peak in polarization fraction and rotation +of the EVPA arise from a shared mechanism, neither arise from a +transition in the absorption conditions of a single component, as was +likely observed in 1989. +MNRAS 000, 1–19 (2023) + +12 +A. K. Hughes et al. +The precession of the jet axis provides a natural mechanism to +explain the rotation of the EVPA. We investigated this possibility +by using the change in the position angles of the dominant ejecta +as a proxy for the precession of the jet axis. The position angles +also exhibited a ∼30◦ rotation, albeit over a longer, ∼2 hr, timescale. +The ∼30◦ rotation begins when the S2 component (PA ∼ 1◦) is the +dominant jet ejection observed by the VLBA (although the core emis- +sion is brighter, see Fig. 3). In the simplest geometries, compression +shocks or velocity-shearing establish dominant field directions par- +allel or perpendicular to the jet flow’s direction of motion (i.e., the +PA; Laing 1980; Jorstad et al. 2007). The approximate orthogonality +(offset by ∼ 10◦) between the initial intrinsic EVPA and the PA dur- +ing the decay of the S2 component is consistent with optically-thick +(optically-thin) synchrotron emission from a magnetic field estab- +lished by compression shocks (velocity shearing). +The rotation coincides with the emergence of a new, dominant +VLBA component (S3) at a position angle of −11.5◦. If the rotation +from an intrinsic EVPA of 80◦ to 50◦ results from the S2-to-S3 +transition, the larger obliquity (∼ 30◦) between the intrinsic EVPA +and the PA of S3 requires a more complex magnetic field origin +(e.g., remnants of helical fields; Gómez et al. 2008). Following S3 +dominance, S5 becomes the dominant ejection, while maintaining a +similar PA of ∼ 13.5◦. The similarity between the PAs of S3 and S5 +is consistent with the stability of the intrinsic EVPA at ∼50◦ between +12:45 and 13:45 UTC, assuming similar intrinsic properties. Alter- +natively, the S6 component has a smaller obliquity when compared +to the late time EVPA (∼ 10◦), and may be a better measure of the +jet orientation, at later times. However, since our observations are +the superposition of multiple overlapping components (including a +bright, unresolved compact core), there may be, in fact, no relation- +ship between the position angles of the resolved components and the +intrinsic EVPAs. +Variability in the EVPA without any change in the jet axis PA +(i.e., rotator events) has been observed in many AGN (see Saikia +& Salter 1988, and references therein), and a couple of BHXBs +(e.g., GRS 1915+105; Fender et al. 2002). These events are thought +to be the result of complex field geometries (e.g., helical magnetic +fields; Gómez et al. 2001) or internal shocks (e.g., Gómez et al. +2008) producing time-varying magnetic fields. Moreover, complex +shock fronts (e.g., conical shock waves) can produce magnetic field +orientations that are neither perpendicular nor parallel to the jet axis +(see Jorstad et al. 2007, and references therein). +Since the VLBA data did not acquire full polarimetric calibrations, +there is no spatially resolved polarimetry that explicitly localizes the +dominant polarized component. In the absence of such detections and +given the multiple scenarios suggested above, we can neither defini- +tively make connections between the VLBA/VLA observations and +the linear polarization properties, nor identify if the polarized emis- +sion originates from an ejected component, a compact steady jet, or a +time-variable combination of the two. This limits the strength of our +claims towards the origin of the polarization and its connection to the +evolution of the Stokes 𝐼 flux density. We note that due to the reduced +sensitivity of spatially-resolved data, without a significant increase in +the polarization fraction (as was observed in M87), the VLBA would +be unable to detect comparably low polarization fractions, even after +including the necessary calibrators. +4.3 Rotation Measure +The rotation measure quantifies the amount of Faraday rotation af- +fecting a linearly polarized emission signal, and is related to the +internal properties of the plasma along the line of sight (i.e., its Fara- +Figure 4. Linear fit to the observed EVPAs during V404 Cyg’s 1989 outburst; +the data was adapted from Han & Hjellming 1992. To account for the wrapping +of the EVPA at large values of 𝜆2 we applied a −2𝜋 correction to the 1.49 GHz +observation. We used scipy.optimize.curve_fit for our linear fit. +day screens). The RM is related to the electron number density, 𝑛𝑒, +the magnetic field oriented parallel to the line of sight (from the +source to the observer), 𝐵||, and the path length 𝑙. Explicitly, the +rotation measure is described by the path integral, +RM = +� +812 +∫ observer +source +𝑛𝑒𝐵||d𝑙 +� +rad m−2, +(5) +where 𝑛𝑒, 𝐵||, and d𝑙 are in units of cm−3, 𝜇G, and kpc, respectively. +The sign of the rotation measure depends on the orientation of the +magnetic field; i.e., when the field lines are parallel (anti-parallel) to +the direction of emission propagation, the sign is positive (negative). +Moreover, for Galactic sources, the total rotation measure can have +significant contributions from both the diffuse interstellar medium +(ISM) and the local environment. Detecting a large local component +necessarily implies a high density, or strongly magnetic environment +to account for the reduced path length when compared to the ISM. +During the previous outburst in 1989, Han & Hjellming (1992) +measured a constant observed EVPA in four frequency bands for the +majority of the (∼50 days) polarization monitoring. The weighted +averages from these observations were; 3 ± 7◦, −44 ± 1◦, −16 ± 1◦, +and −18 ± 2◦, at central frequencies of 1.49, 4.9, 8.4, and 14.9 GHz, +respectively. The observed EVPAs are linear with respect to 𝜆2 (see +Fig. 4), with a slope (i.e., rotation measure) of −151 ± 11 rad m−2. +During the first ∼2–3 days of polarization detections, the EVPAs at +4.9 and 8.4 GHz exhibited a 90◦ rotation. The two-point slope of these +angles (39±4◦ and −60±6◦ at 4.9 and 8.49 GHz, respectively) shows +a consistent rotation measure of −150 ± 50 rad m−2. The matching +rotation measures, even with a changing EVPA, implies a constant +Faraday screen. The magnitude, orientation, and stability of the 1989 +rotation measure is similar to our RM measurement during the 2015 +outburst (−100 ± 15 rad m−2); however, the former is detected over +much longer timescales. The rotation measures during the 1989 and +2015 outbursts are marginally consistent (at the ∼ 2.7𝜎 level). As a +result, we are unable to conclusively identify temporal variability of +the rotation measure (e.g., as was seen in the 1994 outburst of GRO +J1655−40; Hannikainen et al. 2000). Had we identified temporal +variability, we could rule out the scenario that both outbursts are +behind a constant, purely Galactic Faraday screen. +MNRAS 000, 1–19 (2023) + +-1 +-2 +-3 +x +-5 +-6 +0.00 +0.01 +0.02 +EO'O +0.04 +A2 (m7)V404 Cyg’s rapidly evolving polarized jet +13 +Multiple Galactic RM models predict a negative rotation measure +along the line of sight to (and beyond) V404 Cyg; −30 ± 10 rad m−2 +(Oppermann et al. 2012), −40±20 rad m−2 (Oppermann et al. 2015), +and −130 ± 50 rad m−2 (Hutschenreuter & Enßlin 2020). Assuming +a constant Galactic magnetic field, the Galactic RM would be dom- +inated by distances well beyond the ∼2.4 kpc distance to V404 Cyg. +For both standard models of electron distributions (Cordes & Lazio +2003; Yao et al. 2017) the PyGEDM tool (Price et al. 2021) indicates +much larger dispersion measures (which are proxies for the integrated +electron column density) at 10 kpc (269±16 pc cm−3) than at 2.4 kpc +(32 ± 4 pc cm−3). Thus, our measured value of ≈ − 100 rad m−2 sug- +gests, either an inversion (or multiple inversions) of the Galactic +magnetic field along our line of sight, or an intrinsic rotation mea- +sure component local to the source. The morphology of the Galactic +magnetic field is poorly constrained, with different models predicting +radically different structures (see, Haverkorn 2015; Jaffe 2019). As +an example, the “zeroth"-order model by Van Eck et al. (2011) pre- +dicts a parallel Galactic magnetic field within the first ∼4 kpc along +the line-of-sight containing V404 Cyg, inverting to an anti-parallel +orientation at larger distances and producing the net-negative rota- +tion measure. Conversely, the model by Jansson & Farrar (2012), +predicts two large-scale inversions along the line of sight of interest; +an initial anti-parallel magnetic field (and a negative rotation measure +at the position of V404 Cyg), an inversion to a parallel orientation +at intermediate distances, followed by a second inversion back to +an anti-parallel orientation. However, using the standard approxima- +tion, |𝐵||,avg| = |RM/(0.81DM)|, we can estimate a mean parallel +magnetic field magnitude of 3.8 ± 0.7 𝜇G, which is larger than the +total ( +√︃ +𝐵2 +|| + 𝐵2 +⊥) Galactic magnetic field magnitudes predicted by +both Van Eck et al. (2011, ∼ 0.1 𝜇G) and Jansson & Farrar (2012, +∼ 1.0 𝜇G). Given our estimate for the mean parallel magnetic field +strength along the line of sight towards V404 Cyg, there exists three +physical explanations: (i) the mean electron number density is larger +than predicted by the standard dispersion models along this line of +sight; (ii) the mean magnetic field strength within the ISM is stronger +than predicted by Galactic magnetic field models along this line of +sight; or (iii) there is a local Faraday screen that likely resides within +the jets themselves. Here we investigate the source of a (potential) +rotation measure component local to V404 Cyg. +A local rotation measure component is the result of either a fore- +ground Faraday screen (e.g., created by disk outflows) or a rotation +from within the emission regions themselves (e.g., the compact core +or jet ejecta). Muñoz-Darias et al. (2016) detected a strong, contin- +uous wind originating from the accretion disk. Assuming that the +wind creates a foreground Faraday screen with an electron number +density that follows an inverse-square scaling, 𝑛𝑒 ≡ 𝑛0(𝑙/𝑙0)−2, and +a typical ISM magnetic field strength (𝐵||∼ 2 𝜇G; Haverkorn 2015), +equation (5) simplifies to, +RM = 1624 𝑛0𝑙2 +0 +� 1 +𝑙0 +− +1 +𝑙max +� +rad m−2, +(6) +where the wind-fed Faraday screen occupies the space between 𝑙0 +and 𝑙max along our line of sight. We approximate 𝑙max ∼ 𝑣Δ𝑡 ∼ 8 AU +using the measured wind velocity (𝑣 ∼ 2000 km s−1; Muñoz-Darias +et al. 2016) and the time interval between the start of the outburst +and our observations (Δ𝑡 ∼ 7 days). We adopt the VLBA angular +resolution of 1 AU, as a conservative estimate of 𝑙0 for compact core +emission. The jet ejections with well constrained inclination angles +are S2 (∼ 40◦), S3 (∼ 30◦), and S6 (∼ 15◦); all three ejecta have an +angular separation of ∼ 0.5 milliarcseconds during their flux density +peaks (Miller-Jones et al. 2019). The distance to V404 Cyg is 2.39 +kpc, and, therefore, 𝑙0 ∼ 2 − 5 AU for jet ejections. For a wind-fed +Faraday screen to produce our observed rotation measure (|RM| ∼ +100 rad m−2), we require 𝑛0 ∼ (6−15)×106 cm−3. Assuming a 50% +ionized, isotropic, pure hydrogen outflow, launched at a distance of +6 × 105 km from the central black hole, the wind mass loss rate +would need to be �𝑀 ∼ (0.4 − 2) × 10−6 𝑀⊙ yr−1. Muñoz-Darias +et al. (2016) estimated a wind mass loss rate of > 10−13 𝑀⊙ yr−1, +∼7 orders of magnitude smaller then our calculations. The authors +left the estimate as a lower limit because the ionization fraction +(∝ �𝑀−1) may be lower then the assumed value of 𝑓𝑖 = 0.5, and the +launching radius (∝ �𝑀) may be larger then their assumed value of 𝑅 = +6×105 km. A 7 order of magnitude reduction in the ionization fraction +would inhibit Faraday rotation, as the outflow would become neutral. +Furthermore, a 7 order of magnitude increase in the launching radius +corresponds to an distance of 4 × 104 AU, far exceeding the scale +of the system. Therefore, without a highly magnetized wind, or an +extremely anisotropic wind coupled with a favourable line of sight, +disk winds forming a foreground screen cannot be the origin of the +observed rotation measure. +For Faraday rotation internal to the emission environment we +look at the recent model of the compact jet from MAXI J1820+070 +(see, Zdziarski et al. 2022, for a detailed description of the model). +The strength of the magnetic field, and the electron number den- +sity scale according to the power-law relations, 𝐵 = 𝐵0𝜉−𝑏 and +𝑛𝑒 = 𝑛0𝜉−𝑎𝛾−𝑝, where 𝛾 is the lorentz factor of the synchrotron +emitting electrons, and 𝜉 = +𝑧 +𝑧0 = ( 𝜈 +𝜈0 )−𝑞, where 𝑧 is the position +along the jet axis. At 𝑧 > 𝑧0, the jet emits synchrotron radiation; this +leads to a break in the spectrum from optically thick to optically thin +at 𝜈0. +We adopt the following values used in the original paper: 𝑏 = 1.1; +𝑎 = 2.2; 𝑝 = 2; 𝑞 = 0.882; 𝐵0 ∼ 1010 𝜇G, 𝑧0 ∼ 3 × 1010 cm, +𝜈0 ∼ 2.3×104 GHz, 𝑛0 ∼ 3×1014 cm−3, and we adopt a value of 𝛾 = +0.5 × (𝛾min+𝛾max) = 386.5. The model predicts 𝐵 ∼ 3×106 𝜇G and +𝑛𝑒 ∼ 2 × 102 cm−3 at 𝜈 = 6 GHz. Letting, 𝐵|| = 0.5𝐵, and assuming +a uniform Faraday screen, we would require a screen thickness of +𝑑𝑙 ∼ 3×10−10 kpc to account for the rotation measure. To first-order, +this is the same as the radius of the conical jet at position 𝑧, 𝑅 = 𝑧 sin 𝜃, +for the best fit opening angle 𝜃 ∼ 1.5◦. Considering, that the best fit +orbital inclination angle is ∼65◦, it is reasonable to assume the our +line of sight looks partially down the jet axis, and, as a result, 𝑑𝑙 > 𝑅. +Furthermore, The electron number density could be substantially +larger than expected from a typical hard state compact jet if the jet +entrains material from the disk winds. Entrainment is a known source +of internal Faraday depolarization in AGN (e.g., Silpa et al. 2022), +and jet-wind interactions have been observed in the BHXB candidate +SS 443 (Blundell & Hirst 2011). Although we are unable to rule out +that V404 Cyg has a magnetic field oriented perpendicularly to the +line of sight, or significantly different jet parameters when compared +to MAXI J1820+070 (e.g., a weaker magnetic field), to first-order, +it is plausible that the jet itself may act as a strong, local Faraday +screen. +5 SUMMARY AND CONCLUSIONS +In this paper we present our analysis of the multi-frequency (5, 7, +21, and 26 GHz), linear polarization radio data of the BHXB V404 +Cyg during its 2015 outburst. The majority of our results and inter- +pretations focused on the behaviour during the bright flaring activity +on 2015 June 22, however, we also included the upper limits from +six observations during the source’s return to quiescence. Using two +independent polarimetric methods we extracted the fractional po- +larizations, observed/intrinsic EVPAs, and rotation measures from +MNRAS 000, 1–19 (2023) + +14 +A. K. Hughes et al. +the 2015 June 22 data. We tracked the evolution of the polarization +properties on timescales ∼13 min, constituting one of the shortest +timescale polarimetric analyses of a BHXB to date. +By comparing our polarimetric results to the VLA Stokes 𝐼 light +curves modelled by Tetarenko et al. (2017) and the simultaneous +VLBA observations by Miller-Jones et al. (2019), we infer the fol- +lowing properties about the polarization evolution of V404 Cyg: +• V404 Cyg is weakly polarized, with a maximum polarization +fraction that increases with frequency; ∼0.22, 0.25, 0.5, 0.75% for +the 5, 7, 21, and 26 GHz base-bands, respectively. These maxima are +significantly smaller than typically observed in outbursting BHXBs, +suggestive of a complex local environment or complex internal mag- +netic field structure. +• The time-evolution of the linear polarization fraction shows a +frequency-dependent lag, with low frequencies lagging behind their +high frequency counterparts. This behaviour is characteristic of an +emission origin within dynamic components (e.g., expanding ejecta +or propagating shock fronts). +• The maximum polarization fraction is offset from the Stokes 𝐼 +flux density maxima. This suggests an offset between the processes +that maximize each quantity. A secondary peak in fractional polar- +ization at 21 GHz after the second flare in Stokes 𝐼 and the increase +in polarization fraction towards the end of the 21/26 GHz base-band +observations, provide further evidence of a temporal offset. +• The decay of the (brightest) polarization fraction peak coincides +with a rotation of the intrinsic EVPA. We are unable to conclusively +determine if the origin of this feature is the result of an internal change +within the polarized components, or the emergence (and decay) of +polarized components with different magnetic field structures. +• The derived rotation measures show stability in time with an +average value of ∼−100 rad m−2. We investigated the potential of a +strong local component, and although we found it plausible, we are +unable to conclusively rule out a purely Galactic rotation measure. +Overall, our results emphasize the complexity of local (magnetic) +environments during highly energetic outbursts. Although we are +confident that the observed behaviour cannot be ascribed to the sim- +plest interpretation and models (e.g., intrinsic EVPA swings from +changes in the absorption conditions of single components), the lim- +itations of our observations inhibited us from making strong claims +about the origin of the polarized emission. These limitations empha- +size the importance of spatial resolution during polarimetry, which +would enable the identification of the primary source of the po- +larized emission in multi-component outbursts. X-ray polarimetric +observations of black hole X-ray binaries (like those recently done +for the black hole X-ray binary Cygnus X-1; Krawczynski et al. +2022) probe the accretion disk, corona and perhaps some compo- +nent from the synchrotron tail of a jet. For the radio-brightest X-ray +binary outbursts, there is strong potential to combine such observa- +tions with radio through sub-mm polarimetric observations to track +temporally evolving polarization properties across the electromag- +netic spectrum. However, we note that, like the VLA observations +performed here that did not have the angular resolution needed to sep- +arate polarization properties from different ejecta, one must carefully +consider how the polarization properties from different components +will average when making interpretations. If the next ∼1 Jy scale out- +burst of a BHXB is observed with adequate spatial resolution, full +polarization coverage, and sufficient sensitivity, such observations +have the potential to provide invaluable insight into the magnetic +fields that drive accretion-powered jet ejections, emission, and evo- +lution. The new generation of interferometers like the next-generation +VLA, the next-generation EHT, and the Square Kilometre Array, will +combine high sensitivity and good spatial resolution into a single in- +strument, making spatially resolved polarimetry a realistic goal for +future, bright outbursts. +ACKNOWLEDGEMENTS +We extend our sincere thanks to all of the NRAO staff involved in the +scheduling and execution of these observations. We offer a special +thanks to Frank Schnitzel for sharing his expertise of polarization +observation, and calibration, and data reduction with the VLA and +CASA. We thank Cameron Van Eck for helpful discussions regarding +Galactic magnetic field models and the Canadian Initiative for Radio +Astronomy Data Analysis (CIRADA) RM synthesis routine. Finally, +we thank the referee for their insightful and helpful comments. +This research has made use of software provided by CIRADA. +CIRADA is funded by a grant from the Canada Foundation for In- +novation 2017 Innovation Fund (Project 35999), as well as by the +Provinces of Ontario, British Columbia, Alberta, Manitoba, and Que- +bec, in collaboration with the National Research Council of Canada, +the US National Radio Astronomy Observatory and Australia’s Com- +monwealth Scientific and Industrial Research Organisation. +AKH and GRS are supported by NSERC Discovery Grants +RGPIN-2016-06569 and RGPIN-2021-0400. Support for this work +of AJT was provided by NASA through the NASA Hubble Fellow- +ship grant #HST–HF2–51494.001 awarded by the Space Telescope +Science Institute, which is operated by the Association of Universities +for Research in Astronomy, Inc., for NASA, under contract NAS5– +26555. TDR acknowledges financial contribution from the agreement +ASI-INAF n.2017-14-H.0. GEA is the recipient of an Australian Re- +search Council Discovery Early Career Researcher Award (project +number DE180100346). SM is thankful for support from an NWO +(Dutch Research Council) VICI award, grant Nr. 639.043.513. TMB +acknowledges the financial contribution from grant PRIN-INAF 2019 +N.15. RS acknowledges support from grant number 12073029 from +the National Natural Science Foundation of China (NSFC). +AKH and GRS respectfully acknowledge that they perform +the majority of their research from Treaty 6 territory, a tradi- +tional gathering place for diverse Indigenous peoples including the +Cree, Blackfoot, Métis, Nakota Sioux, Iroquois, Dene, Ojibway/ +Saulteaux/Anishinaabe, Inuit, and many others whose histories, lan- +guages, and cultures continue to influence our vibrant community. +The authors also wish to recognize and acknowledge the significant +cultural role and reverence that the summit of Maunakea has always +had within the indigenous Hawaiian community. We are most for- +tunate to have the opportunity to conduct VLBA observations from +this mountain. +DATA AVAILABILITY +Data from the VLA are available through the VLA data archive +(Project ID 15A–504): https://archive.nrao.edu/archive/ +advquery.jsp. 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J., Sikora M., 2022, ApJ, 925, 189 +Zurita C., Casares J., Shahbaz T., 2003, ApJ, 582, 369 +van der Horst A. J., et al., 2013, MNRAS, 436, 2625 +van der Laan H., 1966, Nature, 211, 1131 +1Department of Physics, University of Alberta, CCIS 4-181, Edmon- +ton, AB T6G 2E1, Canada +2International Centre for Radio Astronomy Research- Curtin Uni- +versity, GPO Box U1987, Perth, WA 6845, Australia +3Department of Physics & Astronomy, Texas Tech University, Lub- +bock, TX 79409-1051, USA +4East Asian Observatory, 660 N. A‘oh¯ok¯u Place, University Park, +Hilo, HI 96720, USA +5School of Physics and Astronomy, University of Southampton, +Southampton, SO17 1BJ, UK +6INAF-Osservatorio Astronomico di Brera, via E. Bianchi 46, I- +23807 Merate, Italy +7Department of Astronomy, University of Wisconsin Madison, 475 +N. Charter Street, Madison, WI 53706, USA +8SRON, Netherlands Institute for Space Research, Sorbonnelaan, 2, +NL-3584CA Utrecht, the Netherlands +9Department of Astrophysics/IMAPP, Radboud University, P.O. Box +9010, NL-6500 GL Nijmegen, The Netherlands +10Department of Physics and Astronomy, Wheaton College, Norton, +MA 02766, USA +11Anton Pannekoek Institute for Astronomy, University of Amster- +dam, Science Park 904, NL-1098 XH, Amsterdam, The Netherlands +12Gravitation Astroparticle Physics Amsterdam (GRAPPA) Institute, +University of Amsterdam, Science Park NL-904, 1098 XH Amster- +dam, The Netherlands +13Aurora Technology BV for the European Space Agency, ESAC/ESA, +Camino Bajo del Castillo s/n, Urb. Villafranca del Castillo, 28691, +Villanueva de la Cañada, Madrid, Spain +14Institut de Ciències del Cosmos (ICC), Universitat de Barcelona +(IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain. +15National Radio Astronomy Observatory, Socorro, NM 87801, USA +16Caltech, 1200 E. California Blvd. MC 249-17, Pasadena, CA +91125, USA +17Herzberg Institute of Astrophysics, National Research Council of +Canada, Penticton, BC V2A 6J9, Canada +18Center for Astro, Particle and Planetary Physics, New York Uni- +versity, Abu Dhabi, PO Box 129188, Abu Dhabi, UAE +19INAF/IASF Palermo, via Ugo La Malfa 153, I-90146 Palermo, +Italy +20Department of Astronomy, University of Virginia, P.O. Box 400325, +Charlottesville, VA 22901, USA +21College of Astronomy and Space Sciences, University of the Chi- +nese Academy of Sciences, Beijing 100049, China +22Sydney Institute for Astronomy, School of Physics A28, The Uni- +versity of Sydney, Sydney, NSW 2006, Australia +23INAF - Osservatorio Astrofisico di Torino, Strada Osservatorio 20, +10025, Pino Torinese, Italy +24Institute for Space Sciences, Atomistilor 409, PO Box MG-23, +077125 Bucharest-Magurele, Romania +MNRAS 000, 1–19 (2023) + +V404 Cyg’s rapidly evolving polarized jet +17 +APPENDIX A: STOKES I LIGHT CURVES AND +TEMPORAL DELAYS +The Stokes 𝐼 flux density light curves for all four base-bands share a +common “three-flare” morphology (see Fig. A1), where each flare is +composed of multiple unresolved (by the VLA) jet ejecta (Tetarenko +et al. 2017; Miller-Jones et al. 2019). The temporal delays and longer +rise/decay times for the lower frequency flares are consistent with +the expected behaviour of expanding vdL bubbles. Considering a +single vdl ejection, the flaring results from an evolving optical depth; +the peak flux density occurs near the transition from optically thick +to optically thin emission, and the peak occurs at earlier times for +higher frequencies (van der Laan 1966). Therefore, broadband ob- +servations will mix optically thick (low frequency) and optically thin +(high frequency) emission. Optically thick and thin emission are +orthogonally polarized, and thus their summation causes a (poten- +tially significant) depolarization effect. To quantify this effect, we +measured the delays between flux density peaks (for each of the three +flares) from the cross-correlation function (CCF) of the high time res- +olution (per-spectral window) light curves. Each CCF was generated +using the z-transformed discrete correlation function (ZDCF) tech- +niques of Alexander (1997)5. We measured delays between flares of +∼30–60 min (∼2–4× the imaging window) and ∼7–15 min (∼0.5–1× +the imaging window) comparing the 5-to-26 GHz and 5-to-7 GHz +base-band light curves, respectively. The earliest flare exhibited the +smallest delays between bands. For the polarization fraction, the +small number of time bins inhibited a similar use of the ZDCF +algorithm. However, looking at Figure 1, the delays between the +polarization fraction peaks are ∼2 imaging windows (∼30 min) and +≲1 imaging window (≲15 min), for the 5-to-26 GHz and 5-to-7 GHz +delays, consistent with the Stokes 𝐼 behaviour. The 5-to-26 GHz de- +lays are an appreciable fraction of the lifetime of a single ejection +(≲ 90 min), and, therefore, the full bandwidth (5-to-26 GHz) may +have a non-negligible fraction of orthogonal emission, even when +considering isolated ejecta. +The situation becomes considerably more complicated when con- +sidering the multi-ejecta flares, ejecta collision, and jet precession (as +seen in the 2015 outburst of V404 Cyg). Moreover, we note that mod- +elled ejecta exhibit different delays between frequencies (largely due +to differences in ejecta expansion velocities). As such, the dominant +ejection (in a particular flare) depends on the observing frequency, +thereby introducing another frequency-dependent effect on the ob- +served EVPA. While critical, the VLBA observations only provide +a snapshot at one frequency. Due to the large temporal separations, +we choose to omit the simultaneous linear polarization data from +the 21 and 26 GHz base-bands when extracting EVPAs and rotation +measures. This is why we only consider the simultaneous linear po- +larization data from the 5 and 7 GHz base-bands in § 3.2, § 4.2, and +§ 4.3. We intend to apply temporal corrections to future (broad-band) +outbursts with single (or temporally isolated) ejecta as an investiga- +tion into the effects these delays have on polarization measurements. +APPENDIX B: CALIBRATOR MODEL +For Stokes 𝐼 calibration, casa includes a repository of spatially re- +solved model images for many standard calibrators. Our flux calibra- +tor, 3C48, is included in this repository. Each model image describes +the flux density distribution at a single, band-dependant, reference +5 FORTRAN +code +available +at +http://www.weizmann.ac.il/ +particle/tal/research-activities/software. +frequency (e.g., 𝜈ref = 4.8601 GHz for the 4-8 GHz band). During +calibration, the model is mapped onto the remaining spectral chan- +nels assuming the total flux density follows the flux density scaling +relationships of Perley & Butler (2017). It is assumed that the spatial +distribution of the relative flux densities remains constant across a +band; i.e., for an arbitrary spectral channel with a central frequency 𝜈, +the ratio between the total integrated flux, 𝐼𝜈, and the flux of pixel 𝑖, +𝐼𝑖,𝜈, is independent of frequency, and thus identical to the ratio at the +reference frequency (𝐼𝑖,𝜈/𝐼𝜈 ≡ 𝐼𝑖,𝜈ref/𝐼𝜈ref). This procedure results +in a spatially resolved Stokes 𝐼 model for every spectral channel. +For our Stokes 𝑄 and 𝑈 calibration, we adopted a similar approach +to the default Stokes 𝐼 prescription. We assumed that the spatial +distribution of the linearly polarized flux densities is independent of +frequency and that it has the same spatial distribution as the Stokes 𝐼 +repository images. For each spectral channel with a central frequency +of 𝜈, we calculated the total Stokes 𝑄 and 𝑈 flux densities according +to the following relationships, +𝑄𝜈 = 𝐼𝜈 𝑓𝜈 cos (2𝜒𝜈) , +(B1) +𝑈𝜈 = 𝐼𝜈 𝑓𝜈 sin (2𝜒𝜈) , +(B2) +where 𝑓𝜈 is the linear polarization fraction, and 𝜒𝜈 is the observed +EVPA. We mapped the total flux densities onto each pixel by as- +suming that 𝑈𝑖,𝜈/𝑈𝜈 ≡ 𝑄𝑖,𝜈/𝑄𝜈 ≡ 𝐼𝑖,𝜈ref/𝐼𝜈ref. Our final model +consisted of a spatially resolved Stokes 𝐼, 𝑄, and 𝑈 image (assuming +no circular polarization; i.e., Stokes 𝑉 = 0) for each spectral channel. +We then applied the model to our data using the native casa task ft. +In equation (B1) and (B2), 𝐼𝜈 was calculated according to Perley +& Butler (2017). For 𝑓𝜈 and 𝜒𝜈, we fit the data presented in Perley +& Butler (2013), such that the linear polarization fraction obeys a +third-order log-log polynomial, +log( 𝑓𝜈) = +3 +∑︁ +𝑛=0 +𝑎𝑛 log (𝜈GHz)𝑛 , +(B3) +and the observed EVPAs obey a standard third-order polynomial +(with frequency), +𝜒𝜈 = +3 +∑︁ +𝑛=0 +𝑏𝑛 (𝜈GHz)𝑛 ; +(B4) +where 𝜈 GHz is the observing frequency (in GHz). We decided to fit +the polarization fraction with a 3rd-order polynomial in log-space to +remain consistent with the 𝐼𝜈 fits in Perley & Butler (2017), and 𝜒 +in linear-space to allow for negative EVPAs. We fit the polarization +properties separately for the 5/7 GHz and 21/26 GHz bands, rather +than a single fit as performed by Perley & Butler (2017), following +extensive discussions with NRAO staff (F. Schinzel priv. comm.). +Table B1 contains the third-order polynomial fit for the polarization +calibrator model and Fig. B1 plots the fit over the Perley & Butler +(2013) observations of 3C48. +APPENDIX C: PHASE CALIBRATOR POLARIMETRIC +EVOLUTION +Given the low linear polarization fractions we detected in our V404 +Cyg observations, we checked the relative stability of our polariza- +tion calibrations on short time scales to ensure that the variability we +observed is the result of intrinsic variations and not systematic cali- +bration effects. Therefore, we performed our full polarimetric analy- +sis on the phase calibrator (J2025+3343), grouping scans within the +same bins as used for V404 Cyg when making images. +MNRAS 000, 1–19 (2023) + +18 +A. K. Hughes et al. +Figure A1. High time resolution Stokes 𝐼 light curves for our observing base-bands. In each base-band, we have separately plotted the flux densities of the +8 spectral windows. The key features are the three major flares, with the final flare having a twin-peaked structure in the 21/26 GHz base-bands. The lower +frequency emission is temporally delayed, peaks at lower flux densities, and has broader flares. These properties are consistent with both vdL ejecta and a +shock-in-jet event scenario. +Figure B1. Polarization fraction spectra for 3C48; left: 5/7 GHz base-bands and right: 21/26 GHz base-bands. The shaded regions highlight the range of +frequencies spanned by each observing band. The solid black points are the data from Perley & Butler (2013), and the black line is our polynomial fit. The +logarithmic term in equation (3) is exactly equal to the equivalent frequency in GHz. +MNRAS 000, 1–19 (2023) + +26 GHz +1400 +21 GHz +1200 +7 GHz +Density +5 GHz +Fluy +BDD +Stokes +600 +400 +200. +11:00 +DE:TT +12:00 +DE·ZL +13:00 +DE:ZL +DE:t +Time on 2015 June 22 (UTC)6.0 +8.5 +(%) +Fraction +5.5 +8.0 +5.0 + Polarization +7.5 +4.5 +4.0 +7.0 +3.5 +Linear J +N +6.5 +0'E +2.5 +6.0 +4 +5 +6 +8 +6 +10 +16 +18 +20 +27 +24 +26 +28 +4 +36 +38 +-1.0 +1.05 +-1.10 + (rad) +-1.1 +1.15 +EVPA +-1.2 +-1.20 +-1.25 +Observed +1.3 +1.30 +-1.4 +-1.35 +-1.5 +-1.40 +-1.45 +16 +4 +5 +6 +8 +9 +10 +18 +20 +22 +2426 +28 +32 +34 +36 +38 +Frequency (GHz) +Frequency (GHz)V404 Cyg’s rapidly evolving polarized jet +19 +Table B1. The third-order polynomial fits for Stokes 𝐼, the polarization +fraction spectra, and the intrinsic EVPA of 3C48. The 𝐼𝜈 values are taken +from Perley & Butler (2017). +Quantity +Band +𝑎0 +𝑎1 +𝑎2 +𝑎3 +𝐼𝜈 +5/7 GHz +21/26 GHz +1.3253 +1.3253 +−0.7553 +−0.7553 +−0.1914 +−0.1914 +0.0498 +0.0498 +𝑓𝜈 +5/7 GHz +21/26 GHz +−1.5775 +1.5927 +−1.6671 +−8.9992 +4.7686 +8.5932 +−2.8181 +−2.5275 +Quantity +Band +𝑏0 +𝑏1 +𝑏2 +𝑏3 +𝜒 +5/7 GHz +21/26 GHz +−3.7987 +0.9939 +1.1141 +−0.2480 +−0.1602 +0.0092 +0.0078 +−0.0001 +Figure C2 shows the temporal evolution of the residual rota- +tion measure and residual observed EVPA for both V404 Cyg and +J2025+3343. Here we define the residual as the difference between +the individual time bins, and the weighted average over all time +bins. Visually, J2025+3343 shows both a stable rotation measure +(RM ∼ −750 rad m−2) and observed EVPA (𝜒𝑤 ∼ −35◦). Applying +the same 𝜒2 test as discussed in Section 3.2, neither the rotation +measure (𝜒2 = 27.1/12) nor the observed EVPA (𝜒2 = 2508.3/12) +is consistent with a constant value. However, the linear polarization +detections of J2025+3343 have a much higher signal-to-noise ratio +(S/N > 200), and, as a result, we have likely reached a systematic +threshold. As such, we believe we are underestimating the errors us- +ing an unrestricted S/N scaling (i.e., as S/N → ∞, 𝜎𝜒𝑤 → 0). The +𝜒𝑤 standard deviation for J2025+3343 is ∼ 1.4◦, which is equal to +the smallest 𝜒𝑤 error for V404 Cyg (∼1.4◦) and ∼ 1/2 of our median +error (∼2.6◦). Since V404 Cyg exhibits a ∼ 30◦ degree rotation, the +systematic variability cannot be the cause of the observed evolution. +Figure C2 compares the temporal evolution of the linear polar- +ization fraction, defining the “residual” in the same manner as in +Figure C1. The evolution of J2025+3343 does not track the simulta- +neous evolution of V404 Cyg. The multi-band “jumps”, at ∼ 12:00 +and ∼ 13:45 UTC, correspond to a change of sub-arrays (marked by +the vertical dotted lines). It is unsurprising to see some discontinuity +between the two sub-arrays as each sub-array will have (slightly) +different 𝑢𝑣-coverage, a unique reference antenna, and, as a result, +different calibration solutions. We note: (i) the bin-by-bin variability +of J2025+3343 within a sub-array is significantly smaller than the +jumps; and (ii) although obvious in J2025+3343, we do not observe +similar jumps in our V404 Cyg data — instead, the most significant +evolution occurs absent a change of sub-array; (iii) In all four base- +bands, the variability of V404 Cyg is larger than J2025+3343; (iv) +V404 Cyg shows a common temporal evolution across base-bands, +that is absent in the J2025+3343 data. Therefore, we are confident that +the polarized detections of V404 Cyg are dominated by an intrinsic, +physical evolution. +APPENDIX D: SAMPLE IMAGES +Figure D1 shows a sample set of 𝑃, 𝑄, 𝑈 images at both 5 and 7 GHz. +For this example, despite the clear detection in 𝑃 (top row), we are +unable to detect the source in the Stokes 𝑄 images (middle row). +Due to the intrinsic variability of the source, in other time bins and +frequency ranges the properties of the non-detections may change +(e.g., Stokes 𝑄 is detected but Stokes 𝑈 is not). As a result, we chose +to extract the Stokes 𝑄 and 𝑈 flux densities using forced aperture +photometry. +APPENDIX E: DATA TABLES +The following tables summarize the key observations: Table E1 con- +tains the polarization fraction observations; and Table E2 contains +the EVPAs and rotation measures derived from both polarimetric +routines. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–19 (2023) + +20 +A. K. Hughes et al. +Figure C1. Temporal evolution of the residual rotation measure (top), and observed EVPA (bottom). The red, dashed line represents the phase calibrator +J2025+3343, and the black, solid line, V404Cyg. The observed EVPA of V404 Cyg exhibits a clear evolution that is absent in J2025+3343. +MNRAS 000, 1–19 (2023) + +V404 Cyg +100F +J2025+3343 +ARM (radm-2) +-100 +-200 +-300 +20 +15 +10 +(.) +MXV +10 +11:30 +12:00 +12:30 +13:00 +13:30 +14:00 +Time on 2015 June 22 (UTC)V404 Cyg’s rapidly evolving polarized jet +21 +Figure C2. Temporal evolution of the linear polarization fraction residuals; 26 GHz (top), 21 GHz (2nd from the top), 7 GHz (3rd from the top), 5 GHz (bottom). +The red, dashed line represents the phase calibrator J2025+3343, and the black, solid line, V404 Cyg. The vertical dashed lines highlight the times when the +observing band switched from one sub-array to another; the jumps are the result of this transition. We can see that the linear polarization fraction evolution of +J2025+3343 does not track V404 Cyg, and shows smaller amplitude variability in all base-bands. +MNRAS 000, 1–19 (2023) + +0.4 +V404 Cyg +-*--J2025+3343 +0.2 +00 +-0.2 +26 GHz +0.2 +0.1 +0'0 +0.1 +21 GHz +(%) +0.10 +0.05 +0.00 +-0.05 +7 GHz +0.10 +0.05 +0.00 +-0.05 +5 GHz +11:31 +12:00 +12:28 +12:57 +13:26 +13:55 +14:24 +Time on 2015 June 22 (UTC)22 +A. K. Hughes et al. +Figure D1. Sample images of the 12:09 time bin for both the 5 GHz (left) and 7 GHz (right) base-bands. (top) MFS images (i.e., ∼ 1 GHz bandwidths) of the +linear polarization intensity (𝑃 = +√︁ +𝑄2 + 𝑈2). (middle) Fine spectral resolution Stokes 𝑄 images. (bottom) Fine spectral resolution Stokes 𝑈 images. The +contours show the 3, 4, and 5𝜎 levels, and the color bars are in units of mJy/beam. The example images at fine spectral resolution have central frequencies of +5.209 GHz and 7.545 GHz with bandwidths of 16 MHz and 64 MHz for the 5 and 7 GHz base-bands, respectively. Stokes 𝑄 and 𝑈 are not positive definite, and, +as a result, either may appear as non-detections regardless of the strength of the detection in 𝑃. In this example, the source is not detected in Stokes 𝑄 despite +its detections in both 𝑃 and Stokes 𝑈. This behaviour motivated our use of forced aperture photometry. +MNRAS 000, 1–19 (2023) + +5 GHz +7 GHz +33°52'04" +33°52'04" +0.8 +0.8 +0.7 +0.7 +03" +03" +0.6 +0.6 +-0.5 +0.5 +02" +02" +0.4 +0.4 +0.3 +0.3 +01" +01" +0.2 +0.2 +0.1 +0.1 +00" +00" +20h24m04.0s +s6'E0 +03.85 +03.7s +20h24m04.0s +03.9s +03.8s +03.75 +33°52'04" +0.8 +33°52'04" +0.6 +(J2000) +0.6 +0.4 +03" +03" +0.4 +0.2 +-0.2 +Declination +02" +02" +0.0 +-0.0 +-0.2 +01" +01" +0.4 +-0.2 +0.6 +00" +00" +0.4 +-0.8 +20h24m04.0s +03.9s +03.85 +03.75 +20h24m04.0s +03.9s +03.85 +03.75 +33°52'04" +33°52'04" +1.2 +1.00 +1.0 +03" +0.75 +03" +0.8 +0.50 +0.6 +02" +02" +0.25 +0.4 +0.2 +0.00 +01" +01" +0.0 +-0.25 +-0.2 +0.50 +..00 +00" +-0.4 +-0.75 +20h24m04.0s +03.9s +03.85 +03.75 +20h24m04.0s +03.9s +03.85 +03.75 +Right Ascension (12000)V404 Cyg’s rapidly evolving polarized jet +23 +Table E1. A summary of observations from the base-band integrated polarization fraction images for each base-band; t𝑐𝑡𝑟 is each time bin’s central UTC time +during the 22 June 2015 observations. Sig corresponds to the probability that the detection is intrinsic to the source, and is not a calibration artifact (i.e., the +significance level as defined in the text). See Section 2 for definitions of the remaining parameters. +5 GHz +7 GHz +𝑡ctr (HH:MM) +𝐼𝜆 (mJy) +𝑃𝜆,0 (mJy) +𝑓𝜆 (%) +𝑃𝜆,0 +𝜎𝑄𝑈 +Sig (%) +𝐼𝜆 (mJy) +𝑃𝜆,0 (mJy) +𝑓𝜆 (%) +𝑃𝜆,0 +𝜎𝑄𝑈 +Sig (%) +11:15 +305.3 ± 0.2 +0.24 ± 0.03 +0.080 ± 0.010 +7.6 +83 +275.5 ± 0.4 +0.55 ± 0.03 +0.199 ± 0.012 +16.4 +> 99 +11:29 +363.3 ± 0.2 +0.36 ± 0.03 +0.099 ± 0.009 +10.6 +94 +360.6 ± 0.3 +0.78 ± 0.04 +0.217 ± 0.012 +18.6 +> 99 +11:42 +387.5 ± 0.1 +0.82 ± 0.04 +0.212 ± 0.009 +22.5 +> 99 +381.1 ± 0.2 +0.93 ± 0.04 +0.245 ± 0.011 +23.3 +> 99 +11:54 +379.4 ± 0.1 +0.79 ± 0.03 +0.208 ± 0.009 +22.6 +> 99 +368.8 ± 0.2 +0.86 ± 0.04 +0.233 ± 0.012 +19.8 +> 99 +12:09 +425.7 ± 0.4 +0.85 ± 0.05 +0.200 ± 0.011 +18.3 +> 99 +482.8 ± 0.8 +0.83 ± 0.05 +0.172 ± 0.011 +16.3 +> 99 +12:21 +527.3 ± 0.5 +0.92 ± 0.05 +0.174 ± 0.009 +18.5 +> 99 +634.4 ± 0.7 +1.02 ± 0.06 +0.161 ± 0.010 +15.8 +> 99 +12:33 +621.7 ± 0.5 +0.81 ± 0.05 +0.131 ± 0.009 +14.8 +95 +732.2 ± 0.5 +0.86 ± 0.07 +0.118 ± 0.010 +12.3 +92 +12:45 +663.3 ± 0.2 +0.73 ± 0.05 +0.110 ± 0.008 +13.3 +88 +709.7 ± 0.6 +0.80 ± 0.07 +0.113 ± 0.009 +12.0 +90 +12:57 +637.6 ± 0.3 +0.61 ± 0.06 +0.096 ± 0.009 +10.6 +80 +620.8 ± 0.6 +0.66 ± 0.06 +0.106 ± 0.009 +11.3 +87 +13:11 +592.6 ± 0.3 +0.61 ± 0.05 +0.103 ± 0.008 +12.2 +84 +548.2 ± 0.4 +0.58 ± 0.05 +0.106 ± 0.009 +11.5 +87 +13:25 +542.7 ± 0.3 +0.60 ± 0.05 +0.111 ± 0.009 +12.5 +88 +511.9 ± 0.2 +0.57 ± 0.04 +0.111 ± 0.009 +12.8 +89 +13:39 +514.1 ± 0.2 +0.44 ± 0.05 +0.086 ± 0.009 +9.3 +73 +519.2 ± 0.2 +0.49 ± 0.05 +0.095 ± 0.009 +10.4 +80 +13:53 +504.0 ± 0.2 +0.39 ± 0.04 +0.076 ± 0.009 +8.9 +81 +569.9 ± 0.5 +0.44 ± 0.06 +0.077 ± 0.010 +7.7 +79 +14:07 +538.3 ± 0.2 +0.37 ± 0.04 +0.069 ± 0.007 +9.2 +73 +650.7 ± 0.4 +0.35 ± 0.06 +0.054 ± 0.009 +5.7 +54 +14:20 +587.5 ± 0.4 +0.34 ± 0.04 +0.057 ± 0.008 +7.5 +60 +727.7 ± 0.4 +0.42 ± 0.07 +0.058 ± 0.010 +5.9 +60 +14:32 +630.8 ± 0.2 +0.35 ± 0.05 +0.056 ± 0.008 +7.0 +58 +711.4 ± 0.6 +0.51 ± 0.07 +0.072 ± 0.010 +7.3 +75 +21 GHz +26 GHz +𝑡ctr (HH:MM) +𝐼𝜆 (mJy) +𝑃𝜆,0 (mJy) +𝑓𝜆 (%) +𝑃𝜆,0 +𝜎𝑄𝑈 +Sig (%) +𝐼𝜆 (mJy) +𝑃𝜆,0 (mJy) +𝑓𝜆 (%) +𝑃𝜆,0 +𝜎𝑄𝑈 +Sig (%) +11:15 +516.8 ± 0.9 +2.01 ± 0.10 +0.388 ± 0.020 +19.4 +96 +604.3 ± 1.0 +4.09 ± 0.21 +0.677 ± 0.035 +19.2 +> 99 +11:29 +498.1 ± 1.0 +2.41 ± 0.10 +0.484 ± 0.019 +25.2 +> 99 +482.4 ± 1.5 +3.61 ± 0.20 +0.748 ± 0.042 +17.8 +> 99 +11:42 +457.0 ± 0.8 +1.43 ± 0.09 +0.313 ± 0.020 +15.8 +88 +510.6 ± 1.8 +1.84 ± 0.23 +0.360 ± 0.046 +7.8 +80 +11:54 +749.3 ± 2.0 +2.05 ± 0.14 +0.274 ± 0.018 +15.1 +80 +928.3 ± 2.5 +2.28 ± 0.40 +0.246 ± 0.043 +5.7 +53 +12:09 +1249.0 ± 0.8 +3.61 ± 0.19 +0.289 ± 0.016 +18.5 +88 +1386.3 ± 1.0 +3.26 ± 0.41 +0.235 ± 0.029 +8.0 +61 +12:21 +1162.2 ± 1.4 +2.64 ± 0.18 +0.228 ± 0.015 +14.7 +73 +1194.7 ± 2.1 +3.00 ± 0.39 +0.251 ± 0.032 +7.8 +65 +12:33 +859.9 ± 1.8 +1.26 ± 0.15 +0.147 ± 0.017 +8.5 +42 +781.7 ± 2.1 +2.23 ± 0.24 +0.286 ± 0.031 +9.4 +75 +12:45 +572.5 ± 0.8 +1.11 ± 0.11 +0.195 ± 0.020 +9.9 +61 +485.1 ± 0.7 +1.43 ± 0.15 +0.294 ± 0.032 +9.3 +77 +12:57 +468.3 ± 0.5 +1.31 ± 0.12 +0.279 ± 0.025 +11.4 +86 +409.7 ± 0.3 +1.37 ± 0.14 +0.334 ± 0.035 +9.5 +85 +13:11 +501.1 ± 1.0 +1.15 ± 0.11 +0.229 ± 0.021 +10.9 +73 +534.4 ± 1.9 +1.62 ± 0.18 +0.303 ± 0.034 +9.0 +79 +13:25 +851.7 ± 2.1 +1.65 ± 0.18 +0.194 ± 0.021 +9.2 +61 +1075.1 ± 2.8 +3.27 ± 0.39 +0.304 ± 0.037 +8.3 +79 +13:39 +1223.9 ± 1.5 +2.11 ± 0.28 +0.173 ± 0.023 +7.7 +53 +1462.3 ± 0.9 +5.08 ± 0.58 +0.347 ± 0.040 +8.8 +87 +13:53 +1231.9 ± 0.6 +2.74 ± 0.22 +0.222 ± 0.018 +12.3 +66 +1228.9 ± 0.8 +3.89 ± 0.45 +0.316 ± 0.036 +8.7 +71 +14:07 +1259.3 ± 0.6 +3.20 ± 0.22 +0.254 ± 0.017 +14.5 +75 +1248.9 ± 1.3 +4.59 ± 0.44 +0.368 ± 0.035 +10.5 +81 +14:20 +1021.5 ± 2.6 +3.04 ± 0.20 +0.298 ± 0.019 +15.5 +86 +913.5 ± 3.0 +3.93 ± 0.36 +0.431 ± 0.039 +10.9 +90 +14:32 +568.6 ± 2.0 +2.05 ± 0.15 +0.361 ± 0.027 +13.6 +94 +469.2 ± 1.9 +2.46 ± 0.22 +0.524 ± 0.048 +11.0 +97 +MNRAS 000, 1–19 (2023) + +24 +A. K. Hughes et al. +Table E2. A summary of the RM synthesis and MCMC results, and 𝑡𝑐𝑡𝑟 adopts the same definition as used in Table E1. In this chart we’ve defined the S/N as +the amplitude of the FDF component over the rms error across the FDF; any component with a S/N > 5 was recorded, but only one time bin (at 11:29 UTC) +had a secondary. The lone secondary component had a similar magnitude (|𝑅𝑀 | ∼ 2300 rad m−2) as the systematic errors discussed in Section 3.2. We do not +believe this to be a real signal and have omitted this component from the table. Note that the two components with the most significant deviations from the +weighted mean (∼ − 100 rad m−2) are also the lowest S/N detections (S/N∼ 6). +RM Synthesis +MCMC +𝑡ctr (HH:MM) +RM (rad m−2) +𝜒𝑤 (◦) +𝜒0 (◦) +S/N +RM (rad m−2) +𝜒𝑤 (◦) +𝜒0 (◦) +11:15 +−330+110 +−110 +62+4 +−4 +76+5 +−5 +6.8 +−328+60 +−60 +63+3 +−3 +77+3 +−3 +11:29 +−79+90 +−90 +62+3 +−3 +76+4 +−4 +8.3 +−35+70 +−60 +64+3 +−3 +78+4 +−3 +11:42 +−122+40 +−40 +59+1 +−1 +73+3 +−3 +20.2 +−127+30 +−30 +61+1 +−2 +75+2 +−2 +11:54 +−92+40 +−40 +52+1 +−1 +66+3 +−3 +19.7 +−110+40 +−40 +53+2 +−2 +67+3 +−3 +12:09 +−75+40 +−40 +46+2 +−2 +61+3 +−3 +17.5 +−70+30 +−30 +45+2 +−2 +59+2 +−2 +12:21 +−115+50 +−50 +43+2 +−2 +58+3 +−3 +16.8 +−91+40 +−40 +41+2 +−2 +56+3 +−3 +12:33 +−122+60 +−60 +38+2 +−2 +52+3 +−3 +11.9 +−100+50 +−50 +36+2 +−3 +50+3 +−3 +12:45 +−91+70 +−70 +36+3 +−3 +50+3 +−3 +10.9 +−107+50 +−50 +36+3 +−3 +50+3 +−3 +12:57 +−67+80 +−80 +36+3 +−3 +50+4 +−4 +9.2 +−46+50 +−50 +38+3 +−3 +52+3 +−3 +13:11 +−48+70 +−70 +38+3 +−3 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Timescale Evolution of the Polarized Radio Jet during V404 Cygni’s 2015 Outburst A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes,1★ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sivakoff,1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Macpherson,2 J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Anderson,2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Belloni,6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Heinz,7 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Jonker8,9, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Körding9, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Maitra10, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Markoff11,12, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Migliari13,14, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Mooley15,16, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Rupen17, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Russell18, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Russell19, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sarazin20, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Soria21,22,23, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tudose24 Affiliations are listed at the end of the paper Accepted 2023 January 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Received 2023 January 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' in original form 2022 October 4 ABSTRACT We present a high time resolution, multi-frequency linear polarization analysis of Very Large Array (VLA) radio observations during some of the brightest radio flaring (∼1 Jy) activity of the 2015 outburst of V404 Cygni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The VLA simultaneously captured the radio evolution in two bands (each with two 1 GHz base-bands), recorded at 5/7 GHz and 21/26 GHz, allowing for a broadband polarimetric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Given the source’s high flux densities, we were able to measure polarization on timescales of ∼13 minutes, constituting one of the highest temporal resolution radio polarimetric studies of a black hole X-ray binary (BHXB) outburst to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Across all base-bands, we detect variable, weakly linearly polarized emission (<1%) with a single, bright peak in the time-resolved polarization fraction, consistent with an origin in an evolving, dynamic jet component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We applied two independent polarimetric methods to extract the intrinsic electric vector position angles and rotation measures from the 5 and 7 GHz base-band data and detected a variable intrinsic polarization angle, indicative of a rapidly evolving local environment or a complex magnetic field geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Comparisons to the simultaneous, spatially-resolved observations taken with the Very Long Baseline Array at 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 GHz, do not show a significant connection between the jet ejections and the polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Key words: black hole physics — ISM: jets and outflows — polarization — radio continuum: stars — stars: individual (V404 Cygni, GS 2023+338) — X-rays: binaries 1 INTRODUCTION A black hole X-ray binary (BHXB) is an interacting binary sys- tem composed of a stellar-mass black hole accreting material from a companion star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Standard features of BHXBs are jets and winds, making them ideal candidates for the study of accretion-fed outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The majority of these systems spend most of their lifetimes in quies- cence, accreting small amounts of matter, at low X-ray luminosities (𝐿𝑋 ≲ 1032 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Most of the known systems sporadically enter into bright (𝐿𝑋 > 1035 erg s−1), transient outbursts that last weeks to years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016), allowing for real-time observations of the evolving accretion flow (best measured at X-ray frequencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Belloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tomsick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Kylafis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Plant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014) and relativistic jets (best measured at radio through infrared frequencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Corbel & Fender 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' van der Horst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During an outburst, the morphological evolution of the jet closely correlates with the X-ray properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', accretion states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Mc- Clintock & Remillard 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Belloni 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Fender 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the hard accretion state, an optically thin X-ray corona dominates the X-ray emission and the jet adopts a steady, compact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In ★ E-mail: hughes1@ualberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='ca † NASA Einstein Fellow some systems, the compact jet is observed to persist into quies- cence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Gallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016, we note that, for many BHXBs, jets in quiescence will have flux densities be- low the detection capabilities of most facilities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Compact jet spectra are described by optically-thick, partially self-absorbed synchrotron emission with an inverted or flat spectral index (𝛼 ≳ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' with flux den- sity 𝐹𝜈 ∝ 𝜈𝛼) up to a break frequency (typically in the sub-mm or infrared regime) where the spectra become optically thin (𝛼 ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6) to higher frequency emission (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The flat/inverted spectral index is thought to result from the superposition of mul- tiple spatially-unresolved synchrotron components originating from different positions along the jet axis (Blandford & Königl 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Conversely, in the soft accretion state, thermal emission from the accretion disk dominates the X-ray spectrum, and the radio emission from the compact jet decreases or is fully quenched (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2011, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the hard-to-soft transition, one or more blobs of discrete jet ejecta are typically launched, and these ejecta have been spatially resolved in several sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Mirabel & Rodríguez 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hjellming & Rupen 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hannikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Rushton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The ejection events are attributed to brief periods of highly efficient plasma production at the base of the jet, creating (often adiabatically) expanding plasma knots threaded with complex magnetic fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the van der Laan —vdL— model, van der Laan 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hjellming © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='13281v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='HE] 30 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' & Johnston 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hjellming & Han 1995, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Each ejection has an emission spectrum characterized by a single self-absorbed synchrotron source with a temporally evolving elec- tron/lepton population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As the ejection propagates and expands, the self-absorption turnover transitions to lower frequencies, extending deep into the radio regime and resulting in an observing bandwidth that is optically thin in its entirety (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the radio, these ejections are observed as multi-frequency flares with well-defined rise and decay phases that last minutes to days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to the expansion-driven, evolving optical depth, the lower frequency components are broadened in time and temporally delayed with respect to the higher frequency counterparts (Mirabel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Flaring events from Active Galactic Nuclei (AGN, the large-scale analogous of BHXBs) have also been mod- elled based on the adiabatic expansion of jet plasma (Yusef-Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Falcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Maitra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Michail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Alternative flaring models can also be applied to BHXB observa- tions, such as the “shock-in-jet” picture that is typically associated with AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Marscher & Gear 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Spada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Within this framework, each flare is the result of shocks within a (quasi-) steady jet accelerating particles and temporarily enhancing emission intensities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Türler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Türler 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Malzac 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Most radio jets from BHXBs are described by their photometric, spectral, and (when available) spatial properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, a much smaller fraction of studies explore the linear polarization that results from a synchrotron dominated emission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For optically thin and optically thick synchrotron emission, the maximum expected lin- ear polarization fraction is 𝑓𝜆 = (3𝑝 + 3)/(3𝑝 + 7) × 100 % ≈ 70 % and 𝑓𝜆 = 3/(6𝑝 + 13) × 100 % ≈ 10 %, respectively (assuming a uniform magnetic field and adopting a typical value for the electron energy distribution index, 𝑝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Ginzburg & Syrovatskii 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Longair 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Complex or evolving magnetic fields, disadvanta- geous lines of sight, Faraday depolarization, and the superposition of multiple components are a few mechanisms that can depolarize the observed radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' BHXBs with polarimetric radio analyses typically have linear polarization fractions ≲ 10% with a rare few reaching ∼ 50% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Han & Hjellming 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hannikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Fender 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The polarization fraction measures how “ordered” the local magnetic field is (or appears to be), while the direction of the ob- served electric vector position angle (EVPA) is a measure of the local absorption conditions, jet position angle, magnetic field orientation, as well as Faraday rotation between the emission and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' After measuring and removing the effect of the Faraday rotation, the derived intrinsic EVPA can provide an indirect measure of the jet orientation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In cases where polarized emission is combined with spatially resolved, total intensity observations, polarimetry can directly probe the underlying magnetic field strength and orientation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Stirling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Despite the established observational relationship between the X- ray and radio properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the accretion flow and relativistic jet), the physical mechanisms responsible for the launching and evolution of jets are yet to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Most theories recognize that the local magnetic fields (and their disk/black-hole interactions) play an essential role in extracting energy from the black hole/accretion disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Blandford & Znajek 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Blandford & Payne 1982) and the initial launching and collimation of relativistic jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Vla- hakis & Königl 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Komissarov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These highly energetic processes can leave imprints on the evolving magnetic fields, making time-resolved radio polarimetry, particu- larly around BHXB ejection events, a valuable (yet underutilized) tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Outbursts from BHXBs occur at a moderate frequency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', several times a year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016), with a rare subset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', once per decade) achieving X-ray luminosities near (or exceeding) the Eddington luminosity, and Jansky level radio flux densities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', V404 Cygni’s 1989 outburst reached 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 Jy at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Oosterbroek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Han & Hjellming 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During highly luminous out- bursts, we can study accretion and accretion-rooted phenomena with extraordinary levels of detail, capturing, in real-time, jet ejections at flux densities that allow for a refined spectral and temporal resolution for both total intensity and polarimetric observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' On 2015 June 15, the BHXB V404 Cygni (henceforth V404 Cyg) began one of these rare outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 V404 Cygni First discovered in 1989, V404 Cyg (also known as GS 2023+338) is a low-mass transient BHXB that has undergone four recorded outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Of the four outbursts, two were caught in real-time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the initial discovery with the Ginga satellite (Makino 1989) and the most recent outburst discovered by the Burst Alert Telescope aboard the Neil Gehrels Swift Observatory (Barthelmy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Searches through historical photo plates identified that there were additional outbursts in 1938 and 1956 (Richter 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Observa- tions of the main-sequence companion star revealed an orbital period of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4714 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0001 days and a binary mass function of 𝑓 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='06 𝑀⊙ = 𝑀3 BH sin3 𝑖/(𝑀BH + 𝑀donor)2, where 𝑀BH and 𝑀donor are the masses of the black hole and donor, respectively, and 𝑖 is the orbital inclination angle (Casares & Charles 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The K spectral type of the companion star, coupled with near-infrared spectroscopy (and modeling of the H-band ellipsoidal modulations), infer a BH mass of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 𝑀⊙ with a best fit orbital inclination angle of 67+3 −1 (Khargharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The modelled orbital inclination angle assumes that the optical light curve of the companion star has ≲7% contamination from accretion disk (or jet) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, V404 Cyg has exhibited optical variability in quiescence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Zu- rita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Bernardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016) and, as a result, may have a larger contamination fraction, larger inclination angle, and smaller black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' On the other hand, narrow emission lines suggest that V404 Cyg has a low inclination angle 𝑖 < 40◦ (Casares et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1993) and a higher black hole mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This uncertainty suggests the mass of the black hole is not yet accurate at the 2–7% precision level quoted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' High angular resolution radio parallax measurements determined a source distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='14 kpc (Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2009), making it one of the closest known BHXBs and a superb laboratory for the study of accretion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the 1989 outburst, Han & Hjellming (1992) monitored the radio emission of V404 Cyg between 1989 May 30 and 1991 May 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The monitoring began when the radio light curves were dominated by the tail of a rapidly decaying (decay timescales of ∼5 days) “major synchrotron bubble event".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' At later times, the radio light curves were dominated by a slowly-decaying, nonthermal, optically thick source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', a compact jet) that lasted hundreds of days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Linear polarization was detected for the first 50 days of observations, except for the first observation on 1989 May 30 which did not include adequate polar- ization calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the decay of the synchrotron bubble, the polarization fraction was a few tenths of a percent, before increas- ing to a few percent during the period when the slowly-decaying component dominated the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' On 2015 June 15, V404 Cyg entered its fourth recorded outburst, and a follow-up campaign showed bright multi-wavelength flaring ac- MNRAS 000, 1–19 (2023) V404 Cyg’s rapidly evolving polarized jet 3 tivity in radio through X-ray wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Mooley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Motta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015b,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Maitra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017), and rapid (∼ 15 s) transitions between accretion states (see, Kajava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2020, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' With radio- through-optical flux densities reaching ∼ Jy levels, V404 Cyg became the brightest BHXB outburst observed in the last decade, character- istic of a high, near-Eddington accretion rate, and its close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The source remained in outburst until the end of June, from which it began decaying, eventually reaching quiescence in mid-August (with the source having a brief period of renewed activity in 2015 Decem- ber through 2016 January;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The MASTER Global Robotic Net detected three linear polariza- tion “events” using their optical telescope network (Lipunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In both events, the source exhibited a significant in- crease in the linear polarization fraction, following a (total intensity) flare, followed by a rapid decrease in the linear polarization frac- tion during the rise of another flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The authors favoured a model where decreased X-ray irradiation of the secondary also decreased its optical brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In turn, this makes it easier to detect the po- larized non-thermal emission from the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The authors favoured this model after having discarded the potential that the jet orientation varied on timescales of tens of minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' however, a rapid, variable jet orientation was later confirmed (see below and Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Shahbaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016), detected another linear polarization flare using observations with the Nordic Optical Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This flare occurred during a steady rise of optical flux, and preceded some of the brightest optical flaring of the entire outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, the flare preceded the start of a bright radio flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These authors proposed that the increase in linear polarization could result from multiple ejecta collisions establishing a dominant magnetic field direction perpendicular to the jet axis, and may be the signature for the birth of the ejection that produced the subsequent radio flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During this same outburst, modeling of radio-through-sub-mm observations (Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017) and Very Long Baseline Ar- ray (VLBA) observations (Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019) uncovered short time-scale flaring of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here, the jet ejecta account for most, if not all, of the observed flaring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, we note that detailed modelling of the X-ray emission suggested that the source may have been continuously accreting at an Eddington accretion rate during the brightest phase of the 2015 outburst (Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, the jet ejecta in V404 Cyg are not clearly associated with the same discrete-jet launching process that occurs during hard-to- soft state transitions in BHXBs, which occur around lower accretion rates (Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The VLBA observations directly resolved several of these ejection events on top of a continuous (but vari- able) emission from an empirically defined unresolved radio core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The position angle (PA) of the ejecta varied rapidly with time, a phenomenon that was attributed to the Lense-Thirring precession of the accretion disk (Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Although the authors constrained the period to less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 hr hours, the rapidly varying PAs suggested that the true period was substantially shorter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In this paper, we add to the detailed radio analysis of the 2015 outburst detailed in Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2017) and Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here, our primary focus is the extraction and analysis of V404 Cyg’s (radio) polarization properties — derived from National Science Foundation’s Karl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Jansky Very Large Array (VLA) ob- servations — during some of the outburst’s brightest flaring activity on 2015 June 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The remainder of this paper is structured as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' in Section 2, we introduce our observation and analysis procedure, while in Sections 3 and 4, we present and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Finally, we summarize our findings in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2 OBSERVATIONS AND ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 VLA Data Reduction The details of the primary VLA observations were first discussed in Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' V404 Cyg was observed with the VLA (Project Code: 15A-504) on 2015 June 22 with scans on source between 10:37:24 and 14:38:39 UTC in the 4–8 GHz and 18–26 GHz bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' All observations were made with an 8-bit sampler, comprised of two base-bands, with eight spectral windows of sixty-four 2 MHz channels each, giving a total (unflagged) bandwidth of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='024 GHz per base-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Henceforth, we will refer to each base-band by its characteristic frequency values of (∼) 5, 7, 21, and 26 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The array was in its most extended A configuration, and was split into two sub-arrays of 14 (sub-array 1) and 13 (sub-array 2) an- tennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sub-array 1 observed the sequence (5/7 GHz)-(21/26 GHz)- (5/7 GHz), while sub-array 2 observed the sequence (21/26 GHz)- (5/7 GHz)-(21/26 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Both sub-arrays cycled between V404 Cyg, observed for 88 s per cycle flanked by 32 s observations of a nearby gain calibrator (J2025+3343).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A second epoch was observed during the source’s return to quiescence, taken on 2015 July 2, with scans on source from 10:31:08 to 14:01:32 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The observing bands and sub-array schemes remained consistent with the primary June 22 observations (Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We also analyzed 5 epochs taken between July 11 and August 5, during the source’s return to quiescence (Project Code: SG0196;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016), although we were unable to detect any polarized signal in these latter observations (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We applied standard flagging and calibration to the Stokes 𝐼 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', total continuum flux density) data using the Common Astronomy Software Application package (casa v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We used 3C48 (0137+331) as a flux and absolute (linear) polarization an- gle calibrator, J2025+3343 as a complex gain (aka phase) calibrator, and J2355+4950 as an unpolarized leakage calibrator for both sub- arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to V404 Cyg being weakly polarized, we grouped our po- larization calibration solutions on 16 MHz (8 channel) intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For our Stokes 𝐼 flux calibration model, we used the default casa model repository (Perley & Butler 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, the standard calibration routine for Stokes 𝑄 and 𝑈 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' linearly polarized flux densities) assumes that the polarization calibrators are point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since we observed with the VLA in its most extended configuration, 3C48 was resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The “degree” of resolution ranges from a slightly extended Gaussian at 5/7 GHz to multiple distinct components at 21/26 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we constructed a spatially resolved model image for each observing band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our model contained information on all four Stokes parameters, assuming no circular polarization, and adopted the spatial distribution of the Stokes 𝐼 repository models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A detailed description of our polarized model image can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note that the spatial distribution of the flux densities may differ between Stokes 𝐼, and Stokes 𝑄/𝑈 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' linearly polarized flux densities), and, as a result, our measured polarizations are suscepti- ble to systematic calibration errors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' in particular, for the 21/26 GHz basebands, where our calibrator is significantly resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 Imaging Since V404 Cygni was expected (and found) to be unresolved re- gardless of the chosen visibility weighting, we applied a natural MNRAS 000, 1–19 (2023) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝑢𝑣-weighting scheme to all of our images, maximizing sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, since we also knew that the Stokes 𝐼 flux density was rapidly changing, we generated our analysis images in Stokes 𝐼, 𝑄, and 𝑈, on short timescales of 12 or 14 min (6 or 7 scans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', ∼8 or 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 min on source)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These timescales, some of the shortest ever used in a radio polarimetric analysis of a BHXB, balance cadence and polarized sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We used the wsclean package (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014) to make all of our polarimetric images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We imaged each base-band independently, as well as Stokes 𝐼 separately from 𝑄 and 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In each base-band for every time-bin we had wsclean output a set of images across a user-set number of channels, as well as a single “multi-frequency- synthesis” (MFS) image that stacks all the individual channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We measured the linear polarization intensities (𝑃 = √︁ 𝑄2 + 𝑈2) for each base-band/time-bin pair from the MFS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Any observed EVPA at an arbitrary observing wavelength 𝜆, is related to the intrinsic EVPA, 𝜒0, through the linear relationship, 𝜒(𝜆) = 𝜒0 + RM · 𝜆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The slope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the rotation measure, RM) quantifies the wavelength-dependent Faraday rotation of an EVPA due to linearly polarized light propagating through a magneto-ionic plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since our observables are 𝜒 and 𝜆, the largest detectable rotation measure is inversely proportional to the 𝜆2 channel spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The linearly-spaced frequency channels result in a 𝜆2 channel density that increases with increasing central frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To avoid potential biasing of results by the higher frequency observations, we scaled the imaging frequency bins used for rotation measure analysis to maintain a (roughly) constant 𝜆2 channel spacing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' this resulted in a frequency-space channelization of 16 MHz for the 5 GHz baseband, and 64 MHz for the 7 GHz baseband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to their large temporal delays (≳ 30 min) with respect to the 5/7 GHz base-bands, we chose to omit the 21/26 GHz base-bands from the rotation measure anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The omission will minimize the overlap of optically thick and optically thin emission, as well as any overlap of emission from dif- ferent jet components (see Appendix A for a more comprehensive motivation behind the omission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we did not scale the frequency binning any broader than 64 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The larger Stokes 𝐼 flux densities allowed us to image the total flux density light curves on much shorter timescales (∼10 s) than is required for accurate polarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For each spectral window, we pro- duced a high time-resolution light curve, using the publicly available2 imaging scripts detailed in Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These images crit- ically allow us to compare the simultaneous Stokes 𝐼 flux density, and linear polarization evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We observed an elevated rms noise in each image when compared to the predicted values (see Table 1 for a summary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These effects are most significant in the Stokes 𝐼 images and appear to worsen at higher central frequencies and when larger frequency ranges are used to create a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we implemented a phase self-calibration routine to explore if the elevated Stokes 𝐼 noise is biasing the polarimetric results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our self-calibration routine was bro- ken into three steps that refined the phase calibration solutions on progressively shorter timescales: first, half the length of a source scan, 44 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' then a quarter, 22 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' and ending with solutions on the in- tegration timescale, 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We excluded amplitude self-calibration due to the known Stokes 𝐼 variability within our imaging intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Al- though the phase self-calibration improved the Stokes 𝐼 rms noise, 1 We made the time bins a variable integer number of scans to avoid com- bining scans from different sub-arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note that the ∼20% difference in on-source time has a negligible effect on the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='com/Astroua/AstroCompute_Scripts we were unable to reach the theoretical limit expected from ther- mal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This result is not unexpected: (i) we are averaging over variable emission (spectrally and temporally) during our imaging routines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (ii) the reduction in baseline coverage due to the division into sub-arrays coupled with the bright emission is expected to limit the dynamic range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (iii) completely automated self-calibration, like we employ, can have difficulties achieving high dynamic ranges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (iv) we only image a ≈51′′ × 51′′ field-of-view, and there can be some added noise across the entire image due to our nearby phase calibrator (approximately 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6′from V404 Cyg) — our primary beams range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9′, leading to noise that would be stronger in our lower frequency basebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since the self-calibration and its reduction of the Stokes 𝐼 rms had a negligible effect on both the noise of the Stokes 𝑄 and 𝑈 images and our measured polarimetric parameters, we are confident that the elevated noise is not a significant issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For the remainder of this analysis, we have adopted our self-calibrated results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 Flux Density Extraction We measured the Stokes 𝐼 flux densities and linear polarization inten- sities (from the MFS images) from an image plane analysis using the casa task imfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We fit an elliptical Gaussian component in a small sub-region around the source, fitting for the position, flux density, and shape of the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to the source’s weakly polarized emission, at fine spectral resolutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the 16 MHz channeliza- tion), the Stokes 𝑄 and 𝑈 flux densities are similar in magnitude to (or weaker than) the local peaks in the rms noise (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Often our attempts to freely fit the Stokes 𝑄 and 𝑈 images us- ing imfit did not converge or converged on artificial noise signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we decided to fix the shape of the component, and only fit for the flux density in the region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', we performed forced aperture photometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We set the component shape to be the synthesized beam of each image, and used the position of the 𝑃 peak (for each time bin and base-band) as the position of our aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We extracted the rms of each image using a large annular region centred on the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To check for bias by a non-zero background we subtracted the mean flux density in the rms region from the flux density of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The background subtraction had a negligible effect on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The fine (spectral) resolution images uncovered anomalous chan- nels (∼1–2 per time bin) that were missed during flagging and cali- bration, or corrupted during imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We apply a 𝜎-clipping routine to remove these channels from the Stokes 𝐼 spectrum of each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' After constructing a model spectrum by passing our Stokes 𝐼 data through a narrow Gaussian filter with 𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 data points, corresponding to 5 and 20 MHz at 5 and 7 GHz, respectively, any flux density point that was > 3 residual standard deviations from the model spectrum was flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We continued the routine until the frac- tional difference in residual standard deviations between the current and previous iteration was ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='001%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The channels removed from the total intensity spectra were recorded and subsequently removed from the 𝑄 and 𝑈 spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' No further data manipulation was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 Polarization Properties We derived all polarization properties from the flux densities ex- tracted during image plane analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The polarization intensity im- ages, 𝑃𝜆 = √︃ 𝑄2 𝜆 + 𝑈2 𝜆, for an image with a central wavelength 𝜆, were created from the Stokes 𝑄 and 𝑈 images using the native casa task immath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since 𝑃𝜆 is positive definite, we debiased each po- larization intensity using the correction from George et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) V404 Cyg’s rapidly evolving polarized jet 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Table of imaging properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The highlighted frequency parameters for each base-band are the central frequency of the lowest (𝜈𝑖) and highest (𝜈 𝑓 ) channels, in addition to the imaging bandwidth (Δ𝜈) assuming a typical ∼15% loss during flagging and calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Δ𝑡, is the average time on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The theoretical rms noise (𝜎rms) and the median rms noise for each Stokes parameter (𝜎𝐼 , 𝜎𝑄, 𝜎𝑈) are also highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The high time-resolution images (Δ𝜈 ∼ 110 MHz) were excluded from the self-calibration procedure due to the number of images (∼45000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The theoretical noise estimates were calculated using the VLA exposure calculator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='vla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='edu/ect/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Base-band 𝜈𝑖 (MHz) 𝜈 𝑓 (MHz) Δ𝜈 (MHz) Δ𝑡 (s) 𝜎rms (mJy) 𝜎𝐼 (mJy) 𝜎𝑄 (mJy) 𝜎𝑈 (mJy) 5 GHz 4738 5762 850 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 16 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 110 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 2 — — 7 GHz 6938 7962 850 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='06 64 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='14 110 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 3 — — 21 GHz 20288 21312 850 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='09 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='15 110 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 10 — — 26 GHz 25388 26412 850 520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 110 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 12 — — 𝑃𝜆,0 = √︃ 𝑃2 𝜆 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3𝜎2 𝑄𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To remain consistent with the RM synthe- sis routine (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1), we have chosen 𝜎𝑄𝑈 ≡ 1 2 (𝜎𝑄 + 𝜎𝑈) to parameterize the noise in 𝑃𝜆, noting that 𝜎𝑄 ≈ 𝜎𝑈 for all of our images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The polarization fraction adopts its standard definition, 𝑓𝜆 ≡ 𝑃𝜆,0/𝐼𝜆, and we approximated its error using Gaussian error propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We recognize that the MFS images will experience a degree of bandwidth depolarization due to averaging over an intra- band Faraday rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, at our detected rotation measures (|RM| ∼ 100 rad m−2), even at the lowest frequencies, the amount of depolarization is insignificant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Δ 𝑓𝜆/ 𝑓𝜆 ≲ 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To extract the intrinsic EVPA and rotation measure from each time bin, we applied two independent methods: rotation measure synthesis and a custom Markov-Chain Monte Carlo (MCMC) routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Meaningful RM synthesis results requires a band-averaged, polarized S/N of 𝑃𝜆,0/𝜎𝑄𝑈 ≳ 7 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Brentjens & de Bruyn 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To ensure the significance of each detection, we enforce the 𝑃𝜆,0/𝜎𝑄𝑈 > 7 restriction on the 5/7 GHz base-bands separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our aggressive restriction was motivated by the susceptibility of weakly polarized data to spurious effects from imperfect leakage calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we limited the intrinsic EVPA and rotation measure analysis to the 13 time bins between 11:15 and 13:53 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Data tables including our polarimetric measurements can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 Rotation Measure Synthesis Rotation measure synthesis derives the linear polarization parame- ters of a source through its structure(s) in Faraday space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', its Faraday dispersion function (FDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' see, Burn 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Brentjens & de Bruyn 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012, for a compre- hensive description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We generated each FDF using the rm-tools3 repository, currently developed and maintained by the Canadian Ini- tiative for Radio Astronomy Data Analysis (CIRADA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To mitigate any aliasing at large rotation measures, we fixed the FDF domains at ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5×105 rad m−2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', twice the rotation measure that corresponds to a ∼50% drop in sensitivity at our spectral channelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Further- more, we fixed the bin size at 75 rad m−2, a factor of 20 (twice the median polarized S/N) smaller than the full width at half maximum of the rotation measure synthesis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The package typically quantifies the noise in each FDF (𝜎RM) using the median absolute deviation after masking the strongest rotation measure component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='com/CIRADA-Tools/RM-Tools We chose to use the rms noise as it was a factor of ∼2 larger, and thus, increased our confidence in each detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Any FDF component that satisfied a > 5𝜎RM condition was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the construction of each FDF, the observed EVPAs are de-rotated to their values at the weighted mean of the 𝜆2 channels, with a 1/𝜎2 𝑄𝑈 weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The intrinsic EVPA is calculated from a further de-rotation using the best-fit rotation measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', 𝜒0 = 𝜒𝑤 − RM · 𝜆2𝑤, where 𝜆2𝑤 is the weighted average of all 𝜆2 channels and 𝜒𝑤 is the observed polarization angle at 𝜆2𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 MCMC Since V404 Cyg is weakly polarized, we also employ a simple Bayesian forward model to fit the polarization parameters directly to the Stokes fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Consistency between the two methods is an im- portant check to mitigate the potential that our derived polarization parameters originate from noise, as opposed to an intrinsic signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our fitting functions adopt the following forms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' � 𝑄𝜆 = �𝐼𝜆 �𝑓𝜆 cos � 2𝜒𝑤 + 2RM · (𝜆2 − 𝜆2 𝑤) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' and (1) � 𝑈𝜆 = �𝐼𝜆 �𝑓𝜆 sin � 2𝜒𝑤 + 2RM · (𝜆2 − 𝜆2 𝑤) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2) We chose to fit for 𝜒𝑤, to remain consistent with the RM synthe- sis routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The superscript, 𝜆, in equations (1) and (2) denotes the central wavelength of the spectral channel of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The model parameters for Stokes 𝐼 (�𝐼𝜆) and the linear polarization fraction ( �𝑓𝜆) were excluded from the fitting procedure, due to negligible correla- tion with the quantities of interest (RM and 𝜒𝑤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Instead, the Stokes 𝐼 and polarization fraction models were smoothed using a Savitzky- Golay filter (Savitzky & Golay 1964), retaining the overall structure while removing stochastic variability, and stabilizing the fitting rou- tine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We assumed the sampled flux densities were independently dis- tributed normal random variables, resulting in a log-likelihood func- tion (L) of the following form, log L = − ∑︁ 𝜆 � log √︃ 2𝜋�𝜎2 𝑄,𝜆 + (𝑄𝜆 − � 𝑄𝜆)2 2�𝜎2 𝑄,𝜆 + log √︃ 2𝜋�𝜎2 𝑈,𝜆 + (𝑈𝜆 − � 𝑈𝜆)2 2�𝜎2 𝑈,𝜆 � , (3) where 𝑄𝜆/𝑈𝜆 and � 𝑄𝜆/� 𝑈𝜆 are the measured and modelled flux den- sities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We added two additional modeling parameters, MNRAS 000, 1–19 (2023) 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝜎𝑄,sys and 𝜎𝑈,sys, that are channel independent variances to account for missed systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The variances seen in equation (5) are the sum of the measured rms noise variance and our systematic ad- dition (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', �𝜎2 𝑄,𝜆 ≡ 𝜎2 𝑄,𝜆 + 𝜎2 𝑄,sys).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We used the Markov-Chain Monte Carlo algorithm implemented through Python’s emcee package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' emcee is a pure-Python implemen- tation of Goodman and Weare’s Affine Invariant Markov chain Monte Carlo Ensemble Sampler (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Goodman & Weare 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' a modified version of the classic Metropolis-Hastings algorithm, simultaneously evolving a select number of walkers through parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The number of (sampling) walkers was fixed at five times the number of dimensions, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We chose four broad, uniform, and uninformative priors to reflect the lack of a pri- ori information on V404 Cyg’s polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The systematic variance priors were positive definite, with maximum values chosen to be twice the variance of the measured flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The rotation measure prior adopted the FDF domain, ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 × 105 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A uni- form prior was unable to capture the circularity of the EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, individual walkers frequently would become trapped in the local minima created by the prior’s edges, subsequently inhibiting convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To combat this, we expanded the prior to ±3𝜋/2 rad, while maintaining the initial condition distribution for the physi- cally meaningful range of ±𝜋/2 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We initialized each run with 80 walkers, four times the number intended for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Following an initial set of “burn-in" iterations, we removed the 60 walkers with the lowest posterior probabilities and adopted the remaining 20 as the starting positions for sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' After sampling, we verified that each simulation converged by visually inspecting the walkers over a large number of autocorrelation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We adopted the median of each posterior distribution as the best- fit value of our model, and the ranges between the median and the 15th/ 85th percentiles as the 1𝜎 (−)/(+) uncertainties, noting that the measured uncertainties are purely statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Once again, the intrinsic EVPA was solved for using, 𝜒0 = 𝜒𝑤 − RM · 𝜆2𝑤, and we calculated its error using standard Gaussian error propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3 RESULTS By splitting our ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 hr observation into sixteen ∼13 min time bins, we have measured the temporal evolution of the linear polarization fraction (Figure 1), rotation measure, and intrinsic EVPA (both Fig- ure 2) during the 2015 June 22 flaring events of V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In this section, we present our polarimetric results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note that weak linear polarization fractions should be treated with caution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' in Appendix C, we compare our results to the simultaneous evolution of the phase calibrator ( 𝑓𝜆 ∼ 2%) to ensure that significant changes we see in V404 Cyg arise from physical evolution, and not systematic calibra- tion effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 Linear Polarization Fraction Each base-band showed a weak but variable degree of linear polar- ization with a maximum linear polarization fraction that decreased with decreasing frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e, maxima of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='75% for the 5, 7, 21, and 26 GHz base-bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The maximum linear polarization fraction occurs between the peaks of the first and second Stokes 𝐼 flare, and, like Stokes 𝐼, occurs at later times for lower-frequency observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' There is evidence of a second (much weaker) linear polarization fraction peak in the 21 GHz base-band, between the second and third flare (at ∼ 12:50 UTC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This sec- ondary peak is marginally detected in the 5/7 GHz base-bands, but is consistent with noise at 26 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Additionally, in the 21/26 GHz base-bands, at late times the linear polarization fraction begins to increase alongside the decay of the third flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A similar increase is not observed in the 5 GHz base-band, with a marginal trend seen in 7 GHz, although temporal delays would have likely shifted any peak at these frequencies beyond our observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the 2015 July 2 observations, during V404 Cyg’s return to qui- escence, the Stokes 𝐼 flux densities had decreased to ∼ 4 mJy across all base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we are unable to detect weakly polar- ized emission, and the source showed no polarization with a 99% confidence upper limit on the polarization fraction of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4% for the 5, 7, 21, and 26 GHz base-bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here we calculated upper limits following Vaillancourt (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Further- more, we analyzed 5 subsequent epochs of the 5/7 GHz observations taken between 2015 July 15 and 2015 August 5 (see, Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Of these 5 epochs, data on 2015 August 5 had the most constraining upper limits, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1% (for the 5 and 7 GHz base-bands respectively), with all other epochs having upper limits between 10−25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' While we cannot detect a weakly polarized signal with these observations, a ∼ 5% limit is lower than some past linearly polarized fractions detected in BHXBs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Han & Hjellming 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The S/N of all linear polarization intensities, are ≳ 5 with the strongest detections reaching 𝑃𝜆,0/𝜎𝑄𝑈 ∼ 25 (see Table E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' At all 𝑓𝜆, imperfect leakage calibration systematically increases the ob- served linear polarization fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This effect is not included in the calculations of the S/N of linear polarization intensities or the errors on 𝑓𝜆 that we present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' While our higher 𝑓𝜆 values may be (slightly) overestimated due to imperfect leakage corrections, the lower values could be due to spurious signals and are actually consistent with no linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Following Hales (2017),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the predicted level of spurious linear polarization fraction is Rayleigh distributed with a mean given by 𝑓spur,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='mean ≈ √︂ 𝜋 4𝑁𝑎 � 𝑓 2 true + 𝑁𝑎 [(𝑆/𝑁)𝐼 ]−2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (4) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝑁𝑎 is the number of antenna in each sub-array (𝑁𝑎=11/13 for Sub-array 1/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (𝑆/𝑁)𝐼 is the Stokes 𝐼 signal-to- noise ratio of the leakage calibrator at the frequency of interest (with a 16 MHz leakage solution bandwidth),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' and 𝑓true is the true linear polarization fraction of the leakage calibrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For 𝑓true, we adopted the mean linear polarization fraction from the VLA polarization cal- ibrator catalog4, corresponding to, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='04%, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='17% for the 5/7 and 21/26 GHz bands respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We measured the Stokes 𝐼 signal from an image plane analysis using imfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to the leakage calibrator’s large Stokes 𝐼 flux densities and our sparse 𝑢𝑣-coverage (from a sin- gle scan, 13-element sub-array), the Stokes 𝐼 images are dynamic range limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we chose to use 𝜎𝑄𝑈 as the noise value in (𝑆/𝑁)𝐼 , as opposed to 𝜎𝐼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Stokes 𝑄 and 𝑈 images were not dynamic range limited, and thus would better quantify the instrumen- tal noise that also affects the Stokes 𝐼 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We present the spurious linear polarization parameters in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The differences between the two sub-arrays are the result of elevated noise in sub-array 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note that the minimum linear polarization fraction we detect in each base-band is approximately equal to the predicted values of 𝑓spur,mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Henceforth, we define the significance level (SL) as the proba- bility that a detection is not the result of a purely spurious signal 4 The VLA catalog can be found here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='vla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='edu/ astro/evlapolcal/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='html MNRAS 000, 1–19 (2023) V404 Cyg’s rapidly evolving polarized jet 7 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Table of spurious linear polarization properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' All symbols adopt their definitions as defined in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sub-array 𝑁𝑎 Base-band (𝑆/𝑁 )𝐼 𝑓spur,mean (%) 1 13 5 GHz 1702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='053 7 GHz 1656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='054 21 GHz 516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='18 26 GHz 418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='22 2 11 5 GHz 1331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='067 7 GHz 1335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='066 21 GHz 480 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='19 26 GHz 357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='25 (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1, horizontal-dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The significance levels for all maxima are > 99%, confirming that we have observed an intrinsic polarized signal in each base-band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the SLs for each time bin are tabulated in Table E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Only one scan per sub-array was used to correct leakage, and leakage converts Stokes 𝐼 into 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This leads to a single offset in fractional linear polarization in the absence of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' On the other hand, imperfect leakage can potentially lead to dynamic systematic-error-induced changes in the measured EVPA due to parallactic rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 Rotation Measure and EVPA The derived rotation measures exhibit stochastic variability, with values between −330 ≲ RM ≲ −20 rad m−2 (top panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2), and show a strong agreement between the two polarimetric meth- ods in almost all time bins (the only ≳ 1𝜎 disagreement occurs in the final, lowest S/N time bin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The weighted means of the rotation measure, are −100 ± 16 rad m−2 for the RM synthesis method and −100±12 rad m−2 for the MCMC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We applied a simple vari- ability analysis by calculating the 𝜒2 statistic against a constant RM model equal to the weighted mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The 𝜒2 values of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 (RM Syn- thesis) and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 (MCMC) for 12 degrees of freedom are consistent with a constant rotation measure at probabilities of 56% and 13%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' When we use the default RM synthesis prescription (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', quan- tifying the noise in the dispersion function with an appropriately scaled median absolute deviation rather than the rms), the detection uncertainties reduce by a factor of ∼2, and the RM Synthesis 𝜒2 value become significantly larger than the MCMC value (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5/12), consistent with a variable rotation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, this choice also increases the population of >5𝜎RM components to ≳10 for each FDF, with rotation measure magnitudes ∼103 − 105 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These rotation measures are characteristic of extremely particle-rich lines of sight (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', towards the Galactic centre) and, historically, have not been observed in outbursting BHXBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our phase calibrator, a source with ∼ 2% linear polarization, showed a similar population of secondary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We find it very unlikely that these sources would exist while evading detection during recent Galactic rotation measure analyses (Oppermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hutschenreuter & Enßlin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we propose that these components are ar- tifacts from imperfect 𝜆2 sampling (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the effects of poor/patchy 𝑢𝑣-coverage during synthesis imaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1999) or a sys- tematic effect in the modern RM synthesis routine(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We conclude our decision to use the FDF rms noise is more reflective of the sta- tistical significance of each detection, and that we cannot identify any significant rotation measure variability from V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, for our analyses, we have adopted a constant rotation measure equal to the (inverse-variance) weighted mean of rotation measures across all time bins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', RM = −100 ± 16 (12) rad m−2 for the RM Synthesis (MCMC) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Both the observed EVPA (𝜒𝑤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Second panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2) and intrinsic EVPA (𝜒0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Third panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2) exhibit a clear temporal evolution, with strong agreement between the RM synthesis and MCMC rou- tines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, due to the stable rotation measure, this evolution suggests an intrinsic change in the (polarized) emission environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the SL ≥ 90% regime, both observed and intrinsic EVPA evolve gradually, with a ∼ 30◦ change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The intrinsic EVPA evolves from ∼ 80◦ to ∼ 50◦ between 11:30 and 12:30 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The intrinsic EVPA then stabilized at the ∼ 50◦ for the remaining time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4 DISCUSSION In this section we describe the short timescale evolution of our polari- metric results and compare the observed behaviours to the 1989 out- burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, we correlate this evolution with total intensity light curves, high (spatial) resolution imaging, and optical polarization detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' When analysing the connection between the polarization flaring and the high resolution radio imaging, we limit our discussion to the spatially-resolved VLBA components that dominate the VLBA light curve at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This implicitly assumes that any resolved polarized flux density would track the resolved Stokes 𝐼 flux density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', 𝑃0,𝜆 ∝ 𝐼𝜆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Although we make this assumption, we cannot rule out the possibility that the dominant VLBA components are unpolar- ized and the sub-dominant components (with average Stokes I flux densities ≲ 10% of the dominant counterparts) are the source of the polarized emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 Linear Polarization Fraction The majority of BHXB outbursts have measured linear polariza- tion fractions of ≳2% at 1–10 GHz (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Fender 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015), with rare cases reaching appreciable fractions of the theoretical limits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the ∼50% detections of XTE J1752−223 and Swift J1745−26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Even when considering the typical reduction compared to the theoretical maxima, our measured maximum linear polarization frac- tion (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2% in the 5/7 GHz base-bands) for V404 Cyg is a factor of ∼10 less than a standard, weakly polarized signal during a BHXB outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, we acknowledge that past outbursts with compa- rably weak polarization fractions may not have had sufficient S/N for a clear detection and/or the larger (average) polarization fractions may suffer from a publication bias where strongly polarized outbursts are more often introduced within the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Observers caught a glimpse of a comparably low polarization frac- tion during the monitoring of the 1989 outburst of V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The first day of polarization observations — 1989 June 1, during the de- cay of the “major synchrotron bubble event” (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the ejection of a bright cloud of synchrotron emitting plasma) — recorded the lowest polarization fraction of the entire campaign, measuring 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1% at central frequencies of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the decay of the 1989 outburst, the radio emission exhibited an inverted spectrum and lin- ear polarization fraction of ∼3%, characteristic of a typical compact jet (Han & Hjellming 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The polarized signal was consistently detected for 50 days (between 1989 June 1 and 1989 July 18) before the flux density decayed below the detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In contrast, we did not observe an increase to a few percent polarization frac- tion during the decay of the 2015 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The 2015 July 2 epoch places a 99% confidence interval upper limit of ∼1% on the 5/7 GHz polarization fraction, consistent with the 2015 June 22 observations, MNRAS 000, 1–19 (2023) 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Temporal evolution of the linear polarization fraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 26 GHz (top), 21 GHz (2nd from the top), 7 GHz (4th from top), 5 GHz (5th from top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The two-point spectral indexes of the 21/26 GHz (3rd from top) and the 5/7 GHz (bottom), show the simultaneous evolution of the absorption conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The vertical dashed lines across all panels highlight the time of the maximum fractional polarization in the 26 GHz base-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The horizontal dashed lines highlight the value of the mean spurious linear polarization fraction for each base-band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the discontinuities are the result of elevated noise in Sub-array 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The diamond markers correspond to SL≥ 90% , and the squares to SL< 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The grey curves display the simultaneous Stokes I flux density evolution for each base-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We can see that the linear polarization fraction exhibits a similar frequency-dependent delay as the Stokes 𝐼 light curves, and is offset from the (Stokes 𝐼) maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8 1400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 E 26 GHz 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6E 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5E Stokes Fraction 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 Flux Polarization 00F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 1200 21 GHz isity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 Linear Q0QT mJy) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 600 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='25 700 7 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='20 600 (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='15 500 Stokes Fraction 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='10 OCE Flux tion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='200 E 5 GHz 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='150 (m) ear 500 E 月 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='100 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='075 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='050 OCE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='03 QE-T 12:00 12:30 13:00 13:30 14:00 Time on 2015 Tune 22(UTC)V404 Cyg’s rapidly evolving polarized jet 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Polarization properties measured from the 5 and 7 GHz base-band observations, for both the MCMC and RM synthesis routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The vertical shaded region corresponds to the detections with an average significance level ≥ 90% between the 5 and 7 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The underlying grey curve in each panel is the average Stokes I light curve between the 5 and 7 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' There is strong agreement between the two polarimetric methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (top) The rotation measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the horizontal dotted line shows the weighted average of the rotation measures (∼ − 100 rad m−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2nd from top) The observed EVPA de-rotated to the weighted mean of all 𝜆2 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (3rd from top) The intrinsic EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The horizontal bars show the PAs (+90◦) of the dominant VLBA ejecta identified in Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The length of the bar span the times between the ejection time and when an ejected component is not longer detected, and the darker part of the bar shows when it was the brightest ejected component (excluding the compact core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The vertical size of the bars is fixed at the uncertainty in the PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We adopt the naming conventions from the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (bottom) The average linear polarization fraction between the 5 and 7 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We can see the rotation measure is constant and the EVPA exhibits a ∼ 30◦ rotation between ∼11:30 and 12:30 during the decay of the polarization fraction maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 600 100 l) 500 200 RM 300 400 RM Svnthesis 400 MCMC 300 600 09 500 400 8 40 kes 30 ux 90 D 80 s3 70 S6 60 400 50 300 40F 600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='20 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='10 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 300 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 Time on 2015 June 22 (UTC)10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 GHz VLBA light curve using the data from Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The open black circles represent the total integrated the flux density, the solid black circles represent the core flux density, and all other marker types represent a single spatially-resolved component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We adopt the naming convention from the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The grey shaded region corresponds to the three time bins that encompass the 7 GHz fractional polarization peak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' we shifted the region back in time by 10 minutes to account for the delay between the bands (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We can see that at any point in time the total integrated (VLBA) flux density is a superposition of multiple radio-bright (and potentially polarized) components that are unresolved in our VLA observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' but below the level seen in the compact jet during the 1989 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Furthermore, none of the epochs in Plotkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2017) showed any polarized emission, although we note that the upper limits are significantly larger than the maximum polarization fraction detected during the 1989 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Stokes 𝐼 flux density of V404 Cyg decayed significantly faster in the most recent outburst, taking ∼30 days in 2015, as opposed to ∼300 days in 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2018) suggested that the more rapid decay was the result of the strong winds originating from the accretion disk (detected by Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016) rapidly depleting the disk and leaving less matter to fuel the jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Other factors may have also played a role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' these could include the total mass reservoir built during the quiescent periods prior to the two outbursts — 33/26 years for the 1989/2015 outbursts respectively — or differences in the total mass accreted during the bright outburst phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In contrast, the polarization fraction depends on the structure of the jet(s), and is only indirectly related to the Stokes 𝐼 flux density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', ideal, optically-thick synchrotron emission at 1 Jy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1 mJy would both have a linear polarization fraction of 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, the <1% upper limits on 2015 July 2 when the radio emission was dominated by a compact jet (compared to ∼3% during similar epochs of compact jet dominance in 1989), suggests the most recent outburst had a less-ordered magnetic field in the jet or suffered from higher depolarization due to independent unresolved components within the VLA beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' There are two clear features of the linear polarization fraction: (i) it is continuously weak (<1%) regardless of the time bin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (ii) it evolves in time, with maxima and minima linear polarization fractions (in each base-band) separated by a factor of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 Origin of Low Linear Polarization Fraction The short times between flares, the precession of the jet axis, and the energetics required for such a luminous outburst are character- istic of a complex (magnetic and geometric) environment, and are expected to inhibit strongly polarized emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' At a spatial resolu- tion of ∼1 AU the core emission identified by the VLBA could arise from an unresolved population of ejecta (likely on top of a compact jet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The jet axis precession would cause these ejecta to have vari- able PAs, and, assuming similar internal magnetic fields, variable EVPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The superposition of the unresolved (and resolved) ejecta in our VLA observations will decrease the polarization fraction, un- less all unresolved components have the same polarization fraction and EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The effects of multi-component superposition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', when the coherence length of the magnetic field is significantly smaller than the angular resolution) was seen in the recent Event Horizon Telescope (EHT) observations of M87;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the lower spatial resolution of ALMA reduced multiple components with linear polarizations of ≳20% (resolved with the Event Horizon telescope) to a net polar- ization fraction of ∼2% for the M87 core (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the 2015 outburst of V404 Cyg, the maximum linear polariza- tion fraction in each base-band decreases as the frequency decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here we consider the maximum of each frequency because of the potential time delays between base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A decreasing polariza- tion fraction with decreasing frequency is a common characteristic of Faraday depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In particular, sources with strong Faraday rotation within their emission regions can appear depolarized (in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 we find that the jet itself may be a strong source of Faraday MNRAS 000, 1–19 (2023) c 4 N6 S5 800 N1 + N8 9S (mJy) N2 x N9 S7 N3 S2 o Total N4 S3 Stokes 1 Flux Density 009 400 G Q GGO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 0 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 Time on 2015 June 22(UTC)V404 Cyg’s rapidly evolving polarized jet 11 rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Faraday depolarization has been well established in radio studies of AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2018) and was observed for the candidate BHXB SS 433 (Stirling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' While the complexity of the spectral and temporal evolution of the 2015 outburst of V404 Cyg makes it difficult to determine if we are in fact seeing Faraday depolarization, here we make some simple calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The simplest Faraday screen geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', a single uniform slab of synchrotron emitting plasma Burn 1966;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sokoloff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1998) predicts a depo- larization of Δ 𝑓𝜆/ 𝑓𝜆 ∼ 10% between the 26 and 5 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, a more complex model (see, Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2018) would be required for Faraday depolarization to explain the ∼ 70% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='75% at 26 GHz to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='22% at 5 GHz) depolarization we have observed (such an analysis is beyond the scope of this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We cannot ignore the possibility that during this outburst, the magnetic fields in the jet(s) are intrinsically more disordered than typical BHXB outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The BHXB GRO J1655−40 entered a multi- flaring highly-luminous state during its 1994 outburst, similar to the 2015 outburst of V404 Cyg (although the decay timescales of each flare were significantly longer in GRO J1655−40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, GRO J1655−40 reached a maximum 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 GHz linear polariza- tion fraction of 1–10% with linearly polarized variability as high as Δ 𝑓𝜆 ∼ 4% on timescales less than half a day, suggesting that weakly polarized emission is not an inherent aspect of multi-flaring outbursts (Hjellming & Rupen 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hannikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 Origin of Temporally Evolving Linear Polarization A transition of the absorption conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', from optically-thick to optically-thin synchrotron emission) of a dominant polarized com- ponent will result in a temporally evolving polarization fraction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', as seen in Swift J1745-26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Curran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During these transi- tions, we expect the intrinsic EVPA to rotate by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For optically-thin synchrotron emission, the EVPA and the magnetic field vector are perpendicular (Longair 2011), and for optically thick synchrotron emission, the EVPA tracks the direction of the magnetic field (see, Ginzburg & Syrovatskii 1969, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The EVPA will thus rotate as the source transitions from 𝜏 ∼ 10 to 𝜏 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5, where 𝜏 is the optical depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' this takes about half the rise timescale of a vdL plasmoid (Aller 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We do not observe a ∼90◦ rotation of the intrinsic EVPA, at any time during our monitoring (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, we know that the light curves are a superposition of multiple short-lived (≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 hr) ejecta, and a compact core, further reducing the plausibility of a single component origin for each radio flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' An ensemble of polarized components with evolving optical depths can exhibit a more complex evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As an investigation, we calculated the two-point spectral indexes for the 21/26 GHz and 5/7 GHz VLA observations (see bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We are un- able to disentangle the emission from the multiple unresolved compo- nents (seen in the VLBA), and, as a result, we are measuring the “net” spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, we are measuring a simultaneous spectral index, which may be less appropriate for rapidly evolving ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' An optically thick “net” spectral index (𝛼 > 0) requires that a sub- population of the unresolved components are optically thick (with the inverse being true for optically thin, 𝛼 < 0, spectral indexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The spectral indexes show an evolution in time, exhibiting multiple transitions of the absorption conditions, consistent with an ensemble of evolving components, with both optically thick and optically thin sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Intuitively, one might expect that a negative “net” spectral index measured would correspond to a higher contribution of optically thin synchrotron emission, and, as a result, a higher polar- ization fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The peak polarization fraction does in fact coincide with a negative spectral index;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', 𝛼 ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 in the 5/7 GHz and 21/26 GHz base-bands, respectively (Fig 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Furthermore, the late time rise seen in the 21/26 GHz base-bands (∼14:00–14:30), also coincides with a negative spectral index (𝛼 ∼ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, at ∼12:45 and 13:15 in the 5/7 GHz and 21/26 GHz base-bands, we also have 𝛼 ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During these times the polarization fraction shows a (weak) peak at 21 GHz, with marginal features at 5/7 GHz, and no evolution at 26 GHz (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', a “missing” polarization peak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we are unable to conclusively connect the spectral index to the polarization fraction evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Comparing the short time-scale temporal evolution to the 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 GHz VLBA light curves (reproducing data from Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019 as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3 of this paper), we do not see any clear con- nection between the resolved components and the evolution of the polarization fraction, and cannot distinguish between a polarized core, polarized ejecta, or a combination of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, the “missing” polarization peak coincides with a period of time when the S5 component clearly dominates the VLBA light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' It is pos- sible that S5 was less polarized than the components that preceded and followed its ejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, a complete explanation of the polarization fraction evolution may require a combination of evolv- ing optical depths, and intrinsic differences between the different polarized components launched at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Regarding a po- tential intrinsic evolution, Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2007) expanded upon the shock-in-jet picture outlined in Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2004), suggesting that the collisions between ejecta temporarily disorder the magnetic field lines while producing shock fronts that propagate through the ejecta, reestablishing a dominant field direction at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Shahbaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016) proposed a similar mechanism to explain the behaviour of the polarized optical emission during V404 Cyg’s 2015 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A flare in the optical polarization fraction that preceded a 16 GHz radio flare, was attributed to the compression of the jet’s magnetic field by many small shocks travelling along the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The exis- tence of these shocks is consistent with the detection of sub-second optical flares by Gandhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016) during the same time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The polarization flare was attributed to “a major ejection event” that followed optical flaring that began a couple of hours earlier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', a large outflow imprinted with the recently ordered magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In both of the scenarios proposed by Brocksopp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2007) and Shahbaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016), the ordering of the magnetic field is a result of multiple colliding components, and, as a result, the timescales separating collisions would have to be significantly shorter than the precession period of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This is a plausible theory if the sub- second optical flaring is characteristic of the collision timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Any such model would also need to explain the temporal offset between the Stokes 𝐼 and polarization fraction peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 Intrinsic EVPA In the first few days of the 1989 outburst, the EVPA evolved through a ∼90◦ rotation at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This rotation coincided with the transition from an optically-thin to optically-thick radio spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the 2015 outburst the dominant feature of the intrinsic EVPA evolution is a ∼30◦ rotation that occurs alongside the decay of the maximum polarization (bottom two panels, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This rotation occurs across 6 time bins (80 min) suggesting that a full 90◦ rotation would take ∼4 hr, a timescale longer than the lifetimes of any of the dominant VLBA components (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Under the assumption that the contemporaneous peak in polarization fraction and rotation of the EVPA arise from a shared mechanism, neither arise from a transition in the absorption conditions of a single component, as was likely observed in 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The precession of the jet axis provides a natural mechanism to explain the rotation of the EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We investigated this possibility by using the change in the position angles of the dominant ejecta as a proxy for the precession of the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The position angles also exhibited a ∼30◦ rotation, albeit over a longer, ∼2 hr, timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The ∼30◦ rotation begins when the S2 component (PA ∼ 1◦) is the dominant jet ejection observed by the VLBA (although the core emis- sion is brighter, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the simplest geometries, compression shocks or velocity-shearing establish dominant field directions par- allel or perpendicular to the jet flow’s direction of motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Laing 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Jorstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The approximate orthogonality (offset by ∼ 10◦) between the initial intrinsic EVPA and the PA dur- ing the decay of the S2 component is consistent with optically-thick (optically-thin) synchrotron emission from a magnetic field estab- lished by compression shocks (velocity shearing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The rotation coincides with the emergence of a new, dominant VLBA component (S3) at a position angle of −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' If the rotation from an intrinsic EVPA of 80◦ to 50◦ results from the S2-to-S3 transition, the larger obliquity (∼ 30◦) between the intrinsic EVPA and the PA of S3 requires a more complex magnetic field origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', remnants of helical fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Following S3 dominance, S5 becomes the dominant ejection, while maintaining a similar PA of ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The similarity between the PAs of S3 and S5 is consistent with the stability of the intrinsic EVPA at ∼50◦ between 12:45 and 13:45 UTC, assuming similar intrinsic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Alter- natively, the S6 component has a smaller obliquity when compared to the late time EVPA (∼ 10◦), and may be a better measure of the jet orientation, at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, since our observations are the superposition of multiple overlapping components (including a bright, unresolved compact core), there may be, in fact, no relation- ship between the position angles of the resolved components and the intrinsic EVPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Variability in the EVPA without any change in the jet axis PA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', rotator events) has been observed in many AGN (see Saikia & Salter 1988, and references therein), and a couple of BHXBs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', GRS 1915+105;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Fender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These events are thought to be the result of complex field geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', helical magnetic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2001) or internal shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Gómez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2008) producing time-varying magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, complex shock fronts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', conical shock waves) can produce magnetic field orientations that are neither perpendicular nor parallel to the jet axis (see Jorstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2007, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since the VLBA data did not acquire full polarimetric calibrations, there is no spatially resolved polarimetry that explicitly localizes the dominant polarized component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In the absence of such detections and given the multiple scenarios suggested above, we can neither defini- tively make connections between the VLBA/VLA observations and the linear polarization properties, nor identify if the polarized emis- sion originates from an ejected component, a compact steady jet, or a time-variable combination of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This limits the strength of our claims towards the origin of the polarization and its connection to the evolution of the Stokes 𝐼 flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note that due to the reduced sensitivity of spatially-resolved data, without a significant increase in the polarization fraction (as was observed in M87), the VLBA would be unable to detect comparably low polarization fractions, even after including the necessary calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 Rotation Measure The rotation measure quantifies the amount of Faraday rotation af- fecting a linearly polarized emission signal, and is related to the internal properties of the plasma along the line of sight (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', its Fara- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Linear fit to the observed EVPAs during V404 Cyg’s 1989 outburst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the data was adapted from Han & Hjellming 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To account for the wrapping of the EVPA at large values of 𝜆2 we applied a −2𝜋 correction to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='49 GHz observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We used scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='curve_fit for our linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' day screens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The RM is related to the electron number density, 𝑛𝑒, the magnetic field oriented parallel to the line of sight (from the source to the observer), 𝐵||, and the path length 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Explicitly, the rotation measure is described by the path integral, RM = � 812 ∫ observer source 𝑛𝑒𝐵||d𝑙 � rad m−2, (5) where 𝑛𝑒, 𝐵||, and d𝑙 are in units of cm−3, 𝜇G, and kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The sign of the rotation measure depends on the orientation of the magnetic field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', when the field lines are parallel (anti-parallel) to the direction of emission propagation, the sign is positive (negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, for Galactic sources, the total rotation measure can have significant contributions from both the diffuse interstellar medium (ISM) and the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Detecting a large local component necessarily implies a high density, or strongly magnetic environment to account for the reduced path length when compared to the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the previous outburst in 1989, Han & Hjellming (1992) measured a constant observed EVPA in four frequency bands for the majority of the (∼50 days) polarization monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The weighted averages from these observations were;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 3 ± 7◦, −44 ± 1◦, −16 ± 1◦, and −18 ± 2◦, at central frequencies of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='49, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4, and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The observed EVPAs are linear with respect to 𝜆2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 4), with a slope (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', rotation measure) of −151 ± 11 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During the first ∼2–3 days of polarization detections, the EVPAs at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 GHz exhibited a 90◦ rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The two-point slope of these angles (39±4◦ and −60±6◦ at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='49 GHz, respectively) shows a consistent rotation measure of −150 ± 50 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The matching rotation measures, even with a changing EVPA, implies a constant Faraday screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The magnitude, orientation, and stability of the 1989 rotation measure is similar to our RM measurement during the 2015 outburst (−100 ± 15 rad m−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' however, the former is detected over much longer timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The rotation measures during the 1989 and 2015 outbursts are marginally consistent (at the ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7𝜎 level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we are unable to conclusively identify temporal variability of the rotation measure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', as was seen in the 1994 outburst of GRO J1655−40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hannikainen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Had we identified temporal variability, we could rule out the scenario that both outbursts are behind a constant, purely Galactic Faraday screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 1 2 3 x 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content="02 EO'O 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='04 A2 (m7)V404 Cyg’s rapidly evolving polarized jet 13 Multiple Galactic RM models predict a negative rotation measure along the line of sight to (and beyond) V404 Cyg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' −30 ± 10 rad m−2 (Oppermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2012), −40±20 rad m−2 (Oppermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2015), and −130 ± 50 rad m−2 (Hutschenreuter & Enßlin 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Assuming a constant Galactic magnetic field, the Galactic RM would be dom- inated by distances well beyond the ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 kpc distance to V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For both standard models of electron distributions (Cordes & Lazio 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017) the PyGEDM tool (Price et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2021) indicates much larger dispersion measures (which are proxies for the integrated electron column density) at 10 kpc (269±16 pc cm−3) than at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 kpc (32 ± 4 pc cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Thus, our measured value of ≈ − 100 rad m−2 sug- gests, either an inversion (or multiple inversions) of the Galactic magnetic field along our line of sight, or an intrinsic rotation mea- sure component local to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The morphology of the Galactic magnetic field is poorly constrained, with different models predicting radically different structures (see, Haverkorn 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Jaffe 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As an example, the “zeroth"-order model by Van Eck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2011) pre- dicts a parallel Galactic magnetic field within the first ∼4 kpc along the line-of-sight containing V404 Cyg, inverting to an anti-parallel orientation at larger distances and producing the net-negative rota- tion measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Conversely, the model by Jansson & Farrar (2012), predicts two large-scale inversions along the line of sight of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' an initial anti-parallel magnetic field (and a negative rotation measure at the position of V404 Cyg), an inversion to a parallel orientation at intermediate distances, followed by a second inversion back to an anti-parallel orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, using the standard approxima- tion, |𝐵||,avg| = |RM/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='81DM)|, we can estimate a mean parallel magnetic field magnitude of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 𝜇G, which is larger than the total ( √︃ 𝐵2 || + 𝐵2 ⊥) Galactic magnetic field magnitudes predicted by both Van Eck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2011, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 𝜇G) and Jansson & Farrar (2012, ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 𝜇G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Given our estimate for the mean parallel magnetic field strength along the line of sight towards V404 Cyg, there exists three physical explanations: (i) the mean electron number density is larger than predicted by the standard dispersion models along this line of sight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (ii) the mean magnetic field strength within the ISM is stronger than predicted by Galactic magnetic field models along this line of sight;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' or (iii) there is a local Faraday screen that likely resides within the jets themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here we investigate the source of a (potential) rotation measure component local to V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A local rotation measure component is the result of either a fore- ground Faraday screen (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', created by disk outflows) or a rotation from within the emission regions themselves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', the compact core or jet ejecta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016) detected a strong, contin- uous wind originating from the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Assuming that the wind creates a foreground Faraday screen with an electron number density that follows an inverse-square scaling, 𝑛𝑒 ≡ 𝑛0(𝑙/𝑙0)−2, and a typical ISM magnetic field strength (𝐵||∼ 2 𝜇G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Haverkorn 2015), equation (5) simplifies to, RM = 1624 𝑛0𝑙2 0 � 1 𝑙0 − 1 𝑙max � rad m−2, (6) where the wind-fed Faraday screen occupies the space between 𝑙0 and 𝑙max along our line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We approximate 𝑙max ∼ 𝑣Δ𝑡 ∼ 8 AU using the measured wind velocity (𝑣 ∼ 2000 km s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2016) and the time interval between the start of the outburst and our observations (Δ𝑡 ∼ 7 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We adopt the VLBA angular resolution of 1 AU, as a conservative estimate of 𝑙0 for compact core emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The jet ejections with well constrained inclination angles are S2 (∼ 40◦), S3 (∼ 30◦), and S6 (∼ 15◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' all three ejecta have an angular separation of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 milliarcseconds during their flux density peaks (Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The distance to V404 Cyg is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='39 kpc, and, therefore, 𝑙0 ∼ 2 − 5 AU for jet ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For a wind-fed Faraday screen to produce our observed rotation measure (|RM| ∼ 100 rad m−2), we require 𝑛0 ∼ (6−15)×106 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Assuming a 50% ionized, isotropic, pure hydrogen outflow, launched at a distance of 6 × 105 km from the central black hole, the wind mass loss rate would need to be �𝑀 ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 − 2) × 10−6 𝑀⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Muñoz-Darias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2016) estimated a wind mass loss rate of > 10−13 𝑀⊙ yr−1, ∼7 orders of magnitude smaller then our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The authors left the estimate as a lower limit because the ionization fraction (∝ �𝑀−1) may be lower then the assumed value of 𝑓𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5, and the launching radius (∝ �𝑀) may be larger then their assumed value of 𝑅 = 6×105 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A 7 order of magnitude reduction in the ionization fraction would inhibit Faraday rotation, as the outflow would become neutral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Furthermore, a 7 order of magnitude increase in the launching radius corresponds to an distance of 4 × 104 AU, far exceeding the scale of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, without a highly magnetized wind, or an extremely anisotropic wind coupled with a favourable line of sight, disk winds forming a foreground screen cannot be the origin of the observed rotation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For Faraday rotation internal to the emission environment we look at the recent model of the compact jet from MAXI J1820+070 (see, Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2022, for a detailed description of the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The strength of the magnetic field, and the electron number den- sity scale according to the power-law relations, 𝐵 = 𝐵0𝜉−𝑏 and 𝑛𝑒 = 𝑛0𝜉−𝑎𝛾−𝑝, where 𝛾 is the lorentz factor of the synchrotron emitting electrons, and 𝜉 = 𝑧 𝑧0 = ( 𝜈 𝜈0 )−𝑞, where 𝑧 is the position along the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' At 𝑧 > 𝑧0, the jet emits synchrotron radiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' this leads to a break in the spectrum from optically thick to optically thin at 𝜈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We adopt the following values used in the original paper: 𝑏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝑎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝑝 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='882;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 𝐵0 ∼ 1010 𝜇G, 𝑧0 ∼ 3 × 1010 cm, 𝜈0 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3×104 GHz, 𝑛0 ∼ 3×1014 cm−3, and we adopt a value of 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 × (𝛾min+𝛾max) = 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The model predicts 𝐵 ∼ 3×106 𝜇G and 𝑛𝑒 ∼ 2 × 102 cm−3 at 𝜈 = 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Letting, 𝐵|| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5𝐵, and assuming a uniform Faraday screen, we would require a screen thickness of 𝑑𝑙 ∼ 3×10−10 kpc to account for the rotation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To first-order, this is the same as the radius of the conical jet at position 𝑧, 𝑅 = 𝑧 sin 𝜃, for the best fit opening angle 𝜃 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Considering, that the best fit orbital inclination angle is ∼65◦, it is reasonable to assume the our line of sight looks partially down the jet axis, and, as a result, 𝑑𝑙 > 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Furthermore, The electron number density could be substantially larger than expected from a typical hard state compact jet if the jet entrains material from the disk winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Entrainment is a known source of internal Faraday depolarization in AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Silpa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2022), and jet-wind interactions have been observed in the BHXB candidate SS 443 (Blundell & Hirst 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Although we are unable to rule out that V404 Cyg has a magnetic field oriented perpendicularly to the line of sight, or significantly different jet parameters when compared to MAXI J1820+070 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', a weaker magnetic field), to first-order, it is plausible that the jet itself may act as a strong, local Faraday screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 5 SUMMARY AND CONCLUSIONS In this paper we present our analysis of the multi-frequency (5, 7, 21, and 26 GHz), linear polarization radio data of the BHXB V404 Cyg during its 2015 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The majority of our results and inter- pretations focused on the behaviour during the bright flaring activity on 2015 June 22, however, we also included the upper limits from six observations during the source’s return to quiescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Using two independent polarimetric methods we extracted the fractional po- larizations, observed/intrinsic EVPAs, and rotation measures from MNRAS 000, 1–19 (2023) 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the 2015 June 22 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We tracked the evolution of the polarization properties on timescales ∼13 min, constituting one of the shortest timescale polarimetric analyses of a BHXB to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' By comparing our polarimetric results to the VLA Stokes 𝐼 light curves modelled by Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2017) and the simultaneous VLBA observations by Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (2019), we infer the fol- lowing properties about the polarization evolution of V404 Cyg: V404 Cyg is weakly polarized, with a maximum polarization fraction that increases with frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='22, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='75% for the 5, 7, 21, and 26 GHz base-bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These maxima are significantly smaller than typically observed in outbursting BHXBs, suggestive of a complex local environment or complex internal mag- netic field structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The time-evolution of the linear polarization fraction shows a frequency-dependent lag, with low frequencies lagging behind their high frequency counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This behaviour is characteristic of an emission origin within dynamic components (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', expanding ejecta or propagating shock fronts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The maximum polarization fraction is offset from the Stokes 𝐼 flux density maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This suggests an offset between the processes that maximize each quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A secondary peak in fractional polar- ization at 21 GHz after the second flare in Stokes 𝐼 and the increase in polarization fraction towards the end of the 21/26 GHz base-band observations, provide further evidence of a temporal offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The decay of the (brightest) polarization fraction peak coincides with a rotation of the intrinsic EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We are unable to conclusively determine if the origin of this feature is the result of an internal change within the polarized components, or the emergence (and decay) of polarized components with different magnetic field structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The derived rotation measures show stability in time with an average value of ∼−100 rad m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We investigated the potential of a strong local component, and although we found it plausible, we are unable to conclusively rule out a purely Galactic rotation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Overall, our results emphasize the complexity of local (magnetic) environments during highly energetic outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Although we are confident that the observed behaviour cannot be ascribed to the sim- plest interpretation and models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', intrinsic EVPA swings from changes in the absorption conditions of single components), the lim- itations of our observations inhibited us from making strong claims about the origin of the polarized emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These limitations empha- size the importance of spatial resolution during polarimetry, which would enable the identification of the primary source of the po- larized emission in multi-component outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' X-ray polarimetric observations of black hole X-ray binaries (like those recently done for the black hole X-ray binary Cygnus X-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Krawczynski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2022) probe the accretion disk, corona and perhaps some compo- nent from the synchrotron tail of a jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For the radio-brightest X-ray binary outbursts, there is strong potential to combine such observa- tions with radio through sub-mm polarimetric observations to track temporally evolving polarization properties across the electromag- netic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, we note that, like the VLA observations performed here that did not have the angular resolution needed to sep- arate polarization properties from different ejecta, one must carefully consider how the polarization properties from different components will average when making interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' If the next ∼1 Jy scale out- burst of a BHXB is observed with adequate spatial resolution, full polarization coverage, and sufficient sensitivity, such observations have the potential to provide invaluable insight into the magnetic fields that drive accretion-powered jet ejections, emission, and evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The new generation of interferometers like the next-generation VLA, the next-generation EHT, and the Square Kilometre Array, will combine high sensitivity and good spatial resolution into a single in- strument, making spatially resolved polarimetry a realistic goal for future, bright outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We extend our sincere thanks to all of the NRAO staff involved in the scheduling and execution of these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We offer a special thanks to Frank Schnitzel for sharing his expertise of polarization observation, and calibration, and data reduction with the VLA and CASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We thank Cameron Van Eck for helpful discussions regarding Galactic magnetic field models and the Canadian Initiative for Radio Astronomy Data Analysis (CIRADA) RM synthesis routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Finally, we thank the referee for their insightful and helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This research has made use of software provided by CIRADA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' CIRADA is funded by a grant from the Canada Foundation for In- novation 2017 Innovation Fund (Project 35999), as well as by the Provinces of Ontario, British Columbia, Alberta, Manitoba, and Que- bec, in collaboration with the National Research Council of Canada, the US National Radio Astronomy Observatory and Australia’s Com- monwealth Scientific and Industrial Research Organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' AKH and GRS are supported by NSERC Discovery Grants RGPIN-2016-06569 and RGPIN-2021-0400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Support for this work of AJT was provided by NASA through the NASA Hubble Fellow- ship grant #HST–HF2–51494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', for NASA, under contract NAS5– 26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' TDR acknowledges financial contribution from the agreement ASI-INAF n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2017-14-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' GEA is the recipient of an Australian Re- search Council Discovery Early Career Researcher Award (project number DE180100346).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' SM is thankful for support from an NWO (Dutch Research Council) VICI award, grant Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' TMB acknowledges the financial contribution from grant PRIN-INAF 2019 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' RS acknowledges support from grant number 12073029 from the National Natural Science Foundation of China (NSFC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' AKH and GRS respectfully acknowledge that they perform the majority of their research from Treaty 6 territory, a tradi- tional gathering place for diverse Indigenous peoples including the Cree, Blackfoot, Métis, Nakota Sioux, Iroquois, Dene, Ojibway/ Saulteaux/Anishinaabe, Inuit, and many others whose histories, lan- guages, and cultures continue to influence our vibrant community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The authors also wish to recognize and acknowledge the significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We are most for- tunate to have the opportunity to conduct VLBA observations from this mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' DATA AVAILABILITY Data from the VLA are available through the VLA data archive (Project ID 15A–504): https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='nrao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='edu/archive/ advquery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='jsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We make our flagging and calibration scripts, and the short timescale imaging and analysis, available at 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Bianchi 46, I- 23807 Merate, Italy 7Department of Astronomy, University of Wisconsin Madison, 475 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Charter Street, Madison, WI 53706, USA 8SRON, Netherlands Institute for Space Research, Sorbonnelaan, 2, NL-3584CA Utrecht, the Netherlands 9Department of Astrophysics/IMAPP, Radboud University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Box 9010,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' NL-6500 GL Nijmegen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Netherlands 10Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Wheaton College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Norton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MA 02766,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' USA 11Anton Pannekoek Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' University of Amster- dam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Science Park 904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' NL-1098 XH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Netherlands 12Gravitation Astroparticle Physics Amsterdam (GRAPPA) Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Science Park NL-904,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1098 XH Amster- dam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Netherlands 13Aurora Technology BV for the European Space Agency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' ESAC/ESA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Camino Bajo del Castillo s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Urb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Villafranca del Castillo, 28691, Villanueva de la Cañada, Madrid, Spain 14Institut de Ciències del Cosmos (ICC), Universitat de Barcelona (IEEC-UB), Martí i Franquès 1, E08028 Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 15National Radio Astronomy Observatory, Socorro, NM 87801, USA 16Caltech, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MC 249-17, Pasadena, CA 91125, USA 17Herzberg Institute of Astrophysics, National Research Council of Canada, Penticton, BC V2A 6J9, Canada 18Center for Astro, Particle and Planetary Physics, New York Uni- versity, Abu Dhabi, PO Box 129188, Abu Dhabi, UAE 19INAF/IASF Palermo, via Ugo La Malfa 153, I-90146 Palermo, Italy 20Department of Astronomy, University of Virginia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Box 400325,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' VA 22901,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' USA 21College of Astronomy and Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' University of the Chi- nese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' China 22Sydney Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' School of Physics A28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The Uni- versity of Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' NSW 2006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Australia 23INAF - Osservatorio Astrofisico di Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Strada Osservatorio 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 10025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Pino Torinese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Italy 24Institute for Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Atomistilor 409,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' PO Box MG-23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 077125 Bucharest-Magurele,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Romania MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 1–19 (2023) V404 Cyg’s rapidly evolving polarized jet 17 APPENDIX A: STOKES I LIGHT CURVES AND TEMPORAL DELAYS The Stokes 𝐼 flux density light curves for all four base-bands share a common “three-flare” morphology (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A1), where each flare is composed of multiple unresolved (by the VLA) jet ejecta (Tetarenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Miller-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The temporal delays and longer rise/decay times for the lower frequency flares are consistent with the expected behaviour of expanding vdL bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Considering a single vdl ejection, the flaring results from an evolving optical depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the peak flux density occurs near the transition from optically thick to optically thin emission, and the peak occurs at earlier times for higher frequencies (van der Laan 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, broadband ob- servations will mix optically thick (low frequency) and optically thin (high frequency) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Optically thick and thin emission are orthogonally polarized, and thus their summation causes a (poten- tially significant) depolarization effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' To quantify this effect, we measured the delays between flux density peaks (for each of the three flares) from the cross-correlation function (CCF) of the high time res- olution (per-spectral window) light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Each CCF was generated using the z-transformed discrete correlation function (ZDCF) tech- niques of Alexander (1997)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We measured delays between flares of ∼30–60 min (∼2–4× the imaging window) and ∼7–15 min (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5–1× the imaging window) comparing the 5-to-26 GHz and 5-to-7 GHz base-band light curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The earliest flare exhibited the smallest delays between bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For the polarization fraction, the small number of time bins inhibited a similar use of the ZDCF algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, looking at Figure 1, the delays between the polarization fraction peaks are ∼2 imaging windows (∼30 min) and ≲1 imaging window (≲15 min), for the 5-to-26 GHz and 5-to-7 GHz delays, consistent with the Stokes 𝐼 behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The 5-to-26 GHz de- lays are an appreciable fraction of the lifetime of a single ejection (≲ 90 min), and, therefore, the full bandwidth (5-to-26 GHz) may have a non-negligible fraction of orthogonal emission, even when considering isolated ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The situation becomes considerably more complicated when con- sidering the multi-ejecta flares, ejecta collision, and jet precession (as seen in the 2015 outburst of V404 Cyg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Moreover, we note that mod- elled ejecta exhibit different delays between frequencies (largely due to differences in ejecta expansion velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As such, the dominant ejection (in a particular flare) depends on the observing frequency, thereby introducing another frequency-dependent effect on the ob- served EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' While critical, the VLBA observations only provide a snapshot at one frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to the large temporal separations, we choose to omit the simultaneous linear polarization data from the 21 and 26 GHz base-bands when extracting EVPAs and rotation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This is why we only consider the simultaneous linear po- larization data from the 5 and 7 GHz base-bands in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2, § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2, and § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We intend to apply temporal corrections to future (broad-band) outbursts with single (or temporally isolated) ejecta as an investiga- tion into the effects these delays have on polarization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' APPENDIX B: CALIBRATOR MODEL For Stokes 𝐼 calibration, casa includes a repository of spatially re- solved model images for many standard calibrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our flux calibra- tor, 3C48, is included in this repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Each model image describes the flux density distribution at a single, band-dependant, reference 5 FORTRAN code available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='weizmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='il/ particle/tal/research-activities/software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' frequency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', 𝜈ref = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8601 GHz for the 4-8 GHz band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' During calibration, the model is mapped onto the remaining spectral chan- nels assuming the total flux density follows the flux density scaling relationships of Perley & Butler (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' It is assumed that the spatial distribution of the relative flux densities remains constant across a band;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', for an arbitrary spectral channel with a central frequency 𝜈, the ratio between the total integrated flux, 𝐼𝜈, and the flux of pixel 𝑖, 𝐼𝑖,𝜈, is independent of frequency, and thus identical to the ratio at the reference frequency (𝐼𝑖,𝜈/𝐼𝜈 ≡ 𝐼𝑖,𝜈ref/𝐼𝜈ref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This procedure results in a spatially resolved Stokes 𝐼 model for every spectral channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For our Stokes 𝑄 and 𝑈 calibration, we adopted a similar approach to the default Stokes 𝐼 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We assumed that the spatial distribution of the linearly polarized flux densities is independent of frequency and that it has the same spatial distribution as the Stokes 𝐼 repository images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For each spectral channel with a central frequency of 𝜈, we calculated the total Stokes 𝑄 and 𝑈 flux densities according to the following relationships, 𝑄𝜈 = 𝐼𝜈 𝑓𝜈 cos (2𝜒𝜈) , (B1) 𝑈𝜈 = 𝐼𝜈 𝑓𝜈 sin (2𝜒𝜈) , (B2) where 𝑓𝜈 is the linear polarization fraction, and 𝜒𝜈 is the observed EVPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We mapped the total flux densities onto each pixel by as- suming that 𝑈𝑖,𝜈/𝑈𝜈 ≡ 𝑄𝑖,𝜈/𝑄𝜈 ≡ 𝐼𝑖,𝜈ref/𝐼𝜈ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Our final model consisted of a spatially resolved Stokes 𝐼, 𝑄, and 𝑈 image (assuming no circular polarization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Stokes 𝑉 = 0) for each spectral channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We then applied the model to our data using the native casa task ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In equation (B1) and (B2), 𝐼𝜈 was calculated according to Perley & Butler (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For 𝑓𝜈 and 𝜒𝜈, we fit the data presented in Perley & Butler (2013), such that the linear polarization fraction obeys a third-order log-log polynomial, log( 𝑓𝜈) = 3 ∑︁ 𝑛=0 𝑎𝑛 log (𝜈GHz)𝑛 , (B3) and the observed EVPAs obey a standard third-order polynomial (with frequency), 𝜒𝜈 = 3 ∑︁ 𝑛=0 𝑏𝑛 (𝜈GHz)𝑛 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (B4) where 𝜈 GHz is the observing frequency (in GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We decided to fit the polarization fraction with a 3rd-order polynomial in log-space to remain consistent with the 𝐼𝜈 fits in Perley & Butler (2017), and 𝜒 in linear-space to allow for negative EVPAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We fit the polarization properties separately for the 5/7 GHz and 21/26 GHz bands, rather than a single fit as performed by Perley & Butler (2017), following extensive discussions with NRAO staff (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Schinzel priv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Table B1 contains the third-order polynomial fit for the polarization calibrator model and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' B1 plots the fit over the Perley & Butler (2013) observations of 3C48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' APPENDIX C: PHASE CALIBRATOR POLARIMETRIC EVOLUTION Given the low linear polarization fractions we detected in our V404 Cyg observations, we checked the relative stability of our polariza- tion calibrations on short time scales to ensure that the variability we observed is the result of intrinsic variations and not systematic cali- bration effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we performed our full polarimetric analy- sis on the phase calibrator (J2025+3343), grouping scans within the same bins as used for V404 Cyg when making images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' High time resolution Stokes 𝐼 light curves for our observing base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In each base-band, we have separately plotted the flux densities of the 8 spectral windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The key features are the three major flares, with the final flare having a twin-peaked structure in the 21/26 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The lower frequency emission is temporally delayed, peaks at lower flux densities, and has broader flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' These properties are consistent with both vdL ejecta and a shock-in-jet event scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Polarization fraction spectra for 3C48;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' left: 5/7 GHz base-bands and right: 21/26 GHz base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The shaded regions highlight the range of frequencies spanned by each observing band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The solid black points are the data from Perley & Butler (2013), and the black line is our polynomial fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The logarithmic term in equation (3) is exactly equal to the equivalent frequency in GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 26 GHz 1400 21 GHz 1200 7 GHz Density 5 GHz Fluy BDD Stokes 600 400 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 11:00 DE:TT 12:00 DE·ZL 13:00 DE:ZL DE:t Time on 2015 June 22 (UTC)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 (%) Fraction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 Polarization 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 Linear J N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content="5 0'E 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 4 5 6 8 6 10 16 18 20 27 24 26 28 4 36 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='10 (rad) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='15 EVPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='25 Observed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='45 16 4 5 6 8 9 10 18 20 22 2426 28 32 34 36 38 Frequency (GHz) Frequency (GHz)V404 Cyg’s rapidly evolving polarized jet 19 Table B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The third-order polynomial fits for Stokes 𝐼, the polarization fraction spectra, and the intrinsic EVPA of 3C48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The 𝐼𝜈 values are taken from Perley & Butler (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Quantity Band 𝑎0 𝑎1 𝑎2 𝑎3 𝐼𝜈 5/7 GHz 21/26 GHz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3253 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3253 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7553 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7553 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1914 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0498 𝑓𝜈 5/7 GHz 21/26 GHz −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5775 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5927 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6671 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9992 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7686 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5932 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8181 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5275 Quantity Band 𝑏0 𝑏1 𝑏2 𝑏3 𝜒 5/7 GHz 21/26 GHz −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9939 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1141 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2480 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0078 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0001 Figure C2 shows the temporal evolution of the residual rota- tion measure and residual observed EVPA for both V404 Cyg and J2025+3343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Here we define the residual as the difference between the individual time bins, and the weighted average over all time bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Visually, J2025+3343 shows both a stable rotation measure (RM ∼ −750 rad m−2) and observed EVPA (𝜒𝑤 ∼ −35◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Applying the same 𝜒2 test as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2, neither the rotation measure (𝜒2 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1/12) nor the observed EVPA (𝜒2 = 2508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3/12) is consistent with a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' However, the linear polarization detections of J2025+3343 have a much higher signal-to-noise ratio (S/N > 200), and, as a result, we have likely reached a systematic threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As such, we believe we are underestimating the errors us- ing an unrestricted S/N scaling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', as S/N → ∞, 𝜎𝜒𝑤 → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The 𝜒𝑤 standard deviation for J2025+3343 is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4◦, which is equal to the smallest 𝜒𝑤 error for V404 Cyg (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4◦) and ∼ 1/2 of our median error (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Since V404 Cyg exhibits a ∼ 30◦ degree rotation, the systematic variability cannot be the cause of the observed evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure C2 compares the temporal evolution of the linear polar- ization fraction, defining the “residual” in the same manner as in Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The evolution of J2025+3343 does not track the simulta- neous evolution of V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The multi-band “jumps”, at ∼ 12:00 and ∼ 13:45 UTC, correspond to a change of sub-arrays (marked by the vertical dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' It is unsurprising to see some discontinuity between the two sub-arrays as each sub-array will have (slightly) different 𝑢𝑣-coverage, a unique reference antenna, and, as a result, different calibration solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We note: (i) the bin-by-bin variability of J2025+3343 within a sub-array is significantly smaller than the jumps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' and (ii) although obvious in J2025+3343, we do not observe similar jumps in our V404 Cyg data — instead, the most significant evolution occurs absent a change of sub-array;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (iii) In all four base- bands, the variability of V404 Cyg is larger than J2025+3343;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (iv) V404 Cyg shows a common temporal evolution across base-bands, that is absent in the J2025+3343 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Therefore, we are confident that the polarized detections of V404 Cyg are dominated by an intrinsic, physical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' APPENDIX D: SAMPLE IMAGES Figure D1 shows a sample set of 𝑃, 𝑄, 𝑈 images at both 5 and 7 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' For this example, despite the clear detection in 𝑃 (top row), we are unable to detect the source in the Stokes 𝑄 images (middle row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Due to the intrinsic variability of the source, in other time bins and frequency ranges the properties of the non-detections may change (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', Stokes 𝑄 is detected but Stokes 𝑈 is not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' As a result, we chose to extract the Stokes 𝑄 and 𝑈 flux densities using forced aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' APPENDIX E: DATA TABLES The following tables summarize the key observations: Table E1 con- tains the polarization fraction observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' and Table E2 contains the EVPAs and rotation measures derived from both polarimetric routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Temporal evolution of the residual rotation measure (top), and observed EVPA (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The red, dashed line represents the phase calibrator J2025+3343, and the black, solid line, V404Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The observed EVPA of V404 Cyg exhibits a clear evolution that is absent in J2025+3343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) V404 Cyg 100F J2025+3343 ARM (radm-2) 100 200 300 20 15 10 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=') MXV 10 11:30 12:00 12:30 13:00 13:30 14:00 Time on 2015 June 22 (UTC)V404 Cyg’s rapidly evolving polarized jet 21 Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Temporal evolution of the linear polarization fraction residuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' 26 GHz (top), 21 GHz (2nd from the top), 7 GHz (3rd from the top), 5 GHz (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The red, dashed line represents the phase calibrator J2025+3343, and the black, solid line, V404 Cyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The vertical dashed lines highlight the times when the observing band switched from one sub-array to another;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' the jumps are the result of this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We can see that the linear polarization fraction evolution of J2025+3343 does not track V404 Cyg, and shows smaller amplitude variability in all base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 V404 Cyg -*--J2025+3343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 26 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content="1 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='1 21 GHz (%) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 7 GHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 5 GHz 11:31 12:00 12:28 12:57 13:26 13:55 14:24 Time on 2015 June 22 (UTC)22 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Figure D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Sample images of the 12:09 time bin for both the 5 GHz (left) and 7 GHz (right) base-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (top) MFS images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=', ∼ 1 GHz bandwidths) of the linear polarization intensity (𝑃 = √︁ 𝑄2 + 𝑈2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (middle) Fine spectral resolution Stokes 𝑄 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' (bottom) Fine spectral resolution Stokes 𝑈 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The contours show the 3, 4, and 5𝜎 levels, and the color bars are in units of mJy/beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The example images at fine spectral resolution have central frequencies of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='209 GHz and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='545 GHz with bandwidths of 16 MHz and 64 MHz for the 5 and 7 GHz base-bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Stokes 𝑄 and 𝑈 are not positive definite, and, as a result, either may appear as non-detections regardless of the strength of the detection in 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In this example, the source is not detected in Stokes 𝑄 despite its detections in both 𝑃 and Stokes 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' This behaviour motivated our use of forced aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' MNRAS 000, 1–19 (2023) 5 GHz 7 GHz 33°52\'04" 33°52\'04" 0.' metadata={'source': 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305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='080 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='010 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='6 83 275.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='095 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='009 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 80 13:53 504.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='010 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 79 14:07 538.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='009 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 54 14:20 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='057 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='008 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='5 60 727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='010 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='9 60 14:32 630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='056 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='008 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 58 711.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='0 97 MNRAS 000, 1–19 (2023) 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Hughes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Table E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' A summary of the RM synthesis and MCMC results, and 𝑡𝑐𝑡𝑟 adopts the same definition as used in Table E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' In this chart we’ve defined the S/N as the amplitude of the FDF component over the rms error across the FDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' any component with a S/N > 5 was recorded, but only one time bin (at 11:29 UTC) had a secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' The lone secondary component had a similar magnitude (|𝑅𝑀 | ∼ 2300 rad m−2) as the systematic errors discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' We do not believe this to be a real signal and have omitted this component from the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' Note that the two components with the most significant deviations from the weighted mean (∼ − 100 rad m−2) are also the lowest S/N detections (S/N∼ 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content=' RM Synthesis MCMC 𝑡ctr (HH:MM) RM (rad m−2) 𝜒𝑤 (◦) 𝜒0 (◦) S/N RM (rad m−2) 𝜒𝑤 (◦) 𝜒0 (◦) 11:15 −330+110 −110 62+4 −4 76+5 −5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='8 −328+60 −60 63+3 −3 77+3 −3 11:29 −79+90 −90 62+3 −3 76+4 −4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9FQT4oBgHgl3EQfQDZ4/content/2301.13281v1.pdf'} +page_content='3 −35+70 −60 64+3 −3 78+4 −3 11:42 −122+40 −40 59+1 −1 73+3 −3 20.' metadata={'source': 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b/DdFKT4oBgHgl3EQfZC4_/content/tmp_files/2301.11801v1.pdf.txt @@ -0,0 +1,1090 @@ +arXiv:2301.11801v1 [physics.gen-ph] 12 Jan 2023 +4D Einstein–Gauss–Bonnet gravity coupled to modified +logarithmic nonlinear electrodynamics +Sergey Il’ich Kruglov 1 +Department of Physics, University of Toronto, +60 St. Georges St., Toronto, ON M5S 1A7, Canada +Department of Chemical and Physical Sciences, University of Toronto, +3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada +Abstract +Spherically symmetric solution in 4D Einstein–Gauss–Bonnet grav- +ity coupled to modified logarithmic nonlinear electrodynamics (Mod- +LogNED) is found. +This solution at infinity possesses the charged +black hole Reissner–Nordstr¨om behavior. +We study the black hole +thermodynamics, entropy, shadow, energy emission rate and quasi- +normal modes. It was shown that black holes can possess the phase +transitions and at some range of event horizon radii black holes are +stable. The entropy has the logarithmic correction to the area law. +The shadow radii were calculated for variety of parameters. We found +that there is a peak of the black hole energy emission rate. The real +and imaginary parts of the quasinormal modes frequencies were cal- +culated. The energy conditions of ModLogNED are investigated. +Keywords: Einstein−Gauss−Bonnet gravity; nonlinear electrodynamics; +Hawking temperature; entropy; heat capacity; black hole shadow; energy +emission rate; quasinormal modes +1 +Introduction +Nowadays, there are many theories of gravity that are alternatives to Ein- +stein’s theory [1, 2]. The motivation of generalisations of Einstein’s theory of +General Relativity (GR) is to resolve some problems in cosmology and astro- +physics. One of important modification of GR is the Einstein–Gauss–Bonnet +1E-mail: serguei.krouglov@utoronto.ca +1 + +(EGB) theory [3, 4, 5, 6]. EGB theories do not include extra degrees of free- +dom and field equations have second derivatives of the metric. These theories +also prevent Ostrogradsky instability [7]. The four dimensional (4D) EGB +theory, that includes the Einstein–Hilbert action plus GB term, is a particu- +lar case of the Lovelock theory. It represents the generalization of Einstein’s +GR for higher dimensions and EGB theory results covariant second-order +field equations. The GB part of the action possesses higher order curvature +terms. +It is worth mentioning that at low energy the action of the het- +erotic string theory includes higher order curvature terms [8, 9, 10, 11, 12]. +Therefore, it is of interest to study gravity action with the GB term. The +GB term is a topological invariant in 4D and before a regularization it does +not contribute to the equation of motion. But Glavan and Lin [13] showed +that re-scaling the coupling constant, after the regularization, GB term con- +tributes to the equation of motion. The consistent theory of 4D EGB gravity, +was proposed in [14, 15, 16], is in agreement with the Lovelock theorem [5] +and possesses two dynamical degrees of freedom breaking the temporal dif- +feomorphism invariance. It is worth noting that the theory of [14, 15, 16], +in the spherically-symmetric metrics, gives the solution which is a solution +in the framework of [13] scheme (see [17]). Some aspects of 4D EGB gravity +were considered in [18]. The black hole and wormhole type solutions in the +effective gravity models, including higher curvature terms, were obtained in +[19]. +Here, we study the black hole thermodynamics, the entropy, the shadow, +the energy emission rate and quasinormal modes in the framework of the +ModLogNED model (proposed in [20]) coupled to 4D EGB gravity. It is +worth noting that ModLogNED model is simpler compared with logarithmic +model [21] and generalized logarithmic model [22] because the mass and met- +ric functions here are expressed through simple elementary functions. The +black hole quasinormal modes, deflection angles, shadows and the Hawking +radiation were studied in [23, 24, 25, 26, 27, 28, 29]. +The structure of the paper is as follows. In Sect. 2, we obtain the spher- +ically symmetric solution of black holes in the 4D EGB gravity coupled to +ModLogNED. At infinity the Reissner−Nordstr¨om behavior of the charged +black holes takes place. The black hole thermodynamics is studied in Sect. 3. +We calculate the Hawking temperature, the heat capacity and the entropy. +At some parameters second order phase transitions occur. The entropy in- +cludes the logarithmic correction to Bekenstein–Hawking entropy. In Sect. 4 +the black hole shadow is investigated. We calculate the photon sphere, the +2 + +event horizon, and the shadow radii. The black hole energy emission rate is +investigate in Sect. 5. In Sect. 6 we study quasinormal modes and find com- +plex frequencies. Section 7 is a summary. In Appendix A energy conditions +of ModLogNED model are investigated. +2 +4D EGB model +The action of EGB gravity coupled to nonlinear electrodynamics (NED) in +D-dimensions is given by +I = +� +dDx√−g +� +1 +16πG (R + αLGB) + LNED +� +, +(1) +where G is the Newton’s constant, α has the dimension of (length)2. The +Lagrangian of ModLogNED, proposed in [20], is +LNED = − +√ +2F +8πβ ln +� +1 + β +√ +2F +� +, +(2) +where we use Gaussian units. The parameter β (β ≥ 0) possesses the di- +mension of (length)2, Fµν = ∂µAν − ∂νAµ is the field strength tensor, and +F = (1/4)FµνF µν = (B2 − E2)/2, where B and E are the induction mag- +netic and electric fields, correspondingly. Making use of the limit β → 0 in +Eq. (2), we arrive at the Maxwell’s Lagrangian LM = −F/(4π). The GB +Lagrangian has the structure +LGB = RµναβRµναβ − 4RµνRµν + R2. +(3) +By varying action (1) with respect to the metric we have EGB equations +Rµν − 1 +2gµνR + αHµν = −8πGTµν, +(4) +Hµν = 2 +� +RRµν − 2RµαRα +ν − 2RµανβRαβ − RµαβγRαβγ +ν +� +− 1 +2LGBgµν, +(5) +where Tµν is the stress (energy-momentum) tensor. To obtain the solution +of field equations we need to use an ansatz for the interval. But the va- +lidity of Birkhoff’s theorem [30] for our case of 4D EGB gravity coupled to +ModLogNED model is not proven. Therefore, to simplify the problem we +consider magnetic black holes with the static spherically symmetric metric +3 + +in D dimension. In addition, we assume that components of the interval are +restricted by the relation g11 = g−1 +00 . Thus, we suppose that the metric has +the form +ds2 = −f(r)dt2 + dr2 +f(r) + r2dΩ2 +D−2. +(6) +The dΩ2 +D−2 is the line element of the unit (D − 2)-dimensional sphere. By +following [13] we replace α by α → α/(D − 4) and taking the limit D → 4. +We study the magnetic black holes and find F = q2/(2r4), where q is a +magnetic charge. Then the magnetic energy density becomes [20] +ρ = T 0 +0 = −L = +√ +2F +8πβ ln +� +1 + β +√ +2F +� += +q +8πβr2 ln +� +1 + βq +r2 +� +. +(7) +At the limit D → 4 and from Eq. (4) we obtain +r(2αf(r) − r2 − 2α)f ′(r) − (r2 + αf(r) − 2α)f(r) + r2 − α = 2r4Gρ. +(8) +By virtue of Eq. (7 ) one finds +4π +� r +0 r2ρdr = mM + q +2β +� +r ln +� +1 + βq +r2 +� +− 2 +� +βq arctan +�√βq +r +�� +, +(9) +mM = 4π +� ∞ +0 +r2ρdr = q +2β +� ∞ +0 +ln +� +1 + βq +r2 +� +dr = πq3/2 +2√β , +(10) +where mM is the black hole magnetic mass. Making use of Eqs. (9) and (10) +we obtain the solution to Eq. (8) +f(r) = 1 + r2 +2α + +1 ± +� +1 + 8αG +r3 (m + h(r) + + , +h(r) = mM + q +2β +� +r ln +� +1 + βq +r2 +� +− 2 +� +βq arctan +�√βq +r +�� +, +(11) +where m is the constant of integration (the Schwarzschild mass) and the total +black hole mass is M = m+mM which is the ADM mass. At the limit β → 0 +one has +lim +β→0 h(r) = mM − q2/2r. +4 + +Then making use of Eq. (11), for the negative branch, we obtain +lim +β→0,α→0 f(r) = 1 − 2MG +r ++ Gq2 +r2 , +that corresponds to GR coupled to Maxwell electrodynamics (the Reissner– +Nordstr¨om solution). +It is worth mentioning that for spherically symmetric D-dimensional line +element (6), the Weyl tensor of the D-dimensional spatial part becomes zero +[17]. Therefore, solution (11) corresponds to the consistent theory [14, 15, 16]. +By introducing the dimensionless variable x = r/√βq, Eq. (11) is rewritten +in the form +f(x) = 1 + Cx2 ± C +� +x4 + x(A − Bg(x)), +(12) +where +A = 8αGM +(βq)3/2, B = 4αG +β2 , C = βq +2α, g(x) = 2 arctan +�1 +x +� +− x ln +� +1 + 1 +x2 +� +. +(13) +We will use the negative branch in Eqs. (11) and (12) with the minus sign +of the square root to have black holes without ghosts. As α → 0, r → ∞ the +metric function f(r) (11), for the negative branch, becomes +f(r) = 1 − 2MG +r ++ Gq2 +r2 + O(r−3), +(14) +showing, at infinity, the Reissner−Nordstr¨om behavior of the charged black +holes. The plot of function (12) for a particular chose of parameters, A = 15, +C = 1 (as an example), is depicted in Fig. +1. +The expansion (14) was +observed in other models (see, for example, [31]). According to Fig. 1 there +can be two horizons or one (the extreme) horizon of black holes. +3 +The black hole thermodynamics +To study the black hole thermal stability we will calculate the Hawking tem- +perature +TH(r+) = f ′(r) |r=r+ +4π +, +(15) +5 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +−0.5 +0 +0.5 +1 +1.5 +2 +2.5 +3 +x +f(x) + + +B=23 +B=27.5 +B=32 +Figure 1: The plot of the function f(x) for A = 15, C = 1. +where r+ is the event horizon radius (f(r+) = 0). From Eqs. (12) and (15) +one finds the Hawking temperature +TH(x+) = +1 +4π√βq +�2Cx2 ++ − 1 + BC2x2 ++g′(x+) +2x+(1 + Cx2 ++) +� +, +(16) +g′(x+) = − ln +� +1 + 1 +x2 ++ +� +. +Parameter A was substituted into Eq. (15) from equation f(x+) = 0. The +plot of the dimensionless function TH(x+)√βq versus x+, for the case C = 1, +is represented in Fig. 2. Figure 2 shows that the Hawking temperature is +positive for some interval of event horizon radii. We will calculate the heat +capacity to study the black hole local stability +Cq(x+) = TH +� ∂S +∂TH +� +q += ∂M(x+) +∂TH(x+) = ∂M(x+)/∂x+ +∂TH(x+)/∂x+ +, +(17) +6 + +1 +2 +3 +4 +5 +6 +7 +−0.05 +−0.04 +−0.03 +−0.02 +−0.01 +0 +0.01 +0.02 +0.03 +x+ +TH β1/2 q1/2 + + +B=2 +B=4 +B=6 +Figure 2: The plot of the function TH(x+)√βq at C = 1. +where M(x+) is the black hole gravitational mass as a function of the event +horizon radius. Making use of equation f(x+) = 0 we obtain the black hole +mass +M(x+) = (βq)3/2 +8αG +�1 + 2Cx2 ++ +C2x+ ++ Bg(x+) +� +. +(18) +With the help of Eqs. (16) and (18) one finds +∂M(x+) +∂x+ += (βq)3/2 +8αG +�2Cx2 ++ − 1 +C2x2+ ++ Bg′(x+) +� +, +(19) +∂TH(x+) +∂x+ += +1 +8π√βq +�5Cx2 ++ − 2C2x4 ++ + 1 +x2 ++(1 + Cx2 ++)2 ++BC2[g′(x+)(1 − Cx2 ++) + x+g′′(x+)(1 + Cx2 ++)] +(1 + Cx2+)2 +� +, +(20) +7 + +g′′(x+) = +2 +x+(x2+ + 1). +In accordance with Eq. (17) the heat capacity has a singularity when the +Hawking temperature possesses an extremum (∂TH(x+)/∂x+ = 0). Equa- +tions (16) and (17) show that at one point, x+ = x1, the Hawking temper- +ature and heat capacity become zero and the black hole remnant mass is +formed. In another point x+ = x2 with ∂TH(x+)/∂x+ = 0, the heat capac- +ity has a singularity where the second-order phase transition occurs. Black +holes in the range x2 > x+ > x1 are locally stable but at x+ > x2 black holes +are unstable. Making use of Eqs. (17), (19) and (20) the heat capacity is +depicted in Fig. 3 at C = 1. The Hawking temperature and heat capacity +1.5 +2 +2.5 +3 +3.5 +4 +−5000 +−4000 +−3000 +−2000 +−1000 +0 +1000 +2000 +3000 +x+ +Cqα G/(β2 q2) + + +B=2 +B=4 +B=6 +Figure 3: The plot of the function Cq(x+)αG/(β2q2) at C = 1. +are positive in the range x2 > x+ > x1 and locally stable. +From the first law of black hole thermodynamics dM(x+) = TH(x+)dS + +8 + +φdq we obtain the entropy at the constant charge [32] +S = +� dM(x+) +TH(x+) = +� +1 +TH(x+) +∂M(x+) +∂x+ +dx+. +(21) +From Eqs. (16), (19) and (21) one finds the entropy +S = π(βq)2 +C2αG +� 1 + Cx2 ++ +x+ +dx+ = πr2 ++ +G + 4πα +G ln +� r+ +√βq +� ++ Const, +(22) +with the integration constant Const. The integration constant can be chosen +in the form +Const = 2πα +G ln +�πqβ +G +� +. +(23) +Then making use of Eqs. (22) and (23) we obtain the black hole entropy +S = S0 + 2πα +G ln (S0) , +(24) +with S0 = πr2 ++/G being the Bekenstein–Hawking entropy and with the log- +arithmic correction but without the coupling β. One can find same entropy +(24) in other models [33, 34, 35]. +4 +Black holes shadows +The light gravitational lensing leads to the formation of black hole shadow +and a black circular disk. The Event Horizon Telescope collaboration [36] ob- +served the image of the super-massive black hole M87*. A neutral Schwarzschild +black hole shadow was studied in [37]. We will consider photons moving in the +equatorial plane, ϑ = π/2. With the help of the Hamilton−Jacobi method +one obtains the equation for the photon motion in null curves [38] +H = 1 +2gµνpµpν = 1 +2 +�L2 +r2 − E2 +f(r) + +˙r2 +f(r) +� += 0, +(25) +where pµ is the photon momentum ( ˙r = ∂H/∂pr). The photon energy and +angular momentum are constants of motion, and they are E = −pt and +L = pφ, correspondingly. We can represent Eq. (25) as +V + ˙r2 = 0, +V = f(r) +�L2 +r2 − E2 +f(r) +� +. +(26) +9 + +Photon circular orbit radius rp can be found from equation V (rp) = V ′(r)|r=rp = +0. Making use of Eq. (26) we find +ξ ≡ L +E = +rp +� +f(rp) +, +f ′(rp)rp − 2f(rp) = 0, +(27) +where ξ is the impact parameter. For a distant observer as r0 → ∞, the +shadow radius becomes rs = rp/ +� +f(rp) (rs = ξ). By virtue of Eq. (12) and +equation f(r+) = 0 we obtain parameters A, B and C versus x+ +A = 1 + 2Cx2 ++ +C2x+ ++ Bg(x+), +B = AC2x+ − 2Cx2 ++ − 1 +C2x+g(x+) +, +C = +x2 ++ + +� +x4+ + x+(A − Bg(x+)) +x+(A − Bg(x+)) +, +(28) +with x+ = r+/√βq. The functions (28) plots are depicted in Fig. 4. In +accordance with Fig. 4, Subplot 1, event horizon radius x+ increases when +parameter A increases and Subplot 2 indicates that if parameter B increases, +the event horizon radius decreases. According to Subplot 3 of Fig. 4, when +parameter C increases the event horizon radius x+ also increases. +The photon sphere radii (xp), the event horizon radii (x+), and the shadow +radii (xs) for A = 15 and C = 1 are presented in Table 1. It is worth noting +that the null geodesics radii xp correspond to the maximum of the potential +V (r) (V ′′ ≤ 0) and belong to unstable orbits. +Table 1 shows that when +Table 1: The event horizon, photon sphere and shadow dimensionless radii +for A=15, C=1 +B +9 +13.5 +14 +15 +16.5 +17.5 +18 +19 +x+ +6.763 +6.365 +6.317 +6.219 +6.063 +5.953 +5.896 +5.777 +xp +10.313 +9.806 +9.746 +9.623 +9.431 +9.298 +9.229 +9.088 +xs +18.311 +17.677 +17.603 +17.451 +17.216 +17.054 +16.971 +16.802 +parameter B increases the shadow radius xs decreases. As xs > x+ shadow +radii are defined by rs = xs +√βq. +10 + +0 +2 +4 +6 +8 +0 +10 +20 +30 +x+ +A +Subplot 1: B = 2, 4, 6; C = 1 + + +0 +1 +2 +3 +4 +0 +5 +10 +15 +x+ +B +Subplot 2: A = 8, 9, 10; C = 1 + + +0 +2 +4 +6 +8 +0.2 +0.4 +0.6 +0.8 +1 +x+ +C +Subplot 3: B = 2, 6, 8; A = 20 + + +B=2 +B=4 +B=6 +A=8 +A=9 +A=10 +B=2 +B=6 +B=8 +Figure 4: The plots of the functions A(x+), B(x+), C(x+) +. +It is worth mentioning that currently there is not unique calculation of +the shadow radius of M87* or SgrA* black holes within ModLogNED because +our model possesses four free parameters M, α, β and q (or M, A, B and C) +but from observations one knows only two values: the black hole mass and +the shadow radius. +5 +Black holes energy emission rate +The black hole shadow, for the observer at infinity, is connected with the +high energy absorption cross section [25, 39]. +At very high energies the +absorption cross-section σ ≈ πr2 +s oscillates around the photon sphere. The +11 + +energy emission rate of black holes is given by +d2E(ω) +dtdω += +2π3ω3r2 +s +exp (ω/TH(r+)) − 1, +(29) +where ω is the emission frequency. By using dimensionless variable x+ = +r+/√βq the black hole energy emission rate (29) becomes +� +βqd2E(ω) +dtdω += +2π3̟3x2 +s +exp +� +̟/ ¯TH(x+) +� +− 1 +, +(30) +with ¯TH(x+) = √βqTH(x+) and ̟ = √βqω. The radiation rate versus the +dimensionless emission frequency ¯ω for C = 1, A = 15 and B = 9, 14, 19, +is depicted in Fig. 5. Figure 5 shows that there is a peak of the black hole +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0 +0.005 +0.01 +0.015 +0.02 +0.025 +0.03 +0.035 + ϖ + + +B=9 +B=14 +B=19 +Figure 5: The plot of the function √βq d2E(ω) +dtdω +vs. ̟ for B = 9, 14, 19, A = 15, +C = 1. +energy emission rate. When parameter B increases, the energy emission rate +12 + +peak becomes smaller and corresponds to the lower frequency. The black +hole has a bigger lifetime when parameter B is bigger. +6 +Quasinormal modes +The stability of BHs under small perturbations are characterised by quasi- +normal modes (QNMs) with complex frequencies ω. When Im ω < 0 modes +are stable but if Im ω > 0 modes are unstable. Re ω, in the eikonal limit, is +linked with the black hole radius shadow [40, 41]. Around black holes, the +perturbations by scalar massless fields are described by the effective potential +barrier +V (r) = f(r) +�f ′(r) +r ++ l(l + 1) +r2 +� +, +(31) +with l being the multipole number l = 0, 1, 2.... Equation (31) can be rewrit- +ten in the form +V (x)βq = f(x) +�f ′(x) +x ++ l(l + 1) +x2 +� +. +(32) +Dimensionless variable V (x)βq is depicted in Fig. 6 for A = 15, B = 10, +C = 1 (Subplot 1) and for A = 15, C = 1, l = 5 (Subplot 2). According to +Figure 6, Subplot l, the potential barriers of effective potentials have maxima. +For l increasing the height of the potential increases. Figure 6, Subplot 2, +shows that when the parameter B increases the height of the potential also +increases. The quasinormal frequencies are given by [40, 41] +Re ω = l +rs += +l +� +f(rp) +rp +, +Im ω = −2n + 1 +2 +√ +2rs +� +2f(rp) − r2 +pf ′′(rp), +(33) +where rs is the black hole shadow radius, rp is the black hole photon sphere +radius, and n = 0, 1, 2, ... is the overtone number. The frequencies, at A = 15, +C = 1, n = 5, l = 10, are given in Table 2. Because the imaginary parts +of the frequencies in Table 2 are negative, modes are stable. The real part +Re ω gives the oscillations frequency. +In accordance with Table 2 when +parameter B increasing the real part of frequency √βqRe ω increases and +the absolute value of the frequency imaginary part | √βqIm ω | decreases. +Therefore, when the parameter B increases the scalar perturbations oscillate +with greater frequency and decay lower. +13 + +0 +10 +20 +30 +40 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +x +V(x)qβ +Subplot 1: l=3,5,7; A=15; B=10; C=1 + + +0 +10 +20 +30 +40 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +x +V(x)qβ +Subplot 2: B =9,14,19; A=15; l=5; C=1 + + +l=3 +l=5 +l=7 +B=9 +B=14 +B=19 +Figure 6: The plot of the function V (x)βq for A = 15, C = 1. +7 +Summary +The exact spherically symmetric solution of magnetic black holes is obtained +in 4D EGB gravity coupled to ModLogNED. We studied the thermodynamics +and the thermal stability of magnetically charged black holes. The Hawking +temperature and the heat capacity were calculated. The phase transitions +occur when the Hawking temperature has an extremum. +Black holes are +thermodynamically stable at some range of event horizon radii when the +heat capacity and the Hawking temperature are positive. The heat capacity +has a discontinuity where the second-order phase transitions take place. The +black hole entropy was calculated which has the logarithmic correction. We +calculated the photon sphere radii, the event horizon radii, and the shadow +radii. It was shown that when the model parameter B increases the black +14 + +Table 2: The real and the imaginary parts of the frequencies vs the parameter +B at n = 5, l = 10, A = 15, C = 1 +B +14 +15 +16.5 +17.5 +18 +19 +√βqRe ω +0.568 +0.573 +0.581 +0.586 +0.589 +0.595 +−√βqIm ω +0.2853 +0.2852 +0.2849 +0.2845 +0.2842 +0.2835 +hole energy emission rate decreases and the black hole possesses a bigger +lifetime. We show that when the parameter B increases the scalar pertur- +bations oscillate with greater frequency and decay lower. Other solutions in +4D EGB gravity coupled to NED were found in [33, 34, 35]. +Appendix A +With the spherical symmetry the energy-momentum tensor possesses the +property T t +t += T r +r . Then, the radial pressure is pr = −T r +r += −ρ. The +tangential pressure p⊥ = −T ϑ +ϑ = −T φ +φ +is given by [42] +p⊥ = −ρ − r +2ρ′(r), +(A1) +with the prime being the derivative with respect to the radius r. The Weak +Energy Condition (WEC) is valid when ρ ≥ 0 and ρ + pk ≥ 0 (k=1,2,3) [43], +and then the energy density is positive. According to Eq. (7) ρ ≥ 0. Making +use of Eq. (7) we obtain +ρ′(r) = − q +βr3 ln +� +1 + qβ +r2 +� +− +q2 +r3(r2 + βq) ≤ 0. +(A2) +Therefore WEC, ρ ≥ 0, ρ + pr ≥ 0, ρ + p⊥ ≥ 0, is satisfied. The Dominant +Energy Condition (DEC) takes place if and only if [43] ρ ≥ 0, ρ + pk ≥ 0, +ρ − pk ≥ 0, that includes WEC. One needs only to check the condition +ρ − p⊥ ≥ 0. By virtue of Eqs. (7), (A1) and A(2) one finds +ρ − p⊥ = +q +2βr2 +� +ln +� +1 + qβ +r2 +� +− +qβ +r2 + βq +� +. +(A3) +One can verify that ρ − p⊥ ≥ 0 for any parameters. DEC is satisfied and +therefore the sound speed is less than the speed of light. The Strong Energy +15 + +Condition (SEC) is valid when ρ + �3 +k=1 pk ≥ 0 [43]. From Eqs. (8)-(10) we +obtain +ρ + +3 +� +k=1 +pk = ρ + p⊥ + pr = p⊥ < 0. +(A4) +In accordance with Eq. (A4) SEC is not satisfied. +References +[1] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis. Modified Gravity +and Cosmology, Phys. Rept. 513, 1 (2012) [arXiv:1106.2476]. +[2] C. M. Will. The Confrontation between General Relativity and Experi- +ment. Living Rev. Rel. 17, 4 (2014). +[3] C. Lanczos. Elektromagnetismus als nat¨urliche eigenschaft der rie- +mannschen geometrie, Zeitschrift f¨ur Physik, 73,147 (1932). +[4] C. Lanczos. 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Press, Cambridge UK (1973). +19 + diff --git a/DdFKT4oBgHgl3EQfZC4_/content/tmp_files/load_file.txt b/DdFKT4oBgHgl3EQfZC4_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..55fe4c2b2ca69f8d68c6c18b81f86a8f159b435f --- /dev/null +++ b/DdFKT4oBgHgl3EQfZC4_/content/tmp_files/load_file.txt @@ -0,0 +1,593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf,len=592 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='11801v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='gen-ph] 12 Jan 2023 4D Einstein–Gauss–Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics Sergey Il’ich Kruglov 1 Department of Physics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Georges St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=', Toronto, ON M5S 1A7, Canada Department of Chemical and Physical Sciences, University of Toronto, 3359 Mississauga Road North, Mississauga, Ontario L5L 1C6, Canada Abstract Spherically symmetric solution in 4D Einstein–Gauss–Bonnet grav- ity coupled to modified logarithmic nonlinear electrodynamics (Mod- LogNED) is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' This solution at infinity possesses the charged black hole Reissner–Nordstr¨om behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We study the black hole thermodynamics, entropy, shadow, energy emission rate and quasi- normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It was shown that black holes can possess the phase transitions and at some range of event horizon radii black holes are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The entropy has the logarithmic correction to the area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The shadow radii were calculated for variety of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We found that there is a peak of the black hole energy emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The real and imaginary parts of the quasinormal modes frequencies were cal- culated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The energy conditions of ModLogNED are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Keywords: Einstein−Gauss−Bonnet gravity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' nonlinear electrodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Hawking temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' entropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' heat capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' black hole shadow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' energy emission rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' quasinormal modes 1 Introduction Nowadays, there are many theories of gravity that are alternatives to Ein- stein’s theory [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The motivation of generalisations of Einstein’s theory of General Relativity (GR) is to resolve some problems in cosmology and astro- physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' One of important modification of GR is the Einstein–Gauss–Bonnet 1E-mail: serguei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='krouglov@utoronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='ca 1 (EGB) theory [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' EGB theories do not include extra degrees of free- dom and field equations have second derivatives of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' These theories also prevent Ostrogradsky instability [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The four dimensional (4D) EGB theory, that includes the Einstein–Hilbert action plus GB term, is a particu- lar case of the Lovelock theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It represents the generalization of Einstein’s GR for higher dimensions and EGB theory results covariant second-order field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The GB part of the action possesses higher order curvature terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth mentioning that at low energy the action of the het- erotic string theory includes higher order curvature terms [8, 9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Therefore, it is of interest to study gravity action with the GB term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The GB term is a topological invariant in 4D and before a regularization it does not contribute to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' But Glavan and Lin [13] showed that re-scaling the coupling constant, after the regularization, GB term con- tributes to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The consistent theory of 4D EGB gravity, was proposed in [14, 15, 16], is in agreement with the Lovelock theorem [5] and possesses two dynamical degrees of freedom breaking the temporal dif- feomorphism invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth noting that the theory of [14, 15, 16], in the spherically-symmetric metrics, gives the solution which is a solution in the framework of [13] scheme (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Some aspects of 4D EGB gravity were considered in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole and wormhole type solutions in the effective gravity models, including higher curvature terms, were obtained in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Here, we study the black hole thermodynamics, the entropy, the shadow, the energy emission rate and quasinormal modes in the framework of the ModLogNED model (proposed in [20]) coupled to 4D EGB gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth noting that ModLogNED model is simpler compared with logarithmic model [21] and generalized logarithmic model [22] because the mass and met- ric functions here are expressed through simple elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole quasinormal modes, deflection angles, shadows and the Hawking radiation were studied in [23, 24, 25, 26, 27, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 2, we obtain the spher- ically symmetric solution of black holes in the 4D EGB gravity coupled to ModLogNED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' At infinity the Reissner−Nordstr¨om behavior of the charged black holes takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole thermodynamics is studied in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We calculate the Hawking temperature, the heat capacity and the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' At some parameters second order phase transitions occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The entropy in- cludes the logarithmic correction to Bekenstein–Hawking entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4 the black hole shadow is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We calculate the photon sphere, the 2 event horizon, and the shadow radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole energy emission rate is investigate in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 6 we study quasinormal modes and find com- plex frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Section 7 is a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In Appendix A energy conditions of ModLogNED model are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 2 4D EGB model The action of EGB gravity coupled to nonlinear electrodynamics (NED) in D-dimensions is given by I = � dDx√−g � 1 16πG (R + αLGB) + LNED � , (1) where G is the Newton’s constant, α has the dimension of (length)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Lagrangian of ModLogNED, proposed in [20], is LNED = − √ 2F 8πβ ln � 1 + β √ 2F � , (2) where we use Gaussian units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The parameter β (β ≥ 0) possesses the di- mension of (length)2, Fµν = ∂µAν − ∂νAµ is the field strength tensor, and F = (1/4)FµνF µν = (B2 − E2)/2, where B and E are the induction mag- netic and electric fields, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of the limit β → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (2), we arrive at the Maxwell’s Lagrangian LM = −F/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The GB Lagrangian has the structure LGB = RµναβRµναβ − 4RµνRµν + R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (3) By varying action (1) with respect to the metric we have EGB equations Rµν − 1 2gµνR + αHµν = −8πGTµν, (4) Hµν = 2 � RRµν − 2RµαRα ν − 2RµανβRαβ − RµαβγRαβγ ν � − 1 2LGBgµν, (5) where Tµν is the stress (energy-momentum) tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' To obtain the solution of field equations we need to use an ansatz for the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' But the va- lidity of Birkhoff’s theorem [30] for our case of 4D EGB gravity coupled to ModLogNED model is not proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Therefore, to simplify the problem we consider magnetic black holes with the static spherically symmetric metric 3 in D dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In addition, we assume that components of the interval are restricted by the relation g11 = g−1 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Thus, we suppose that the metric has the form ds2 = −f(r)dt2 + dr2 f(r) + r2dΩ2 D−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (6) The dΩ2 D−2 is the line element of the unit (D − 2)-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' By following [13] we replace α by α → α/(D − 4) and taking the limit D → 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We study the magnetic black holes and find F = q2/(2r4), where q is a magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Then the magnetic energy density becomes [20] ρ = T 0 0 = −L = √ 2F 8πβ ln � 1 + β √ 2F � = q 8πβr2 ln � 1 + βq r2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (7) At the limit D → 4 and from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (4) we obtain r(2αf(r) − r2 − 2α)f ′(r) − (r2 + αf(r) − 2α)f(r) + r2 − α = 2r4Gρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (8) By virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (7 ) one finds 4π � r 0 r2ρdr = mM + q 2β � r ln � 1 + βq r2 � − 2 � βq arctan �√βq r �� , (9) mM = 4π � ∞ 0 r2ρdr = q 2β � ∞ 0 ln � 1 + βq r2 � dr = πq3/2 2√β , (10) where mM is the black hole magnetic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (9) and (10) we obtain the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (8) f(r) = 1 + r2 2α \uf8eb \uf8ed1 ± � 1 + 8αG r3 (m + h(r) \uf8f6 \uf8f8 , h(r) = mM + q 2β � r ln � 1 + βq r2 � − 2 � βq arctan �√βq r �� , (11) where m is the constant of integration (the Schwarzschild mass) and the total black hole mass is M = m+mM which is the ADM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' At the limit β → 0 one has lim β→0 h(r) = mM − q2/2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4 Then making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (11), for the negative branch, we obtain lim β→0,α→0 f(r) = 1 − 2MG r + Gq2 r2 , that corresponds to GR coupled to Maxwell electrodynamics (the Reissner– Nordstr¨om solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth mentioning that for spherically symmetric D-dimensional line element (6), the Weyl tensor of the D-dimensional spatial part becomes zero [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Therefore, solution (11) corresponds to the consistent theory [14, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' By introducing the dimensionless variable x = r/√βq, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (11) is rewritten in the form f(x) = 1 + Cx2 ± C � x4 + x(A − Bg(x)), (12) where A = 8αGM (βq)3/2, B = 4αG β2 , C = βq 2α, g(x) = 2 arctan �1 x � − x ln � 1 + 1 x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (13) We will use the negative branch in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (11) and (12) with the minus sign of the square root to have black holes without ghosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' As α → 0, r → ∞ the metric function f(r) (11), for the negative branch, becomes f(r) = 1 − 2MG r + Gq2 r2 + O(r−3), (14) showing, at infinity, the Reissner−Nordstr¨om behavior of the charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The plot of function (12) for a particular chose of parameters, A = 15, C = 1 (as an example), is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The expansion (14) was observed in other models (see, for example, [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 1 there can be two horizons or one (the extreme) horizon of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 3 The black hole thermodynamics To study the black hole thermal stability we will calculate the Hawking tem- perature TH(r+) = f ′(r) |r=r+ 4π , (15) 5 1 2 3 4 5 6 7 8 9 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 3 x f(x) B=23 B=27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 B=32 Figure 1: The plot of the function f(x) for A = 15, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' where r+ is the event horizon radius (f(r+) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (12) and (15) one finds the Hawking temperature TH(x+) = 1 4π√βq �2Cx2 + − 1 + BC2x2 +g′(x+) 2x+(1 + Cx2 +) � , (16) g′(x+) = − ln � 1 + 1 x2 + � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Parameter A was substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (15) from equation f(x+) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The plot of the dimensionless function TH(x+)√βq versus x+, for the case C = 1, is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Figure 2 shows that the Hawking temperature is positive for some interval of event horizon radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We will calculate the heat capacity to study the black hole local stability Cq(x+) = TH � ∂S ∂TH � q = ∂M(x+) ∂TH(x+) = ∂M(x+)/∂x+ ∂TH(x+)/∂x+ , (17) 6 1 2 3 4 5 6 7 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='03 x+ TH β1/2 q1/2 B=2 B=4 B=6 Figure 2: The plot of the function TH(x+)√βq at C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' where M(x+) is the black hole gravitational mass as a function of the event horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of equation f(x+) = 0 we obtain the black hole mass M(x+) = (βq)3/2 8αG �1 + 2Cx2 + C2x+ + Bg(x+) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (18) With the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (16) and (18) one finds ∂M(x+) ∂x+ = (βq)3/2 8αG �2Cx2 + − 1 C2x2+ + Bg′(x+) � , (19) ∂TH(x+) ∂x+ = 1 8π√βq �5Cx2 + − 2C2x4 + + 1 x2 +(1 + Cx2 +)2 +BC2[g′(x+)(1 − Cx2 +) + x+g′′(x+)(1 + Cx2 +)] (1 + Cx2+)2 � , (20) 7 g′′(x+) = 2 x+(x2+ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (17) the heat capacity has a singularity when the Hawking temperature possesses an extremum (∂TH(x+)/∂x+ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Equa- tions (16) and (17) show that at one point, x+ = x1, the Hawking temper- ature and heat capacity become zero and the black hole remnant mass is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In another point x+ = x2 with ∂TH(x+)/∂x+ = 0, the heat capac- ity has a singularity where the second-order phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Black holes in the range x2 > x+ > x1 are locally stable but at x+ > x2 black holes are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (17), (19) and (20) the heat capacity is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 3 at C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Hawking temperature and heat capacity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 4 −5000 −4000 −3000 −2000 −1000 0 1000 2000 3000 x+ Cqα G/(β2 q2) B=2 B=4 B=6 Figure 3: The plot of the function Cq(x+)αG/(β2q2) at C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' are positive in the range x2 > x+ > x1 and locally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' From the first law of black hole thermodynamics dM(x+) = TH(x+)dS + 8 φdq we obtain the entropy at the constant charge [32] S = � dM(x+) TH(x+) = � 1 TH(x+) ∂M(x+) ∂x+ dx+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (21) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (16), (19) and (21) one finds the entropy S = π(βq)2 C2αG � 1 + Cx2 + x+ dx+ = πr2 + G + 4πα G ln � r+ √βq � + Const, (22) with the integration constant Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The integration constant can be chosen in the form Const = 2πα G ln �πqβ G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (23) Then making use of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (22) and (23) we obtain the black hole entropy S = S0 + 2πα G ln (S0) , (24) with S0 = πr2 +/G being the Bekenstein–Hawking entropy and with the log- arithmic correction but without the coupling β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' One can find same entropy (24) in other models [33, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4 Black holes shadows The light gravitational lensing leads to the formation of black hole shadow and a black circular disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Event Horizon Telescope collaboration [36] ob- served the image of the super-massive black hole M87*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' A neutral Schwarzschild black hole shadow was studied in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We will consider photons moving in the equatorial plane, ϑ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' With the help of the Hamilton−Jacobi method one obtains the equation for the photon motion in null curves [38] H = 1 2gµνpµpν = 1 2 �L2 r2 − E2 f(r) + ˙r2 f(r) � = 0, (25) where pµ is the photon momentum ( ˙r = ∂H/∂pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The photon energy and angular momentum are constants of motion, and they are E = −pt and L = pφ, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We can represent Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (25) as V + ˙r2 = 0, V = f(r) �L2 r2 − E2 f(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (26) 9 Photon circular orbit radius rp can be found from equation V (rp) = V ′(r)|r=rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (26) we find ξ ≡ L E = rp � f(rp) , f ′(rp)rp − 2f(rp) = 0, (27) where ξ is the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' For a distant observer as r0 → ∞, the shadow radius becomes rs = rp/ � f(rp) (rs = ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' By virtue of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (12) and equation f(r+) = 0 we obtain parameters A, B and C versus x+ A = 1 + 2Cx2 + C2x+ + Bg(x+), B = AC2x+ − 2Cx2 + − 1 C2x+g(x+) , C = x2 + + � x4+ + x+(A − Bg(x+)) x+(A − Bg(x+)) , (28) with x+ = r+/√βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The functions (28) plots are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In accordance with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4, Subplot 1, event horizon radius x+ increases when parameter A increases and Subplot 2 indicates that if parameter B increases, the event horizon radius decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' According to Subplot 3 of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 4, when parameter C increases the event horizon radius x+ also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The photon sphere radii (xp), the event horizon radii (x+), and the shadow radii (xs) for A = 15 and C = 1 are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth noting that the null geodesics radii xp correspond to the maximum of the potential V (r) (V ′′ ≤ 0) and belong to unstable orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Table 1 shows that when Table 1: The event horizon, photon sphere and shadow dimensionless radii for A=15, C=1 B 9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 14 15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 18 19 x+ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='763 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='365 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='317 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='219 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='063 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='953 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='896 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='777 xp 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='313 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='806 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='746 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='623 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='431 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='298 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='229 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='088 xs 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='311 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='677 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='603 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='451 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='216 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='054 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='971 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='802 parameter B increases the shadow radius xs decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' As xs > x+ shadow radii are defined by rs = xs √βq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 10 0 2 4 6 8 0 10 20 30 x+ A Subplot 1: B = 2, 4, 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' C = 1 0 1 2 3 4 0 5 10 15 x+ B Subplot 2: A = 8, 9, 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' C = 1 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='8 1 x+ C Subplot 3: B = 2, 6, 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' A = 20 B=2 B=4 B=6 A=8 A=9 A=10 B=2 B=6 B=8 Figure 4: The plots of the functions A(x+), B(x+), C(x+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It is worth mentioning that currently there is not unique calculation of the shadow radius of M87* or SgrA* black holes within ModLogNED because our model possesses four free parameters M, α, β and q (or M, A, B and C) but from observations one knows only two values: the black hole mass and the shadow radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 5 Black holes energy emission rate The black hole shadow, for the observer at infinity, is connected with the high energy absorption cross section [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' At very high energies the absorption cross-section σ ≈ πr2 s oscillates around the photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The 11 energy emission rate of black holes is given by d2E(ω) dtdω = 2π3ω3r2 s exp (ω/TH(r+)) − 1, (29) where ω is the emission frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' By using dimensionless variable x+ = r+/√βq the black hole energy emission rate (29) becomes � βqd2E(ω) dtdω = 2π3̟3x2 s exp � ̟/ ¯TH(x+) � − 1 , (30) with ¯TH(x+) = √βqTH(x+) and ̟ = √βqω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The radiation rate versus the dimensionless emission frequency ¯ω for C = 1, A = 15 and B = 9, 14, 19, is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Figure 5 shows that there is a peak of the black hole 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='035 ϖ B=9 B=14 B=19 Figure 5: The plot of the function √βq d2E(ω) dtdω vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' ̟ for B = 9, 14, 19, A = 15, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' energy emission rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' When parameter B increases, the energy emission rate 12 peak becomes smaller and corresponds to the lower frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole has a bigger lifetime when parameter B is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 6 Quasinormal modes The stability of BHs under small perturbations are characterised by quasi- normal modes (QNMs) with complex frequencies ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' When Im ω < 0 modes are stable but if Im ω > 0 modes are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Re ω, in the eikonal limit, is linked with the black hole radius shadow [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Around black holes, the perturbations by scalar massless fields are described by the effective potential barrier V (r) = f(r) �f ′(r) r + l(l + 1) r2 � , (31) with l being the multipole number l = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='. Equation (31) can be rewrit- ten in the form V (x)βq = f(x) �f ′(x) x + l(l + 1) x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (32) Dimensionless variable V (x)βq is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 6 for A = 15, B = 10, C = 1 (Subplot 1) and for A = 15, C = 1, l = 5 (Subplot 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' According to Figure 6, Subplot l, the potential barriers of effective potentials have maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' For l increasing the height of the potential increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Figure 6, Subplot 2, shows that when the parameter B increases the height of the potential also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The quasinormal frequencies are given by [40, 41] Re ω = l rs = l � f(rp) rp , Im ω = −2n + 1 2 √ 2rs � 2f(rp) − r2 pf ′′(rp), (33) where rs is the black hole shadow radius, rp is the black hole photon sphere radius, and n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' is the overtone number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The frequencies, at A = 15, C = 1, n = 5, l = 10, are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Because the imaginary parts of the frequencies in Table 2 are negative, modes are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The real part Re ω gives the oscillations frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' In accordance with Table 2 when parameter B increasing the real part of frequency √βqRe ω increases and the absolute value of the frequency imaginary part | √βqIm ω | decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Therefore, when the parameter B increases the scalar perturbations oscillate with greater frequency and decay lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 13 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='18 x V(x)qβ Subplot 1: l=3,5,7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' A=15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' B=10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' C=1 0 10 20 30 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='12 x V(x)qβ Subplot 2: B =9,14,19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' A=15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' l=5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' C=1 l=3 l=5 l=7 B=9 B=14 B=19 Figure 6: The plot of the function V (x)βq for A = 15, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 7 Summary The exact spherically symmetric solution of magnetic black holes is obtained in 4D EGB gravity coupled to ModLogNED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We studied the thermodynamics and the thermal stability of magnetically charged black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Hawking temperature and the heat capacity were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The phase transitions occur when the Hawking temperature has an extremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Black holes are thermodynamically stable at some range of event horizon radii when the heat capacity and the Hawking temperature are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The heat capacity has a discontinuity where the second-order phase transitions take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The black hole entropy was calculated which has the logarithmic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We calculated the photon sphere radii, the event horizon radii, and the shadow radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' It was shown that when the model parameter B increases the black 14 Table 2: The real and the imaginary parts of the frequencies vs the parameter B at n = 5, l = 10, A = 15, C = 1 B 14 15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='5 18 19 √βqRe ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='595 −√βqIm ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2835 hole energy emission rate decreases and the black hole possesses a bigger lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' We show that when the parameter B increases the scalar pertur- bations oscillate with greater frequency and decay lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Other solutions in 4D EGB gravity coupled to NED were found in [33, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Appendix A With the spherical symmetry the energy-momentum tensor possesses the property T t t = T r r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Then, the radial pressure is pr = −T r r = −ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The tangential pressure p⊥ = −T ϑ ϑ = −T φ φ is given by [42] p⊥ = −ρ − r 2ρ′(r), (A1) with the prime being the derivative with respect to the radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Weak Energy Condition (WEC) is valid when ρ ≥ 0 and ρ + pk ≥ 0 (k=1,2,3) [43], and then the energy density is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (7) ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Making use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (7) we obtain ρ′(r) = − q βr3 ln � 1 + qβ r2 � − q2 r3(r2 + βq) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (A2) Therefore WEC, ρ ≥ 0, ρ + pr ≥ 0, ρ + p⊥ ≥ 0, is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Dominant Energy Condition (DEC) takes place if and only if [43] ρ ≥ 0, ρ + pk ≥ 0, ρ − pk ≥ 0, that includes WEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' One needs only to check the condition ρ − p⊥ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' By virtue of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (7), (A1) and A(2) one finds ρ − p⊥ = q 2βr2 � ln � 1 + qβ r2 � − qβ r2 + βq � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (A3) One can verify that ρ − p⊥ ≥ 0 for any parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' DEC is satisfied and therefore the sound speed is less than the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Strong Energy 15 Condition (SEC) is valid when ρ + �3 k=1 pk ≥ 0 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (8)-(10) we obtain ρ + 3 � k=1 pk = ρ + p⊥ + pr = p⊥ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (A4) In accordance with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' (A4) SEC is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' References [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Clifton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Ferreira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Padilla, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Skordis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Modified Gravity and Cosmology, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' 513, 1 (2012) [arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content='2476].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' The Confrontation between General Relativity and Experi- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} 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property of the riemann-christoffel tensor in four dimensions, Annals of Mathematics, 842–850 (1938).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Lovelock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Divergence-free tensorial concomitants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' Aequationes math- ematicae, 4, 127 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdFKT4oBgHgl3EQfZC4_/content/2301.11801v1.pdf'} 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b/DtAzT4oBgHgl3EQfif1c/content/tmp_files/2301.01500v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d69c08b7d9dd70128e1e07ced8faea1c3845ab4 --- /dev/null +++ b/DtAzT4oBgHgl3EQfif1c/content/tmp_files/2301.01500v1.pdf.txt @@ -0,0 +1,611 @@ +arXiv:2301.01500v1 [nucl-th] 4 Jan 2023 +The Langevin approach for fission of heavy and +super-heavy nuclei∗ +F.A.Ivanyuk, S.V.Radionov +Institute for Nuclear Research, Kyiv, Ukraine +C.Ishizuka, S.Chiba +Tokyo Institute of Technology, Tokyo, Japan +In this contribution, we present the main relations of the Langevin +approach to the description of fission or fusion-fission reactions. The results +of Langevin calculations are shown for the mass distributions of fission +fragments of super-heavy elements and used for the investigation of memory +effects in nuclear fission. +1. Introduction +We describe the nuclear fission process by the four-dimensional set of +the Langevin equations for the shape degrees of freedom with the shape +given by the two-center shell model (TCSM) shape parametrization. The +potential energy is calculated within the macroscopic-microscopic method. +The collective mass, M, and friction, γ, tensors are defined in macroscopic +(Werner-Wheller and wall-and-window formula) or microscopic (linear re- +sponse theory) approaches. +We start calculations from the ground state shape with zero collective +velocities and solve equations until the neck radius of the nucleus turns zero +(scission point). At the scission point, the solutions of Langevin equations +supply complete information about the system, its shape, excitation energy, +and collective velocities. +This information makes it possible to calculate +the mass distributions, the total kinetic energy, and the excitation energies +of fission fragments. The results of numerous previous calculations are in +reasonable agreement with the available experimental data. +∗ Presented at the Zakopane Conference on Nuclear Physics, Zakopane, Poland, 28 +August - 4 September 2022 +(1) + +2 +preprint +printed on January 5, 2023 +Below in this contribution, we present the calculated results for the mass +distributions of super-heavy nuclei and clarify the impact of memory effects +on the fission width of heavy nuclei. +The physics of super-heavy elements (SHE) has a long history. The ex- +istence of the “island of stability” was predicted at the end of the 1960s +[1]. Nevertheless, it took almost 30 years until the alpha-decay of the ele- +ment with Z=114 was observed experimentally at Flerov Nuclear Reactions +Laboratory in Dubna [2]. +With the development of experimental facility, it became possible not +only to fix the fact of formation of SHE, but examine their properties. +One of the first property of interest – the process of fission of SHEs. For +the successful planning and carrying out of experiments, it is crucial to +understand what kind of fission fragments mass distribution (FFMD) one +should expect in the result of the fission of SHEs. The two double magic +nuclei 132Sn and 208Pb may contribute. Both have the shell correction in +the ground state of the same magnitude. +In order to clarify what kind of FFMD one could expect in the fission of +SHEs, we have carried out the calculations of FFMD for a number of SHEs. +The results are given in Section 3. +Another problem we address in this contribution is the influence of mem- +ory effects on the probability of the fission process. Commonly one uses the +Markovian approximation to Langevin approach in which all quantities are +defined at the same moment. This approximation provides reasonable re- +sults, but its accuracy is not well established. In publications, one can find +statements that the memory effects have a significant influence on the fusion +or fission processes and the statements that memory effects are very small. +To clarify this uncertainty, we have calculated the fission width using +the Langevin approach with memory effects included in a wide range of im- +portant parameters: the excitation energy E∗ of the system, the damping +parameter η, the relaxation time τ. The details and results of the calcula- +tions are given in Section 4. +2. The Langevin approach for the fission process +Within the Langevin approach, the fission process is described by solving +the equations for the time evolution of the shape of nuclear surface of the fis- +sioning system. For the shape parametrization, we use that of the two-center +shell model (TCSM) [3] with 4 deformation parameters qµ = z0/R0, δ1, δ2, α. +Here z0/R0 refers to the distance between the centers of left and right os- +cillator potentials, R0 being the radius of spherical nucleus with the mass +number A. The parameters δi describe the deformation of the right and left +fragment tips. The fourth parameter α is the mass asymmetry and the fifth + +preprint +printed on January 5, 2023 +3 +parameter of the TCSM shape parametrization ǫ was kept constant, ǫ=0.35, +in all our calculations. +The first-order differential equations (Langevin equations) for the time +dependence of collective variables qµ and the conjugated momenta pµ are: +dqµ +dt += +� +m−1� +µν pν, +(1) +dpµ +dt += −∂F(q, T) +∂qµ +− 1 +2 +∂m−1 +νσ +∂qµ +pνpσ − γµνm−1 +νσ pσ + Rµ(t). +In Eqs. (1) the F(q, T) is the temperature-dependent free energy of the +system, and γµν and (m−1)µν are the friction and inverse of mass tensors. +The free energy F(q, T) is calculated within the shell correction method. +The single particle energies are calculated with the deformed Woods-Saxon +potential fitted to the mentioned above TCSM shapes. +The collective inertia tensor mµν is calculated by the Werner-Wheeler +approximation and for the friction tensor γµν we used the wall-and-window +formula. The random force Rµ(t) is the product of the temperature-depen- +dent strength factors gµν and the white noise ξν(t), Rµ(t) = gµνξν(t). The +factors gµν are related to the temperature and friction tensor via the Einstein +relation, +gµσgσν = Tγµν +(2) +The temperature T is kept constant, aT 2 = E∗, or adjusted to the local +excitation energy on each step of integration by the relation, +aT 2 = E∗ − p2(t)/2M − [Epot(q) − Epot(qgs)]. +(3) +Here qgs is the ground state deformation. More details are given in our +earlier publications [4, 5, 6, 7]. +Initially, the momenta pµ are set to zero, and calculations are started +from the ground state deformation. Such calculations are continued until the +trajectories reach the ”scission point”, defined as the point in deformation +space where the neck radius turns zero. +3. Fission fragments mass distributions of super-heavy nuclei +In order to understand what kind of mass distributions one can expect +from the solution of Langevin equations for super-heavy nuclei, we looked +first at the potential energy of fissioning nuclei. Fig. 1 shows the potential +energy Edef of nuclei 296Lv and 302120 at zero temperature as a function +of elongation (the distance R12 between the centers of mass of left and +right parts of a nucleus) and the mass asymmetry (fragment mass number). + +4 +preprint +printed on January 5, 2023 +In the top part of Fig. 1 the energy was minimized with respect to the +deformation parameters δ1 and δ2. One sees the bottom of potential energy +leading to almost symmetric mass splitting. There is also a hint on the mass +asymmetric valley at AF close to AF =208. +1.0 +1.5 +2.0 +100 +150 +200 +302120, δ1= - 0.2, δ2= 0.2 +R12 / R0 +Fragment mass number +-60 +-52 +-44 +-36 +-28 +-20 +-12 +-4.0 +4.0 +10 +1.0 +1.5 +2.0 +100 +150 +200 +302120, δ1,δ2 - min. +Fragment mass number +-60 +-52 +-44 +-36 +-28 +-20 +-12 +-4.0 +4.0 +10 +1.0 +1.5 +2.0 +50 +100 +150 +200 +296Lv, δ1= - 0.2, δ2= 0.2 +R12 / R0 +Fragment mass number +-60 +-52 +-44 +-36 +-28 +-20 +-12 +-4.0 +4.0 +10 +1.0 +1.5 +2.0 +50 +100 +150 +200 +Fragment mass number +-60 +-52 +-44 +-36 +-28 +-20 +-12 +-4.0 +4.0 +10 +296Lv, δ1,δ2 - min. +Fig. 1. (top) The potential energy of 296Lv and 302120 at T = 0 minimized with +respect to deformation parameters δ1 and δ2 (bottom), and at fixed values δ1 = +−0.2 and δ2 = 0.2. +If the trajectories followed the bottom of potential energy, the mass +distributions would be symmetric. However, it is well known that the tra- +jectories may deviate substantially from the bottom of the potential valley +due to dynamic effects. We calculate the trajectories in four-dimensional +deformation space. In this space, the local minima could lead away from +the bottom of the potential valley. An example is shown in the bottom part +of Fig. 1. Here we show the potential energy for fixed δ1= - 0.2 and δ2=0.2. +One clearly sees another valley, leading to strongly mass asymmetric split- +ting. +In Fig. 2, we show the fission fragment mass distributions of super-heavy +nuclei from 276Hs to 308122 as a function of fragment mass number AF . The +FFMDs of nuclei from 276Cn to 308122 have three or four peak structures. +The main component is the symmetric peak, split into two components in +some isotopes. The peaks of lighter fragments are located around AF =140. + +preprint +printed on January 5, 2023 +5 +5 +10 + + Fission from the ground state, ---- E +*=10 MeV, ---- E +*=20 MeV, ---- E +*=30 MeV +N = 168 170 172 174 176 178 180 182 184 186 +Z = 108 110 112 114 116 118 120 122 +5 +10 + +5 +10 + +5 +10 + +0 +5 +10 +5 +10 +15 + +140 +5 +10 +15 + +5 +10 +15 +20 + +40 140 +05 +10 +15 +20 + +40 140 +F r a g m e n t m a s s n u m b e r +Yield (%) +40 14040 14040 14040 14040 14040 14040 14040 140 + +Fig. 2. The fission fragment mass distributions of super-heavy nuclei from 276Hs to +308122 calculated for the excitation energies E∗=10, 20 and 30 MeV as a function +of the fragment mass number +One can also see the strongly asymmetric peak at the mass number +close to AF =208. The strength of the (almost) symmetric and asymmetric +components in FFMD of SHEs depends on the proton and neutron num- +bers of the compound nucleus. For 276Cn, the contribution of a strongly +asymmetric peak is tiny. This contribution becomes larger for more heavy +SHE. In some elements of SHEs with Z =116-122, the symmetric and mass- +asymmetric peaks are of the same magnitude. More details can be found in +[8]. +The similar strongly mass-asymmetric peaks in FFMD of SHEs were +also found recently in [9] within the Langevin approach with the so call +Fourier shape parametrization. + +6 +preprint +printed on January 5, 2023 +4. The memory effects in nuclear fission +In order to investigate the role of memory effects in nuclear fission, we +exploit a simple one-dimensional model with the potential energy given by +the two-parabolic potential (Kramers potential), see Fig. 3. +Epot(q) = 2Vbq(q − q0)/q2 +0, 0 < q < q0; 2Vb(q − q0)(2q0 − q)q2 +0, q0 < q < 2q0. +(4) +The potential (4) depends on two parameters, the barrier height Vb and the +barrier width q0. We have fixed the barrier height Vb = 6 MeV, which is +close to the value of the fission barrier of actinide nuclei. The width of the +barrier is somewhat uncertain. It depends on the definition of the collective +coordinate q and the model for the potential energy. For simplicity, we have +put here q0 = 1.0. +For the potential (4) one can define the stiffness C = d2Epot/dq2 and +the frequency of harmonic vibrations ω0 = +� +C/M. In the present work, +we fix ¯hω0 =1.0 MeV, which is close to the frequency of collective vibra- +tions calculated for 224Th in [10] within the microscopic linear response +theory. Then, for the mass parameter we will have the deformation and +temperature-independent value, +M = C/ω2 +0 = 4Vb/(ω2 +0q2 +0). +(5) +For the friction coefficient ¯γ we use a slightly modified approximation of +[10], +¯γ/M = 0.6(T 2 + ¯h2ω2 +0/π2))/(1 + T 2/40). +(6) +For the temperature, we consider here two options: constant temperature +regime and constant energy regime. In a constant temperature regime, the +temperature is time-independent, related to the initial excitation energy E∗ +by the Fermi-gas relation, aT 2 = E∗, where a is the level density parameter +of T¨oke and Swiatecki [11]. +The fission width calculated in a constant +temperature regime will be denoted as Γf(T). +At small excitations, the temperature varies with deformation and time, +and there is no reason to consider it constant. So, it should be adjusted +to the local excitation energy on each integration step by the relation (3). +Correspondingly, fission width calculated in a constant energy regime is +denoted as Γf(E). +The fission width, Γf, is defined assuming the exponential decay of the +number of ”particles” in the potential well, +P(t) = e−Γf t/¯h → Γf = −¯h ln[P(t)]/t. +(7) +By solving the Langevin equations one will get the set of time moments tb, +at which some trajectories would cross the barrier. From this information, +one can find the probability P(t) and the fission width Γf, see [12]. + +preprint +printed on January 5, 2023 +7 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +5 +10 +Epot (MeV) +q + A=236, E +*=Vb +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1000 +2000 +3000 + + +Γf (eV) +η + Γf(T) + ΓLV + ΓHV +T=1.5 MeV +Fig. 3. (left) The two-parabolic potential (4) and few examples of the dynamical +trajectories. (right) The fission width as the solution of Eqs. (1, 4, 7) calculated at +constant temperature (open dots), and the Kramers approximations (8) for high +and low damping limits. +The Markovian fission width Γf(T) calculated by Eqs. (1, 4, 7) is plotted +as function of the damping parameter η in the right part of Fig. 3. +To +present the results in a broader range of parameters, the damping parameter +η ≡ ¯γ/2Mω0 in these calculations was considered as a free parameter. +For the comparison, in Fig.3 we also show the Kramers decay width +ΓHV , ΓLV in limits of high and low viscosity (friction) [13], +ΓHV = ¯hω0 +2π e−Vb/T ( +� +1 + η2 − η) , +ΓLV = ¯h¯γ +M +Vb +T e−Vb/T . +(8) +As one can see, the dependence of Γf(T) on η is rather complicated. The +fission width Γf(T) grows as function of η in low damping region (η < 0.1). +For η > 0.2, the fission width Γf(T) decreases as function of η. +In nuclear systems, the Markovian assumption is often too restrictive. +We thus have to generalize the above Langevin equations to allow for finite +memory effects. They read as [14], +dq/dt = p(t)/M, +(9) +dp +dt = −∂Epot +∂q +− +� t +0 +dt′γ(t − t′)p(t′)/M + ζ , +γ(t − t′) ≡ ¯γe− t−t′ +τ /τ , +where τ is the memory (or relaxation) time. +The extension consists in +allowing the friction to have a memory time, i.e., the friction reacts on past +stages of the system, what is called a retarded friction. +The random numbers ζ in (9) are the normally distributed random num- +bers with the properties < ζ(t) >= 0, < ζ(t)ζ(t′) >= Tγ(t−t′). In the limit +ω0τ << 1, one recovers the Markovian limit of nuclear fission dynamics, i.e., + +8 +preprint +printed on January 5, 2023 +when the friction force is simply given by γ ˙q(t). The random numbers ζ(t) +in (9) satisfy the equation +dζ(t)/dt = −ζ(t)/τ + R(t)/τ , +(10) +and are used in the description of the so-called Ornstein-Uhlenbeck pro- +cesses. +In the top part of Fig. 4 the calculated fission width Γf(E) is shown as +a function of the damping parameter η both for small and large excitation +energies, E∗=10, 25 and 60 MeV, for few values of the relaxation time. +Besides τ = 0, we choose in calculations below the two values of τ, τ = +5 · 10−22 sec and τ = 10−21 sec. +0 +1 +2 +0 +5 +10 +Γf (eV) +η + Γf(E) + Γeff(T) +E*=10 MeV, Tin=0.6 MeV +0 +1 +2 +0 +100 +200 +300 +E*=25 MeV, Tin=1.0 MeV +η +0 +1 +2 +0 +1000 +2000 +3000 +E*=60 MeV. Tin=1.5 MeV +η + τ=0 + τ=5 10 +-22 sec + τ=10 10 +-22 sec +0 +5 +10 +0 +5 +10 + Γf(E) + Γeff(T) +Γf (eV) +τ (10 +-22 sec) +0 +5 +10 +0 +100 +200 +300 +τ (10 +-22 sec) +0 +5 +10 +0 +1000 +2000 +3000 +τ (10 +-22 sec) + η=0.1 + η=0.5 + η=1.0 +Fig. 4. (top) The dependence of the fission width Γf(E) (solid) and the approxima- +tion (11) (dashed) on the damping parameter η for few values of the relaxation time +τ, τ=0, τ = 5 · 10−22 sec, τ = 10−21 sec and the initial excitation energies E∗ +in=10, +25 and 60 MeV. (bottom) The dependence of the fission width Γf(E) (solid) and +the approximation (11) (dashed) on the relaxation time τ for a few values of the +damping parameter η, η=0.1, 0.5 and 1.0. +The results of Langevin calculations satisfying the energy conservation +condition are shown in Fig. 4 by solid lines. The fission width Γf(E) grows + +preprint +printed on January 5, 2023 +9 +as a function of η and decreases as a function of τ in low damping region. +The tendency is the opposite in the high damping region; the fission width +Γf falls as a function of η and increases as a function of τ. Such dependence +is common both for small and large excitation energies. +In the bottom part of Fig. 4, the fission width Γf(E) (solid lines) is shown +as a function of the relaxation time τ for a few fixed values of the damping +parameter η. +The bottom part of Fig. 4 confirms the above conclusion: +the dependence of fission width Γf on η and τ is opposite in low and high +damping regions. +For the comparison, we show by dashed lines in Fig. 4 the available +analytical approximation for Γf(T, τ) [14, 15, 16], +1 +Γeff += +1 +ΓLV ++ +1 +ΓHV +, +ΓLV (τ) = ΓLV (0) +1 + ω2 +0τ 2 , +ΓHV (τ) = ¯hλ +2π e−Vb/T , (11) +where λ is the largest positive solution of the secular equation, +λ3 + λ2/τ + (¯γ/Mτ − ω2 +0)λ − ω2 +0/τ = 0 . +(12) +As can be seen, the results of Langevin calculations for Γf(E) are smaller +than the analytical estimate (11) both in low and high damping limits. The +ratio Γf(E)/Γeff is close to 1 at E∗=60 MeV and close to 0.1 at E∗=10 +MeV. +5. Summary +The calculated mass distributions of fission fragments of super-heavy +nuclei from 268Hs to 308122 demonstrate a three-four peaks structure of mass +distributions. In light super-heavies, we see the dominant mass symmetric +peak at AF ≈ 140. With increasing mass and charge numbers of fissioning +nuclei, the highly asymmetric peaks at AH ≈ 208 appears. In 290−296Lv +and 290−296Og, the three peaks in FFMD are approximately of the same +magnitude at E*=10 MeV. +The investigation of memory effects in nuclear fission is carried out. The +calculations presented here offer complete information on the dependence +of fission probability on all essential parameters, the relaxation time τ, the +damping parameter η, and the excitation energy E*. +It turned out that the fission width Γf(E) calculated under the constant +energy requirement is generally smaller than that calculated in the constant +temperature regime, Γf(T), or the Bohr-Wheeler approximation. +The dependence of the fission width Γf(E) on the relaxation time τ is +very sensitive to the damping parameter η. In the low viscosity region, the +fission width Γf(E) grows as a function of η and decreases as a function of τ. + +10 +preprint +printed on January 5, 2023 +In the high-viscosity region, the tendency is the opposite. Such dependence +is common both for small and large excitation energies. +Acknowledgements. The authors are grateful to Prof. K.Pomorski +for the valuable discussions and presentation of our results at the Zakopane +Conference +REFERENCES +[1] S.G.Nilsson, C.F. Tsang, A. Sobiczewski et al, Nucl. Phys. A 131, 1 (1969). +[2] Yu.Ts. Oganessian, A.V. Yeremin, A.G. Popeko et al, Nature 400, 242 (1999). +[3] J. Maruhn and W. Greiner, Zeit. f. Phys. 251, 431 (1972). +[4] M.D. Usang, F.A. Ivanyuk, C. Ishisuka, and S. Chiba, Phys. Rev. C 94, 044602 +(2016). +[5] C. Ishizuka, M.D. Usang, F.A. Ivanyuk et al, Phys. Rev. C 96, 064616 (2017). +[6] M.D. Usang, F.A. Ivanyuk, C. Ishizuka, and S. Chiba, Phys. Rev. C 96, 064617 +(2017). +[7] M.D. Usang, F.A. Ivanyuk, C. Ishisuka, and S. Chiba, Scientific Reports 9, +1525 (2019). +[8] C. Ishisuka, X. Zhang, M. D. Usang, F. A. Ivanyuk, and S. Chiba, Phys. Rev. +C 101, 011601(R) (2020). +[9] P.V. Kostryukov, A. Dobrowolski, B. Nerlo-Pomorska et al, Chin. Phys. C 45, +124108 (2021). +[10] H. Hofmann, F. A. Ivanyuk, C. Rummel, and S. Yamaji, Phys. Rev. C 64, +054316 (2001). +[11] J. T¨oke, W. J. Swiatecki, Nucl. Phys. A 372, 141 (1981). +[12] F.A. Ivanyuk, S.V. Radionov, C. ishizuka and S. Chiba, Nucl. Phys. A 1028, +122526 (2022). +[13] H. A. Kramers, Physica VII, 284 (1940). +[14] Y. Abe, S. Ayik, P.-G. Reinhard, and E. Suraud, Phys. Rep. 275, 49 (1996). +[15] R.F. Grote and J.T. Hynes, Jour. Chem. Phys. 73, 2715 (1980). +[16] D.Boilley, Y.Lallouet, Jour. Stat. Phys. 125, 477 (2006). + diff --git a/DtAzT4oBgHgl3EQfif1c/content/tmp_files/load_file.txt b/DtAzT4oBgHgl3EQfif1c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89ed434d56d4ba107e4e99316913d505073038cd --- /dev/null +++ b/DtAzT4oBgHgl3EQfif1c/content/tmp_files/load_file.txt @@ -0,0 +1,348 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf,len=347 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='01500v1 [nucl-th] 4 Jan 2023 The Langevin approach for fission of heavy and super-heavy nuclei∗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='Ivanyuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='Radionov Institute for Nuclear Research, Kyiv, Ukraine C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='Ishizuka, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='Chiba Tokyo Institute of Technology, Tokyo, Japan In this contribution, we present the main relations of the Langevin approach to the description of fission or fusion-fission reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The results of Langevin calculations are shown for the mass distributions of fission fragments of super-heavy elements and used for the investigation of memory effects in nuclear fission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Introduction We describe the nuclear fission process by the four-dimensional set of the Langevin equations for the shape degrees of freedom with the shape given by the two-center shell model (TCSM) shape parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The potential energy is calculated within the macroscopic-microscopic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The collective mass, M, and friction, γ, tensors are defined in macroscopic (Werner-Wheller and wall-and-window formula) or microscopic (linear re- sponse theory) approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' We start calculations from the ground state shape with zero collective velocities and solve equations until the neck radius of the nucleus turns zero (scission point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' At the scission point, the solutions of Langevin equations supply complete information about the system, its shape, excitation energy, and collective velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' This information makes it possible to calculate the mass distributions, the total kinetic energy, and the excitation energies of fission fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The results of numerous previous calculations are in reasonable agreement with the available experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' ∗ Presented at the Zakopane Conference on Nuclear Physics, Zakopane, Poland, 28 August - 4 September 2022 (1) 2 preprint printed on January 5, 2023 Below in this contribution, we present the calculated results for the mass distributions of super-heavy nuclei and clarify the impact of memory effects on the fission width of heavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The physics of super-heavy elements (SHE) has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The ex- istence of the “island of stability” was predicted at the end of the 1960s [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Nevertheless, it took almost 30 years until the alpha-decay of the ele- ment with Z=114 was observed experimentally at Flerov Nuclear Reactions Laboratory in Dubna [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' With the development of experimental facility, it became possible not only to fix the fact of formation of SHE, but examine their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' One of the first property of interest – the process of fission of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For the successful planning and carrying out of experiments, it is crucial to understand what kind of fission fragments mass distribution (FFMD) one should expect in the result of the fission of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The two double magic nuclei 132Sn and 208Pb may contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Both have the shell correction in the ground state of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In order to clarify what kind of FFMD one could expect in the fission of SHEs, we have carried out the calculations of FFMD for a number of SHEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The results are given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Another problem we address in this contribution is the influence of mem- ory effects on the probability of the fission process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Commonly one uses the Markovian approximation to Langevin approach in which all quantities are defined at the same moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' This approximation provides reasonable re- sults, but its accuracy is not well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In publications, one can find statements that the memory effects have a significant influence on the fusion or fission processes and the statements that memory effects are very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' To clarify this uncertainty, we have calculated the fission width using the Langevin approach with memory effects included in a wide range of im- portant parameters: the excitation energy E∗ of the system, the damping parameter η, the relaxation time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The details and results of the calcula- tions are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The Langevin approach for the fission process Within the Langevin approach, the fission process is described by solving the equations for the time evolution of the shape of nuclear surface of the fis- sioning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For the shape parametrization, we use that of the two-center shell model (TCSM) [3] with 4 deformation parameters qµ = z0/R0, δ1, δ2, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Here z0/R0 refers to the distance between the centers of left and right os- cillator potentials, R0 being the radius of spherical nucleus with the mass number A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The parameters δi describe the deformation of the right and left fragment tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fourth parameter α is the mass asymmetry and the fifth preprint printed on January 5, 2023 3 parameter of the TCSM shape parametrization ǫ was kept constant, ǫ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='35, in all our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The first-order differential equations (Langevin equations) for the time dependence of collective variables qµ and the conjugated momenta pµ are: dqµ dt = � m−1� µν pν, (1) dpµ dt = −∂F(q, T) ∂qµ − 1 2 ∂m−1 νσ ∂qµ pνpσ − γµνm−1 νσ pσ + Rµ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (1) the F(q, T) is the temperature-dependent free energy of the system, and γµν and (m−1)µν are the friction and inverse of mass tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The free energy F(q, T) is calculated within the shell correction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The single particle energies are calculated with the deformed Woods-Saxon potential fitted to the mentioned above TCSM shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The collective inertia tensor mµν is calculated by the Werner-Wheeler approximation and for the friction tensor γµν we used the wall-and-window formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The random force Rµ(t) is the product of the temperature-depen- dent strength factors gµν and the white noise ξν(t), Rµ(t) = gµνξν(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The factors gµν are related to the temperature and friction tensor via the Einstein relation, gµσgσν = Tγµν (2) The temperature T is kept constant, aT 2 = E∗, or adjusted to the local excitation energy on each step of integration by the relation, aT 2 = E∗ − p2(t)/2M − [Epot(q) − Epot(qgs)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (3) Here qgs is the ground state deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' More details are given in our earlier publications [4, 5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Initially, the momenta pµ are set to zero, and calculations are started from the ground state deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Such calculations are continued until the trajectories reach the ”scission point”, defined as the point in deformation space where the neck radius turns zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Fission fragments mass distributions of super-heavy nuclei In order to understand what kind of mass distributions one can expect from the solution of Langevin equations for super-heavy nuclei, we looked first at the potential energy of fissioning nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1 shows the potential energy Edef of nuclei 296Lv and 302120 at zero temperature as a function of elongation (the distance R12 between the centers of mass of left and right parts of a nucleus) and the mass asymmetry (fragment mass number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4 preprint printed on January 5, 2023 In the top part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1 the energy was minimized with respect to the deformation parameters δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' One sees the bottom of potential energy leading to almost symmetric mass splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' There is also a hint on the mass asymmetric valley at AF close to AF =208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 100 150 200 302120, δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2, δ2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2 R12 / R0 Fragment mass number 60 52 44 36 28 20 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 100 150 200 302120, δ1,δ2 - min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Fragment mass number 60 52 44 36 28 20 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 50 100 150 200 296Lv, δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2, δ2= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2 R12 / R0 Fragment mass number 60 52 44 36 28 20 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 50 100 150 200 Fragment mass number 60 52 44 36 28 20 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 10 296Lv, δ1,δ2 - min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (top) The potential energy of 296Lv and 302120 at T = 0 minimized with respect to deformation parameters δ1 and δ2 (bottom), and at fixed values δ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2 and δ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' If the trajectories followed the bottom of potential energy, the mass distributions would be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' However, it is well known that the tra- jectories may deviate substantially from the bottom of the potential valley due to dynamic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' We calculate the trajectories in four-dimensional deformation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In this space, the local minima could lead away from the bottom of the potential valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' An example is shown in the bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Here we show the potential energy for fixed δ1= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2 and δ2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' One clearly sees another valley, leading to strongly mass asymmetric split- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 2, we show the fission fragment mass distributions of super-heavy nuclei from 276Hs to 308122 as a function of fragment mass number AF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The FFMDs of nuclei from 276Cn to 308122 have three or four peak structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The main component is the symmetric peak, split into two components in some isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The peaks of lighter fragments are located around AF =140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' preprint printed on January 5, 2023 5 5 10 Fission from the ground state, ---- E =10 MeV, ---- E =20 MeV, ---- E =30 MeV N = 168 170 172 174 176 178 180 182 184 186 Z = 108 110 112 114 116 118 120 122 5 10 5 10 5 10 0 5 10 5 10 15 140 5 10 15 5 10 15 20 40 140 05 10 15 20 40 140 F r a g m e n t m a s s n u m b e r Yield (%) 40 14040 14040 14040 14040 14040 14040 14040 140 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fission fragment mass distributions of super-heavy nuclei from 276Hs to 308122 calculated for the excitation energies E∗=10, 20 and 30 MeV as a function of the fragment mass number One can also see the strongly asymmetric peak at the mass number close to AF =208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The strength of the (almost) symmetric and asymmetric components in FFMD of SHEs depends on the proton and neutron num- bers of the compound nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For 276Cn, the contribution of a strongly asymmetric peak is tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' This contribution becomes larger for more heavy SHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In some elements of SHEs with Z =116-122, the symmetric and mass- asymmetric peaks are of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' More details can be found in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The similar strongly mass-asymmetric peaks in FFMD of SHEs were also found recently in [9] within the Langevin approach with the so call Fourier shape parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 6 preprint printed on January 5, 2023 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The memory effects in nuclear fission In order to investigate the role of memory effects in nuclear fission, we exploit a simple one-dimensional model with the potential energy given by the two-parabolic potential (Kramers potential), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Epot(q) = 2Vbq(q − q0)/q2 0, 0 < q < q0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 2Vb(q − q0)(2q0 − q)q2 0, q0 < q < 2q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (4) The potential (4) depends on two parameters, the barrier height Vb and the barrier width q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' We have fixed the barrier height Vb = 6 MeV, which is close to the value of the fission barrier of actinide nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The width of the barrier is somewhat uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' It depends on the definition of the collective coordinate q and the model for the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For simplicity, we have put here q0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For the potential (4) one can define the stiffness C = d2Epot/dq2 and the frequency of harmonic vibrations ω0 = � C/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In the present work, we fix ¯hω0 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 MeV, which is close to the frequency of collective vibra- tions calculated for 224Th in [10] within the microscopic linear response theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Then, for the mass parameter we will have the deformation and temperature-independent value, M = C/ω2 0 = 4Vb/(ω2 0q2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (5) For the friction coefficient ¯γ we use a slightly modified approximation of [10], ¯γ/M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='6(T 2 + ¯h2ω2 0/π2))/(1 + T 2/40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (6) For the temperature, we consider here two options: constant temperature regime and constant energy regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In a constant temperature regime, the temperature is time-independent, related to the initial excitation energy E∗ by the Fermi-gas relation, aT 2 = E∗, where a is the level density parameter of T¨oke and Swiatecki [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fission width calculated in a constant temperature regime will be denoted as Γf(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' At small excitations, the temperature varies with deformation and time, and there is no reason to consider it constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' So, it should be adjusted to the local excitation energy on each integration step by the relation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Correspondingly, fission width calculated in a constant energy regime is denoted as Γf(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fission width, Γf, is defined assuming the exponential decay of the number of ”particles” in the potential well, P(t) = e−Γf t/¯h → Γf = −¯h ln[P(t)]/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (7) By solving the Langevin equations one will get the set of time moments tb, at which some trajectories would cross the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' From this information, one can find the probability P(t) and the fission width Γf, see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' preprint printed on January 5, 2023 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 0 5 10 Epot (MeV) q A=236, E =Vb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 0 1000 2000 3000 Γf (eV) η Γf(T) ΓLV ΓHV T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 MeV Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (left) The two-parabolic potential (4) and few examples of the dynamical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (right) The fission width as the solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (1, 4, 7) calculated at constant temperature (open dots), and the Kramers approximations (8) for high and low damping limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The Markovian fission width Γf(T) calculated by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (1, 4, 7) is plotted as function of the damping parameter η in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' To present the results in a broader range of parameters, the damping parameter η ≡ ¯γ/2Mω0 in these calculations was considered as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For the comparison, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='3 we also show the Kramers decay width ΓHV , ΓLV in limits of high and low viscosity (friction) [13], ΓHV = ¯hω0 2π e−Vb/T ( � 1 + η2 − η) , ΓLV = ¯h¯γ M Vb T e−Vb/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (8) As one can see, the dependence of Γf(T) on η is rather complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fission width Γf(T) grows as function of η in low damping region (η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='2, the fission width Γf(T) decreases as function of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In nuclear systems, the Markovian assumption is often too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' We thus have to generalize the above Langevin equations to allow for finite memory effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' They read as [14], dq/dt = p(t)/M, (9) dp dt = −∂Epot ∂q − � t 0 dt′γ(t − t′)p(t′)/M + ζ , γ(t − t′) ≡ ¯γe− t−t′ τ /τ , where τ is the memory (or relaxation) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The extension consists in allowing the friction to have a memory time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=', the friction reacts on past stages of the system, what is called a retarded friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The random numbers ζ in (9) are the normally distributed random num- bers with the properties < ζ(t) >= 0, < ζ(t)ζ(t′) >= Tγ(t−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In the limit ω0τ << 1, one recovers the Markovian limit of nuclear fission dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=', 8 preprint printed on January 5, 2023 when the friction force is simply given by γ ˙q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The random numbers ζ(t) in (9) satisfy the equation dζ(t)/dt = −ζ(t)/τ + R(t)/τ , (10) and are used in the description of the so-called Ornstein-Uhlenbeck pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In the top part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4 the calculated fission width Γf(E) is shown as a function of the damping parameter η both for small and large excitation energies, E∗=10, 25 and 60 MeV, for few values of the relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Besides τ = 0, we choose in calculations below the two values of τ, τ = 5 · 10−22 sec and τ = 10−21 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 0 1 2 0 5 10 Γf (eV) η Γf(E) Γeff(T) E*=10 MeV, Tin=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='6 MeV 0 1 2 0 100 200 300 E*=25 MeV, Tin=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 MeV η 0 1 2 0 1000 2000 3000 E*=60 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Tin=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 MeV η τ=0 τ=5 10 22 sec τ=10 10 22 sec 0 5 10 0 5 10 Γf(E) Γeff(T) Γf (eV) τ (10 22 sec) 0 5 10 0 100 200 300 τ (10 22 sec) 0 5 10 0 1000 2000 3000 τ (10 22 sec) η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='1 η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 η=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (top) The dependence of the fission width Γf(E) (solid) and the approxima- tion (11) (dashed) on the damping parameter η for few values of the relaxation time τ, τ=0, τ = 5 · 10−22 sec, τ = 10−21 sec and the initial excitation energies E∗ in=10, 25 and 60 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (bottom) The dependence of the fission width Γf(E) (solid) and the approximation (11) (dashed) on the relaxation time τ for a few values of the damping parameter η, η=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The results of Langevin calculations satisfying the energy conservation condition are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4 by solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The fission width Γf(E) grows preprint printed on January 5, 2023 9 as a function of η and decreases as a function of τ in low damping region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The tendency is the opposite in the high damping region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' the fission width Γf falls as a function of η and increases as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Such dependence is common both for small and large excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In the bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4, the fission width Γf(E) (solid lines) is shown as a function of the relaxation time τ for a few fixed values of the damping parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4 confirms the above conclusion: the dependence of fission width Γf on η and τ is opposite in low and high damping regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' For the comparison, we show by dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 4 the available analytical approximation for Γf(T, τ) [14, 15, 16], 1 Γeff = 1 ΓLV + 1 ΓHV , ΓLV (τ) = ΓLV (0) 1 + ω2 0τ 2 , ΓHV (τ) = ¯hλ 2π e−Vb/T , (11) where λ is the largest positive solution of the secular equation, λ3 + λ2/τ + (¯γ/Mτ − ω2 0)λ − ω2 0/τ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' (12) As can be seen, the results of Langevin calculations for Γf(E) are smaller than the analytical estimate (11) both in low and high damping limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The ratio Γf(E)/Γeff is close to 1 at E∗=60 MeV and close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='1 at E∗=10 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Summary The calculated mass distributions of fission fragments of super-heavy nuclei from 268Hs to 308122 demonstrate a three-four peaks structure of mass distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In light super-heavies, we see the dominant mass symmetric peak at AF ≈ 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' With increasing mass and charge numbers of fissioning nuclei, the highly asymmetric peaks at AH ≈ 208 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In 290−296Lv and 290−296Og, the three peaks in FFMD are approximately of the same magnitude at E*=10 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The investigation of memory effects in nuclear fission is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The calculations presented here offer complete information on the dependence of fission probability on all essential parameters, the relaxation time τ, the damping parameter η, and the excitation energy E*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' It turned out that the fission width Γf(E) calculated under the constant energy requirement is generally smaller than that calculated in the constant temperature regime, Γf(T), or the Bohr-Wheeler approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The dependence of the fission width Γf(E) on the relaxation time τ is very sensitive to the damping parameter η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' In the low viscosity region, the fission width Γf(E) grows as a function of η and decreases as a function of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' 10 preprint printed on January 5, 2023 In the high-viscosity region, the tendency is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Such dependence is common both for small and large excitation energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' The authors are grateful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='Pomorski for the valuable discussions and presentation of our results at the Zakopane Conference REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAzT4oBgHgl3EQfif1c/content/2301.01500v1.pdf'} +page_content='G.' metadata={'source': 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index 0000000000000000000000000000000000000000..111625618efcc2371c3c53822a0c19628ea7051c --- /dev/null +++ b/ItE4T4oBgHgl3EQfhQ0t/content/tmp_files/2301.05123v1.pdf.txt @@ -0,0 +1,659 @@ +Physical Layer Security Techniques Applied to +Vehicle-to-Everything Networks +Leonardo B. da Silva, Evelio M. G. Fernández and Ândrei Camponogara +Abstract— Physical Layer Security (PLS) is an emerging con- +cept in the field of secrecy for wireless communications that can +be used alongside cryptography to prevent unauthorized devices +from eavesdropping a legitimate transmission. It offers low com- +putational cost and overhead by injecting an interfering signal +in the wiretap channels of potential eavesdroppers. This paper +discusses the benefits of the Artificial Noise and Cooperative +Jamming techniques in the context of Vehicle-to-everything (V2X) +networks, which require secure data exchange with small latency. +The simulations indicate that messages can be safely delivered +even with devices that have low available power. +Keywords— Wireless communication networks, Physical Layer +Security, secrecy, Vehicle-to-everything, Artificial Noise, Cooper- +ative Jamming. +I. INTRODUCTION +Urban mobility is one of the main focuses of the Internet +of Things (IoT) when applied to smart cities, due to the +necessity for more responsive and safe traffic control. Gener- +ally, the solutions proposed in this scope involve the wireless +communication between not only the vehicles themselves, +but also with pedestrians, infrastructure, and networks. This +paradigm is known as Vehicle-to-everything (V2X) and it can +be standardized by protocols such as C-ITS (Cellular Intelli- +gent Transportation System) and WAVE (Wireless Access for +Vehicular Environment) that are based on the IEEE 802.11p +amendment, and the Cellular-V2X (C-V2X) that implements +the 5G standard from 3GPP (3rd Generation Partnership +Project) [1]. +A. Problem Outline +Due to the ever-changing location of most of the involved +communication nodes and the time-sensitive nature of the +data involved (brake position, vehicle speed, traffic volume, +accident reports, etc), the transmission needs not only to occur +at high rates, but also offer reliability through high secrecy, low +packet loss, and small delay. Furthermore, those nodes have +to be affordable to justify their implementation on a city-wide +scale, thus having low power consumption and the most cost- +efficient embedded processing unit possible [2]. +Since the main source of information security in today’s +landscape is provided through cryptography, the secrecy con- +straint can negatively affect most of these criteria. As a result +L. +B. +da +Silva, +E. +M. +G. +Fernandez, +Â. +Camponogara, +Electri- +cal Engineering Department, Federal University of Paraná (UFPR), Cu- +ritiba, PR, Brazil, e-mails: leonardobarbosa@ufpr.br, evelio@ufpr.br and +andrei.camponogara@ufpr.br. This study was financed in part by the Coorde- +nação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – +Finance Code 001. +of the growth in the availability of portable and connected +equipment with high processing capabilities, the safety mea- +sures implemented need to match this computational power +with proportionally longer and more complex keys to not be +vulnerable to brute-force attacks from well-equipped malicious +devices [2], [3]. This approach, however, is not sustainable, +because it produces increasingly long authentication routines, +due to the raise in computational overhead and processing cost +as a result of the implemented security algorithms. +B. Overview of the proposed solution +To counterbalance this issue, this paper studies the use of +Physical Layer Security (PLS) techniques as an additional +protection to increase the secrecy of wireless communications +in a V2X environment. As the name suggests, PLS is applied +at the Physical Layer, making it an alternative that can be used +with low processing cost when compared with cryptography, +which is more oriented towards the computational side of the +network stack on the Application Layer [1]. +Since cryptography techniques provide security in different +sections of the wireless protocols, PLS is proposed as a +complement to them, rather than a replacement [1]. Through +the use of both approaches on the same node, it is possible to +offer high secrecy without the necessity of infinitely growing +key complexity. +The PLS has its origins on the analytical proposal of +Wyner’s wiretap channel [4], where it is described a commu- +nication between two legitimate nodes that is spied on by an +eavesdropper through an unauthorized channel called wiretap. +In the modern literature, these devices are usually referred to +as a transmitter called Alice, an authorized receiver Bob, and +the set of K eavesdroppers named Eves. +In the wiretap channel model shown in Fig. 1, the original +message m is encoded and transmitted by Alice as the signal +sa, that reaches Bob through the main channel hAB. The +received signal yB is then decoded by Bob, obtaining the +estimated message ˆm. Additionally, the k-th Eve can intercept +sa through the wiretap channel hAE,k, obtaining the signal +yE,k that when decoded produces z. +The main focus of PLS is to guarantee that the mutual +information between m and z is as close to zero as possible. +When this condition is met, even if z is know, it is impossible +for Eve to infer the contents of the original message. +Wyner then presents a set of parameters that enable the use +of the physical imperfections of the channel, such as noise +and fading, to provide information secrecy by raising the level +of confusion on undesired nodes. Rendering them unable to +distinguish between the message and the interference. +arXiv:2301.05123v1 [eess.SP] 12 Jan 2023 + +Fig. 1: The wiretap channel generic model based on [4] +Currently, plenty of techniques to provide security at the +physical layer level have been proposed in the literature [3]. +This paper will focus on two approaches first presented in [5]: +• Artificial Noise (AN): This approach uses a portion of +the transmitter node’s power to inject artificially gener- +ated noise in the eavesdropper’s channel; +• Cooperative Jamming (CJ): This approach expands the +AN model by proposing a connected network where +nearby relay nodes (Charlies) send a jamming signal to +the eavesdropper’s channel. +To demonstrate the viability of AN and CJ applications in +a V2X network, it is common to create stochastic geometric +models that randomly generate streets and distribute com- +munication nodes in a predefined area to represent an urban +mobility scenario [6], [7]. When implementing these methods, +metrics such as the Signal-to-Interference Ratio (SIR) are used +to define the threshold of confusion necessary to provide se- +crecy at the physical layer. The SIR on each eavesdropper can +then be evaluated to determine the secrecy outage probability +(SOP) of the data transmission with different densities of the +involved nodes in the simulated network. +In this paper, Section II describes the stochastic algorithms +implemented to model a V2X network that includes streets +and communication nodes (vehicular and planar). Section III +presents the analytical basis of the AN and CJ techniques, +while also introducing the SIR and SOP metrics. In Section +IV, the results of numerical simulations are shown to illustrate +the benefits of the considered PLS techniques on the generated +V2X networks. Finally, Section V states some final remarks. +Notation: IN is an identity matrix of order N, Poisson(n) +is a Poisson distribution with mean number of arrivals n, +CN(m, n) is a complex normal distribution with average m +and covariance n, exp(n) is an exponential distribution with +mean n and Gamma(m, n) is the gamma distribution with +form m and scale n. +II. THE V2X NETWORK MODEL +As mentioned previously, vehicular networks are dynamic, +with devices changing location constantly. Thus, a determin- +istic model is not well-suited for this application. A common +alternative is the use of stochastic geometry to represent this +random spatial nature through a variety of different processes +to distribute the streets and communication nodes within the +desired coverage area [8]. +A viable option is the use of Poisson processes, as they are +memoryless counting processes for integer arrivals [9]. In other +words, each set of elements generated will be independent +with a Poisson distributed integer number of uniformly spaced +nodes. The intensity of the arrivals in these processes are +represented by λ and the expected number of elements is +the product of the said intensity and the Lebesgue measure, +which in this context is essentially the spatial measurement +associated with the object that the points will be distributed on. +For instance, the Lebesgue measure to populate a circle is its +area and for a line is the length. One realization of the resulting +spatial model derived from the use of different variations of +the Poisson processes is represented in Fig. 2. +Fig. 2: +Spatial simulation of the modeled V2X network. +The color green indicates the Charlies implemented in CJ +techniques and the Eves are in red. The planar devices are +generated by PPPs represented by circles (◦) with intensity λ += 10−6/m2 for both node types. Through a PLP, the streets +(blue lines) have been modeled with an intensity of λl = 10−3 +/m, and the vehicular devices are originated from PLP-driven +Cox Processes indicated with triangles (△) of intensity u = +10−3/m for both Charlies and Eves. A single Alice is indicated +with a black × at the origin. +In this model, the wireless devices of pedestrians and +connected infrastructure are considered free to be positioned in +the whole area A of the modeled network, which is a circle of +radius r = 3 km. Thus, these “planar nodes“ are generated by +2-D Poisson Point Processes (PPP) and the expected amount +of elements is given by Poisson(λ · A). The set of planar +nodes is indicated by Φ, thus the planar Eves and Charlies +are respectively represented by ΦE and ΦC. +The streets are represented by uniformly distributed lines +with density µl = λl/π generated by a Poisson Line Process +(PLP) Φl based on the second method of the Bertrand paradox +[10], in which a set of expected Poisson(µl·2πr) midpoints are +created [11], each with a random radius P ∈ [0, r) and angle +θ ∈ [0, 2π). From these coordinates, a segment perpendicular +to P is traced between two points at the edge of the circle +of radius r. This effectively means that a pair of 1-D PPP +points are created in the perimeter of the circular area for +each modeled street. +On those PLP-generated lines, a Cox process of intensity u + +YB +TRANSMITTER +MAIN CHANNEL +RECEIVER +m - +(ALICE) +hAB +m +(BOB) +Ye,k +WIRETAP CHANNEL +k-thEAVESDROPPER +>Z +(EVE)3000 +X +Alice node +Planar Charlies +Q +Planar Eves +Vehicular Charlies +O +A +Vehicular Eves +2000 +O +Q +1000 +O +F 0 +2 +X +O +-1000 +C +-2000 +3000 +4000 +3000 +-2000 +-1000 +0 +1000 +2000 +3000 +4000is implemented, which is used to create the “vehicular nodes“ +on each segment [12]. These elements represent vehicles +whose spatial distribution are constrained to a street by a 1-D +PPP. Considering a street of length l, the number of vehicles +in it is given by Poisson(u · l). +The set of vehicular Eves and Charlies on each street l are +respectively denoted by ψE and ψC. Based on these, the total +nodes of each type can be obtained by evaluating the sets on +the whole range of Φl [6], resulting in ΨE = {ψE(l)}l∈Φl for +Eves and ΨC = {ψC(l)}l∈Φl for Charlies. +Furthermore, a single deterministic transmitter (Alice) is +included at the origin of the circle. This point is selected to +simplify the distance calculations between a legitimate device +and the Eve nodes, which can be planar or vehicular. This +measurement is one of the parameters for the SIR calculations, +that are considered to determine the effectiveness of the PLS. +For the CJ case, auxiliary nodes (Charlies) are also modeled, +some as planar and others as vehicular devices. Note that the +distance between Charlies and Eves influences the power of +the interference injected on the unauthorized channels as part +of the jamming technique. +III. PLS TECHNIQUES +The PLS techniques presented in this paper are part of the +key-less-based class [2], which implements secure information +transmission by making the unauthorized channel’s capacity +(CE) lower than that of the legitimate channel’s (CB). This re- +lationship can be presented by evaluating these values through +the Shannon-Hartley theorem, which produces the secrecy +capacity (CS) metric as +CS = CB − CE = log2(1 + γB) − log2(1 + γE), +(1) +where γB and γE are, respectively, the SIRs of Bob and Eve. +Based on this expression, it can be inferred that in order to +guarantee that CB is sufficiently larger than CE, the value of +γE must be as low as possible. The approach utilized by AN +and CJ is the injection of artificially generated interference in +the eavesdropper channels. +Typically, this injection is implemented with multi-antenna +networks, as it enables the use of beamforming to selectively +direct the transmission to legitimate receivers with minimum +noise and high efficiency [3]. The unintended receivers on +the other hand, intercept a signal that contains the secret +message as well as AN. Therefore, secrecy is provided when +the distinction between them by the Eves is improbable. +The wireless channels in this paper are modeled with com- +plex normal distributions (CN) which implies in a Rayleigh +fading model. This decision provides simpler analytical equa- +tions and also proposes a more pessimistic scenario, in which +there is no Line-of-Sight (LoS) available. By evaluating the +metrics in these worst-case conditions, it is possible to verify +that even then the secrecy can be guaranteed. +A. Artificial Noise +In the AN scenario, the legitimate communication is es- +tablished between a single transmitter Alice and a receiver +Bob. Additional nodes (both planar and vehicular) that try to +obtain Alice’s signal are then considered eavesdroppers and +their channels will be affected by the AN. +The signal transmitted by the Alice node with NA antennas +is composed of two terms: the first contains a message x +intended for Bob and the second is based on a zero-forcing +vector for the unauthorized devices [13], i.e, +sa = +� +φPt +ha +∥ha∥x + +� +(1 − φ)Pt +NA − 1 Wana, +(2) +where ha/∥ha∥ is the beamforming vector with the normaliza- +tion of the Alice’s channel estimation ha ∈ CNA×1, that will +be modeled as CN(0, INA). The AN is formed by the null- +space orthonormal basis Wa ∈ CNA×(NA−1) and the noise +signal na ∈ C(NA−1)×1. +The distribution of the available power, Pt, between the +two terms of (2) is controlled by φ ∈ {0,1}. φ = 0 means that +all power is allocated to noise generation and no message is +sent. Conversely, when φ = 1 the AN is not active and Pt is +allocated entirely for data transmission. +B. Cooperative Jamming +The Cooperative Jamming extends the AN case, maintaining +the single Alice-Bob authorized transmission with multiple +Eves, however, adding auxiliary nodes in the network. These +devices, typically called Charlies, can also be either planar or +vehicular, just like the Eves. In contrast, they are responsible +for providing additional security by sending jamming signals +that further decrease the channel quality of the Eves. +For simplicity, it is considered that only Alice will transmit +messages in the scenarios evaluated in this paper. Hence, the +signals sent by the Charlie nodes are made of only the AN +(zero-forcing) portion, as follows +sc = +� +Pc +NC − 1Wcnc, +(3) +where NC is the number of antennas of each Charlie and PC +is the power available for jamming. Notice that since these +nodes are not transmitting messages, all the available power +is directed towards CJ. Additionally, Wc ∈ CNC×(NC−1) is +the null space orthonormal matrix and nc ∈ C(NC−1)×1 is the +artificial noise component. +C. Received Signals +By considering that the channel estimation ha is precisely +the main channel established between Alice and Bob, hAB, +it is implied that the receiver node is not affected by the +interference from AN or CJ. That happens because the or- +thonormal basis Wa and Wc are null when applied to the +authorized channels, resulting in the relationships h† +ABWa = 0 +and h† +ABWc = 0, respectively. Therefore, the signal received +by Bob can be expressed as +yB = +� +φPt ∥ha∥ D−α/2 +AB +x, +(4) +where DAB is the distance between the devices and α > 2 +is the path loss exponent considering an NLoS scenario. The + +distances are obtained through simple trigonometry based on +the coordinates randomly generated by the stochastic processes +described in Section II. +For the signal intercepted by the eavesdroppers, it is eval- +uated a set of K = (ΦE + ΨE) Eves, containing both planar +and vehicular nodes. Similar considerations are adopted for +the Charlies in the CJ scenario, resulting in C = (ΦC + ΨC). +As discussed when sa was presented, Alice sends a signal +containing the secret information and AN. Since authorized +Alice-Eves channels are not expected in the beamforming +sense, the orthonormal basis are not null, thus the Eves +receive interference. When the Cooperative Jamming is taken +into consideration, Eves are also affected by the interference +generated by the nearby Charlies through the sc signals. With +that in mind, the signal obtained by the k-th Eve is given by +yE,k = +� +φPt h† +AE,k D−α/2 +AE,k x ++ +� +(1 − φ)Pt +NA − 1 h† +AE,k Wa D−α/2 +AE,k na ++ +� +c ∈C +� +Pc +NC − 1 h† +c,k Wc D−α/2 +c,k +nc , +(5) +which is composed of essentially three terms. The first is the +intercepted secret message itself, the second term is the AN +signal generated by Alice, and the third term is a sum of all +the interference injected by the Charlie nodes. Since CJ only +affects the last term of (5), the AN scenario can be obtained +by simply adopting that the sum in this term is equal to zero. +From (4) and (5), it is possible to determine the SIR of Bob +and the K Eves. Thus, the SIR of Bob can be determined as +γB = Ptφ ∥ha∥2 D−α +AB, +(6) +and the SIR for each Eve can be obtained from (5) as follows +γE,k = +Pt φ +���h† +AE,k ha/∥ha∥ +��� +2 +D−α +AE,k +Pt (1−φ) +NA−1 +���h† +AE,k Wa +��� +2 +D−α +AE,k + Ic +, +(7) +where Ic is the sum of the interference injected by the Charlies +given by +Ic = +� +c ∈ C +Pc +Nc − 1∥h† +c,k Wc∥2 D−α +ck , +(8) +which is non-zero only in the CJ scenario. The products h† +AE,k· +ha/∥ha∥ and h† +AE,k ·Wa from the Alice-Eve channel and also +h† +ck·Wc from Charlie-Eve produce independent identically dis- +tributed CN random variables with unitary variance [6]. This +enables the approximations +���h† +AE,k(ha/∥ha∥) +��� +2 +∼ exp(1), +∥h† +AE,kWa∥2 ∼ Gamma(NA − 1, 1) and ∥h† +c,k Wc∥2 ∼ +Gamma(NC − 1, 1). +D. Performance metric +Considering that Alice transmits codewords at a rate Rb with +a secrecy rate RS ≤ CS, the redundancy rate can be defined +as Re = Rb − RS. Then a secrecy outage event occurs when +the channel capacity of any Eve is higher than the redundancy +rate that Alice can provide, i.e., CE > Re. +In a multiple passive Eves scenario, whose Channel State +Information (CSI) are unknown, the secrecy performance is +addressed in terms of the Secrecy Outage Probability (SOP), +since the only available information about the Alice-Eve +channel is its statistics. Thus, the SOP is defined as +SOP = 1 − Pr +� +max +k∈K γE,k < β +� +, +(9) +which is the complement of the probability that the highest +SIR among all Eves is less than the threshold β = 2Re − 1. +This means that higher values of secrecy can be obtained by +implementing the aforementioned PLS techniques to reduce +γE,k as much as possible. +IV. NUMERICAL RESULTS +Various simulations with different parameters were per- +formed to evaluate the relationship between the SOP and the +decrease of the SIR for the k-th Eve. Since the V2X network +model is randomly generated, the coordinates of each node +and street change with each run. To provide more consistent +results, the curves presented below are the average of multiple +realizations of each simulation configuration. +Fig. 3 illustrates the SOP for different Pt and Pc values, +ranging from 10 mW (10 dBm) to 1 W (30 dBm). As expected, +when the devices have more power available for interference, +the SOP is greatly reduced. However, for the AN scenario +secrecy is still not guaranteed when φ grows. For CJ, the SOP +increases in a much slower rate due to the larger amount of +nodes jamming the signal received by the Eves. +(a) Artificial Noise +(b) Cooperative Jamming +Fig. 3: SOP versus φ (25 realizations) for the AN and CJ with +different available power {0.01, 0.1, 1} W. β = 0 dB, α = 3, +NA = NC = 4, λE = λC = 10−6/m2 , µE = µC = 10−3/m, r += 3 km. +Through the simulation results presented in Fig. 4, it can be +easily noted that as β increases the SOP decreases, because + +1.0 +0.8 +0.6 +SOP +S +0.4 +0.2 +Pt = 0.01 W +Pt = 0.10 W +Pt = 1.00 W +0.0 +0.00 +0.25 +0.50 +0.75 +1.00 +Φ1.0 +Pt = Pc = 0.01 W +Pt = Pc = 0.10 W +Pt = Pc = 1.00 W +0.8 +0.6 +SOP +0.4 +0.2 +0.0 +0.00 +0.25 +0.50 +0.75 +1.00the criteria for secrecy failure is becoming more selective. +Furthermore, φ have an opposing effect when compared to +β, suggesting that for higher threshold values to guarantee +low SOP, more power needs to be allocated to interference. +Because of that, in applications where the devices have limited +power (such as IoT and V2X), CJ is a more economic approach +as long as there are sufficient nearby auxiliary nodes. +Fig. 4: +SOP versus β (50 realizations) for the AN and CJ +with different power allocation ratios {0.4, 0.6, 0.8}. α = 3, +Pt = Pc = 20 dBm, NA = NC = 4, λE = λC = 10−6/m2 , +µE = µC = 10−3/m, r = 3 km. +In Fig. 5, it is evaluated the influence that the proportion +of Charlies to Eves have on the SOP. This is achieved by +implementing different values of intensities (λ and u) for +the Poisson processes that generate these nodes. The SOP +grows rapidly in the AN, indicating that the available power +is insufficient to guarantee secrecy with the given Eve density. +For the CJ cases, however, as the number of Charlie nodes +rises, the SOP starts to reduce, making the communication +viable even for higher values of φ. When there are more +Charlies than Eves it is shown that very little power needs +to be applied in each device to provide a low SOP. +V. CONCLUSION +In this paper, a stochastic geometric approach was presented +as a method to randomly generate V2X network models. The +coordinates of these elements were then used to evaluate the +effectiveness of PLS techniques in different realizations of +vehicular networks subjected to path loss with NLoS. +Both AN and CJ were introduced based on the analytical +signals that the involved nodes transmit. Next, expressions +were obtained for the SIR of Bob and the k-th Eve. Finally, +the SOP was computed to evaluate the level of information +security provided by the presented PLS techniques. +Based on numerical results, it can be concluded that PLS +can provide additional security for the V2X networks with +relative low power cost, specially when both the techniques +are combined. It is also noted that in the CJ scenario, when +Fig. 5: +SOP versus φ (25 realizations) for the AN and CJ +with different λC/λE ratios {0.1, 0.5, 1, 5, 10}. β = 0 dB, α += 3, Pt = Pc = 10 dBm, NA = NC = 4, λE = 10−6/m2 , µE += 10−3/m, r = 3 km. +there are more Charlies in the proximity, the security increases. +Therefore, the urban networks are the most benefited by this +technique, since it is expected a higher density of wireless +devices in the same area in these environments. +REFERENCES +[1] B. M. ElHalawany, A. A. El-Banna and K. Wu, “Physical-Layer Se- +curity and Privacy for Vehicle-to-Everything”, IEEE Communications +Magazine, vol. 57, n. 10, pp. 84-90, 2019. +[2] J. M. Hamamreh, H. M. Furqan and H. Arslan, “Classifications and +Applications of Physical Layer Security Techniques for Confidentiality: +A Comprehensive Survey”, IEEE Communications Surveys & Tutorials, +vol. 21, n. 2, pp. 1773-1828, 2019. +[3] A. Sanenga, G. A. Mapunda, T. M. L. Jacob, L. Marata, B. Basutli and +J. M. Chuma, “An Overview of Key Technologies in Physical Layer +Security”, Entropy, vol. 22, n. 11, MDPI, 2020. +[4] A. D. Wyner, “The wire-tap channel”, The Bell System Technical +Journal, vol. 54, n. 8, pp. 1355-1387, 1975. +[5] R. Negi and S. Goel, “Secret communication using artificial noise”, +VTC-2005-Fall. 2005 IEEE 62nd Vehicular Technology Conference, vol. +3, pp. 1906-1910, 2005. +[6] C. Wang, Z. Li, X. Xia, J. Shi, J. Si, and Y. Zou, “Physical Layer Security +Enhancement Using Artificial Noise in Cellular Vehicle-to-Everything +(C-V2X) Networks”, IEEE Transactions on Vehicular Technology, vol. +69, n. 12, pp. 15253-15268, 2020. +[7] B. Qiu and C. Jing, “Performance Analysis for Cooperative Jamming +and Artificial Noise Aided Secure Transmission Scheme in Vehicular +Communication Network”, Research Square Platform LLC, 2020. +[8] M. Haenggi, Stochastic Geometry for Wireless Networks. Cambridge: +Cambridge University Press, 2012. +[9] R. D. Yates and D. J. Goodman, “Probability and Stochastic Processes: A +Friendly Introduction for Electrical and Computer Engineers”, Nashville, +TN: John Wiley & Sons, 2005. +[10] J. Bertrand, Calcul des probabilités. Gauthier-Villars, 1889. +[11] V. V. Chetlur and H. S. Dhillon, “Coverage Analysis of a Vehicular +Network Modeled as Cox Process Driven by Poisson Line Process”, +IEEE Transactions on Wireless Communications, vol. 17, n. 7, 2018. +[12] C. Choi and F. Baccelli, “Poisson Cox Point Processes for Vehicular +Networks”, IEEE Transactions on Vehicular Technology, vol. 67, n. 10, +pp. 10160-10165, 2018. +[13] L. Hu, H. Wen, B. Wu, F. Pan, R. Liao, H. Song, J. Tang, and X. +Wang, “Cooperative Jamming for Physical Layer Security Enhancement +in Internet of Things”, IEEE Internet of Things Journal, vol. 5, n. 1, +2018. + +100 +10-1 +SOP +10~2 +10-3 +-10 +-5 +0 +5 +10 +15 +β +AN: Φ = 0.4 +AN: Φ = 0.6 +- +AN: @ = 0.8 +CJ: Φ = 0.4 +CJ: Φ = 0.6 +- +CJ: Φ = 0.81.0 +0.8 +0.6 +SOP +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +AN +CJ: Charlies/Eves = 0.5 +CJ: Charlies/Eves = 5.0 +CJ: Charlies/Eves = 0.1 +CJ: Charlies/Eves = 1.0 +CJ: Charlies/Eves = 10.0 \ No newline at end of file diff --git a/ItE4T4oBgHgl3EQfhQ0t/content/tmp_files/load_file.txt b/ItE4T4oBgHgl3EQfhQ0t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..10eae7ed7c8393de8103c829a98a388883739606 --- /dev/null +++ b/ItE4T4oBgHgl3EQfhQ0t/content/tmp_files/load_file.txt @@ -0,0 +1,333 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf,len=332 +page_content='Physical Layer Security Techniques Applied to Vehicle-to-Everything Networks Leonardo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' da Silva, Evelio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Fernández and Ândrei Camponogara Abstract— Physical Layer Security (PLS) is an emerging con- cept in the field of secrecy for wireless communications that can be used alongside cryptography to prevent unauthorized devices from eavesdropping a legitimate transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' It offers low com- putational cost and overhead by injecting an interfering signal in the wiretap channels of potential eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This paper discusses the benefits of the Artificial Noise and Cooperative Jamming techniques in the context of Vehicle-to-everything (V2X) networks, which require secure data exchange with small latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The simulations indicate that messages can be safely delivered even with devices that have low available power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Keywords— Wireless communication networks, Physical Layer Security, secrecy, Vehicle-to-everything, Artificial Noise, Cooper- ative Jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' INTRODUCTION Urban mobility is one of the main focuses of the Internet of Things (IoT) when applied to smart cities, due to the necessity for more responsive and safe traffic control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Gener- ally, the solutions proposed in this scope involve the wireless communication between not only the vehicles themselves, but also with pedestrians, infrastructure, and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This paradigm is known as Vehicle-to-everything (V2X) and it can be standardized by protocols such as C-ITS (Cellular Intelli- gent Transportation System) and WAVE (Wireless Access for Vehicular Environment) that are based on the IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='11p amendment, and the Cellular-V2X (C-V2X) that implements the 5G standard from 3GPP (3rd Generation Partnership Project) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Problem Outline Due to the ever-changing location of most of the involved communication nodes and the time-sensitive nature of the data involved (brake position, vehicle speed, traffic volume, accident reports, etc), the transmission needs not only to occur at high rates, but also offer reliability through high secrecy, low packet loss, and small delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Furthermore, those nodes have to be affordable to justify their implementation on a city-wide scale, thus having low power consumption and the most cost- efficient embedded processing unit possible [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Since the main source of information security in today’s landscape is provided through cryptography, the secrecy con- straint can negatively affect most of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' As a result L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' da Silva, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Fernandez, Â.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Camponogara, Electri- cal Engineering Department, Federal University of Paraná (UFPR), Cu- ritiba, PR, Brazil, e-mails: leonardobarbosa@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='br, evelio@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='br and andrei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='camponogara@ufpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This study was financed in part by the Coorde- nação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' of the growth in the availability of portable and connected equipment with high processing capabilities, the safety mea- sures implemented need to match this computational power with proportionally longer and more complex keys to not be vulnerable to brute-force attacks from well-equipped malicious devices [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This approach, however, is not sustainable, because it produces increasingly long authentication routines, due to the raise in computational overhead and processing cost as a result of the implemented security algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Overview of the proposed solution To counterbalance this issue, this paper studies the use of Physical Layer Security (PLS) techniques as an additional protection to increase the secrecy of wireless communications in a V2X environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' As the name suggests, PLS is applied at the Physical Layer, making it an alternative that can be used with low processing cost when compared with cryptography, which is more oriented towards the computational side of the network stack on the Application Layer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Since cryptography techniques provide security in different sections of the wireless protocols, PLS is proposed as a complement to them, rather than a replacement [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Through the use of both approaches on the same node, it is possible to offer high secrecy without the necessity of infinitely growing key complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The PLS has its origins on the analytical proposal of Wyner’s wiretap channel [4], where it is described a commu- nication between two legitimate nodes that is spied on by an eavesdropper through an unauthorized channel called wiretap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In the modern literature, these devices are usually referred to as a transmitter called Alice, an authorized receiver Bob, and the set of K eavesdroppers named Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In the wiretap channel model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 1, the original message m is encoded and transmitted by Alice as the signal sa, that reaches Bob through the main channel hAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The received signal yB is then decoded by Bob, obtaining the estimated message ˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Additionally, the k-th Eve can intercept sa through the wiretap channel hAE,k, obtaining the signal yE,k that when decoded produces z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The main focus of PLS is to guarantee that the mutual information between m and z is as close to zero as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' When this condition is met, even if z is know, it is impossible for Eve to infer the contents of the original message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Wyner then presents a set of parameters that enable the use of the physical imperfections of the channel, such as noise and fading, to provide information secrecy by raising the level of confusion on undesired nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Rendering them unable to distinguish between the message and the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='05123v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='SP] 12 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 1: The wiretap channel generic model based on [4] Currently, plenty of techniques to provide security at the physical layer level have been proposed in the literature [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This paper will focus on two approaches first presented in [5]: Artificial Noise (AN): This approach uses a portion of the transmitter node’s power to inject artificially gener- ated noise in the eavesdropper’s channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Cooperative Jamming (CJ): This approach expands the AN model by proposing a connected network where nearby relay nodes (Charlies) send a jamming signal to the eavesdropper’s channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' To demonstrate the viability of AN and CJ applications in a V2X network, it is common to create stochastic geometric models that randomly generate streets and distribute com- munication nodes in a predefined area to represent an urban mobility scenario [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' When implementing these methods, metrics such as the Signal-to-Interference Ratio (SIR) are used to define the threshold of confusion necessary to provide se- crecy at the physical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The SIR on each eavesdropper can then be evaluated to determine the secrecy outage probability (SOP) of the data transmission with different densities of the involved nodes in the simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In this paper, Section II describes the stochastic algorithms implemented to model a V2X network that includes streets and communication nodes (vehicular and planar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Section III presents the analytical basis of the AN and CJ techniques, while also introducing the SIR and SOP metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In Section IV, the results of numerical simulations are shown to illustrate the benefits of the considered PLS techniques on the generated V2X networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Finally, Section V states some final remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Notation: IN is an identity matrix of order N, Poisson(n) is a Poisson distribution with mean number of arrivals n, CN(m, n) is a complex normal distribution with average m and covariance n, exp(n) is an exponential distribution with mean n and Gamma(m, n) is the gamma distribution with form m and scale n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' THE V2X NETWORK MODEL As mentioned previously, vehicular networks are dynamic, with devices changing location constantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Thus, a determin- istic model is not well-suited for this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' A common alternative is the use of stochastic geometry to represent this random spatial nature through a variety of different processes to distribute the streets and communication nodes within the desired coverage area [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' A viable option is the use of Poisson processes, as they are memoryless counting processes for integer arrivals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In other words, each set of elements generated will be independent with a Poisson distributed integer number of uniformly spaced nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The intensity of the arrivals in these processes are represented by λ and the expected number of elements is the product of the said intensity and the Lebesgue measure, which in this context is essentially the spatial measurement associated with the object that the points will be distributed on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For instance, the Lebesgue measure to populate a circle is its area and for a line is the length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' One realization of the resulting spatial model derived from the use of different variations of the Poisson processes is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 2: Spatial simulation of the modeled V2X network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The color green indicates the Charlies implemented in CJ techniques and the Eves are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The planar devices are generated by PPPs represented by circles (◦) with intensity λ = 10−6/m2 for both node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Through a PLP, the streets (blue lines) have been modeled with an intensity of λl = 10−3 /m, and the vehicular devices are originated from PLP-driven Cox Processes indicated with triangles (△) of intensity u = 10−3/m for both Charlies and Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' A single Alice is indicated with a black × at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In this model, the wireless devices of pedestrians and connected infrastructure are considered free to be positioned in the whole area A of the modeled network, which is a circle of radius r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Thus, these “planar nodes“ are generated by 2-D Poisson Point Processes (PPP) and the expected amount of elements is given by Poisson(λ · A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The set of planar nodes is indicated by Φ, thus the planar Eves and Charlies are respectively represented by ΦE and ΦC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The streets are represented by uniformly distributed lines with density µl = λl/π generated by a Poisson Line Process (PLP) Φl based on the second method of the Bertrand paradox [10], in which a set of expected Poisson(µl·2πr) midpoints are created [11], each with a random radius P ∈ [0, r) and angle θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' From these coordinates, a segment perpendicular to P is traced between two points at the edge of the circle of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This effectively means that a pair of 1-D PPP points are created in the perimeter of the circular area for each modeled street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' On those PLP-generated lines, a Cox process of intensity u YB TRANSMITTER MAIN CHANNEL RECEIVER m - (ALICE) hAB m (BOB) Ye,k WIRETAP CHANNEL k-thEAVESDROPPER >Z (EVE)3000 X Alice node Planar Charlies Q Planar Eves Vehicular Charlies O A Vehicular Eves 2000 O Q 1000 O F 0 2 X O 1000 C 2000 3000 4000 3000 2000 1000 0 1000 2000 3000 4000is implemented, which is used to create the “vehicular nodes“ on each segment [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' These elements represent vehicles whose spatial distribution are constrained to a street by a 1-D PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Considering a street of length l, the number of vehicles in it is given by Poisson(u · l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The set of vehicular Eves and Charlies on each street l are respectively denoted by ψE and ψC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Based on these, the total nodes of each type can be obtained by evaluating the sets on the whole range of Φl [6], resulting in ΨE = {ψE(l)}l∈Φl for Eves and ΨC = {ψC(l)}l∈Φl for Charlies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Furthermore, a single deterministic transmitter (Alice) is included at the origin of the circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This point is selected to simplify the distance calculations between a legitimate device and the Eve nodes, which can be planar or vehicular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This measurement is one of the parameters for the SIR calculations, that are considered to determine the effectiveness of the PLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For the CJ case, auxiliary nodes (Charlies) are also modeled, some as planar and others as vehicular devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Note that the distance between Charlies and Eves influences the power of the interference injected on the unauthorized channels as part of the jamming technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' PLS TECHNIQUES The PLS techniques presented in this paper are part of the key-less-based class [2], which implements secure information transmission by making the unauthorized channel’s capacity (CE) lower than that of the legitimate channel’s (CB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This re- lationship can be presented by evaluating these values through the Shannon-Hartley theorem, which produces the secrecy capacity (CS) metric as CS = CB − CE = log2(1 + γB) − log2(1 + γE), (1) where γB and γE are, respectively, the SIRs of Bob and Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Based on this expression, it can be inferred that in order to guarantee that CB is sufficiently larger than CE, the value of γE must be as low as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The approach utilized by AN and CJ is the injection of artificially generated interference in the eavesdropper channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Typically, this injection is implemented with multi-antenna networks, as it enables the use of beamforming to selectively direct the transmission to legitimate receivers with minimum noise and high efficiency [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The unintended receivers on the other hand, intercept a signal that contains the secret message as well as AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Therefore, secrecy is provided when the distinction between them by the Eves is improbable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The wireless channels in this paper are modeled with com- plex normal distributions (CN) which implies in a Rayleigh fading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This decision provides simpler analytical equa- tions and also proposes a more pessimistic scenario, in which there is no Line-of-Sight (LoS) available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' By evaluating the metrics in these worst-case conditions, it is possible to verify that even then the secrecy can be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Artificial Noise In the AN scenario, the legitimate communication is es- tablished between a single transmitter Alice and a receiver Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Additional nodes (both planar and vehicular) that try to obtain Alice’s signal are then considered eavesdroppers and their channels will be affected by the AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The signal transmitted by the Alice node with NA antennas is composed of two terms: the first contains a message x intended for Bob and the second is based on a zero-forcing vector for the unauthorized devices [13], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='e, sa = � φPt ha ∥ha∥x + � (1 − φ)Pt NA − 1 Wana, (2) where ha/∥ha∥ is the beamforming vector with the normaliza- tion of the Alice’s channel estimation ha ∈ CNA×1, that will be modeled as CN(0, INA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The AN is formed by the null- space orthonormal basis Wa ∈ CNA×(NA−1) and the noise signal na ∈ C(NA−1)×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The distribution of the available power, Pt, between the two terms of (2) is controlled by φ ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' φ = 0 means that all power is allocated to noise generation and no message is sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Conversely, when φ = 1 the AN is not active and Pt is allocated entirely for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Cooperative Jamming The Cooperative Jamming extends the AN case, maintaining the single Alice-Bob authorized transmission with multiple Eves, however, adding auxiliary nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' These devices, typically called Charlies, can also be either planar or vehicular, just like the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In contrast, they are responsible for providing additional security by sending jamming signals that further decrease the channel quality of the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For simplicity, it is considered that only Alice will transmit messages in the scenarios evaluated in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Hence, the signals sent by the Charlie nodes are made of only the AN (zero-forcing) portion, as follows sc = � Pc NC − 1Wcnc, (3) where NC is the number of antennas of each Charlie and PC is the power available for jamming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Notice that since these nodes are not transmitting messages, all the available power is directed towards CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Additionally, Wc ∈ CNC×(NC−1) is the null space orthonormal matrix and nc ∈ C(NC−1)×1 is the artificial noise component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Received Signals By considering that the channel estimation ha is precisely the main channel established between Alice and Bob, hAB, it is implied that the receiver node is not affected by the interference from AN or CJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' That happens because the or- thonormal basis Wa and Wc are null when applied to the authorized channels, resulting in the relationships h† ABWa = 0 and h† ABWc = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Therefore, the signal received by Bob can be expressed as yB = � φPt ∥ha∥ D−α/2 AB x, (4) where DAB is the distance between the devices and α > 2 is the path loss exponent considering an NLoS scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The distances are obtained through simple trigonometry based on the coordinates randomly generated by the stochastic processes described in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For the signal intercepted by the eavesdroppers, it is eval- uated a set of K = (ΦE + ΨE) Eves, containing both planar and vehicular nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Similar considerations are adopted for the Charlies in the CJ scenario, resulting in C = (ΦC + ΨC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' As discussed when sa was presented, Alice sends a signal containing the secret information and AN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Since authorized Alice-Eves channels are not expected in the beamforming sense, the orthonormal basis are not null, thus the Eves receive interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' When the Cooperative Jamming is taken into consideration, Eves are also affected by the interference generated by the nearby Charlies through the sc signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' With that in mind, the signal obtained by the k-th Eve is given by yE,k = � φPt h† AE,k D−α/2 AE,k x + � (1 − φ)Pt NA − 1 h† AE,k Wa D−α/2 AE,k na + � c ∈C � Pc NC − 1 h† c,k Wc D−α/2 c,k nc , (5) which is composed of essentially three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The first is the intercepted secret message itself, the second term is the AN signal generated by Alice, and the third term is a sum of all the interference injected by the Charlie nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Since CJ only affects the last term of (5), the AN scenario can be obtained by simply adopting that the sum in this term is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' From (4) and (5), it is possible to determine the SIR of Bob and the K Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Thus, the SIR of Bob can be determined as γB = Ptφ ∥ha∥2 D−α AB, (6) and the SIR for each Eve can be obtained from (5) as follows γE,k = Pt φ ���h† AE,k ha/∥ha∥ ��� 2 D−α AE,k Pt (1−φ) NA−1 ���h† AE,k Wa ��� 2 D−α AE,k + Ic , (7) where Ic is the sum of the interference injected by the Charlies given by Ic = � c ∈ C Pc Nc − 1∥h† c,k Wc∥2 D−α ck , (8) which is non-zero only in the CJ scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The products h† AE,k· ha/∥ha∥ and h† AE,k ·Wa from the Alice-Eve channel and also h† ck·Wc from Charlie-Eve produce independent identically dis- tributed CN random variables with unitary variance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This enables the approximations ���h† AE,k(ha/∥ha∥) ��� 2 ∼ exp(1), ∥h† AE,kWa∥2 ∼ Gamma(NA − 1, 1) and ∥h† c,k Wc∥2 ∼ Gamma(NC − 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Performance metric Considering that Alice transmits codewords at a rate Rb with a secrecy rate RS ≤ CS, the redundancy rate can be defined as Re = Rb − RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Then a secrecy outage event occurs when the channel capacity of any Eve is higher than the redundancy rate that Alice can provide, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=', CE > Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In a multiple passive Eves scenario, whose Channel State Information (CSI) are unknown, the secrecy performance is addressed in terms of the Secrecy Outage Probability (SOP), since the only available information about the Alice-Eve channel is its statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Thus, the SOP is defined as SOP = 1 − Pr � max k∈K γE,k < β � , (9) which is the complement of the probability that the highest SIR among all Eves is less than the threshold β = 2Re − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This means that higher values of secrecy can be obtained by implementing the aforementioned PLS techniques to reduce γE,k as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' NUMERICAL RESULTS Various simulations with different parameters were per- formed to evaluate the relationship between the SOP and the decrease of the SIR for the k-th Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Since the V2X network model is randomly generated, the coordinates of each node and street change with each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' To provide more consistent results, the curves presented below are the average of multiple realizations of each simulation configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 3 illustrates the SOP for different Pt and Pc values, ranging from 10 mW (10 dBm) to 1 W (30 dBm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' As expected, when the devices have more power available for interference, the SOP is greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' However, for the AN scenario secrecy is still not guaranteed when φ grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For CJ, the SOP increases in a much slower rate due to the larger amount of nodes jamming the signal received by the Eves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' (a) Artificial Noise (b) Cooperative Jamming Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 3: SOP versus φ (25 realizations) for the AN and CJ with different available power {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='1, 1} W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' β = 0 dB, α = 3, NA = NC = 4, λE = λC = 10−6/m2 , µE = µC = 10−3/m, r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Through the simulation results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 4, it can be easily noted that as β increases the SOP decreases, because 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='6 SOP S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='2 Pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='01 W Pt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='10 W Pt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00 Φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='0 Pt = Pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='01 W Pt = Pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='10 W Pt = Pc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='6 SOP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='00the criteria for secrecy failure is becoming more selective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Furthermore, φ have an opposing effect when compared to β, suggesting that for higher threshold values to guarantee low SOP, more power needs to be allocated to interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Because of that, in applications where the devices have limited power (such as IoT and V2X), CJ is a more economic approach as long as there are sufficient nearby auxiliary nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 4: SOP versus β (50 realizations) for the AN and CJ with different power allocation ratios {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' α = 3, Pt = Pc = 20 dBm, NA = NC = 4, λE = λC = 10−6/m2 , µE = µC = 10−3/m, r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 5, it is evaluated the influence that the proportion of Charlies to Eves have on the SOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' This is achieved by implementing different values of intensities (λ and u) for the Poisson processes that generate these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The SOP grows rapidly in the AN, indicating that the available power is insufficient to guarantee secrecy with the given Eve density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' For the CJ cases, however, as the number of Charlie nodes rises, the SOP starts to reduce, making the communication viable even for higher values of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' When there are more Charlies than Eves it is shown that very little power needs to be applied in each device to provide a low SOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' CONCLUSION In this paper, a stochastic geometric approach was presented as a method to randomly generate V2X network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' The coordinates of these elements were then used to evaluate the effectiveness of PLS techniques in different realizations of vehicular networks subjected to path loss with NLoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Both AN and CJ were introduced based on the analytical signals that the involved nodes transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Next, expressions were obtained for the SIR of Bob and the k-th Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Finally, the SOP was computed to evaluate the level of information security provided by the presented PLS techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Based on numerical results, it can be concluded that PLS can provide additional security for the V2X networks with relative low power cost, specially when both the techniques are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' It is also noted that in the CJ scenario, when Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 5: SOP versus φ (25 realizations) for the AN and CJ with different λC/λE ratios {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content='5, 1, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' β = 0 dB, α = 3, Pt = Pc = 10 dBm, NA = NC = 4, λE = 10−6/m2 , µE = 10−3/m, r = 3 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' there are more Charlies in the proximity, the security increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Therefore, the urban networks are the most benefited by this technique, since it is expected a higher density of wireless devices in the same area in these environments.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Tang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' Wang, “Cooperative Jamming for Physical Layer Security Enhancement in Internet of Things”, IEEE Internet of Things Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 5, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 1, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} +page_content=' 100 10-1 SOP 10~2 10-3 10 5 0 5 10 15 β AN: Φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE4T4oBgHgl3EQfhQ0t/content/2301.05123v1.pdf'} 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However, the formalism for these ap- +proaches generally assumes features to be shared across views to be captured +coherently. We consider the problem of learning a unified representation from +partial observations, where useful features may be present in only some of the +views. We approach this through a probabilistic formalism enabling views to map +to representations with different levels of uncertainty in different components; +these views can then be integrated with one another through marginalisation over +that uncertainty. Our approach, Partial Observation Experts Modelling (POEM), +then enables us to meta-learn consistent representations from partial observations. +We evaluate our approach on an adaptation of a comprehensive few-shot learn- +ing benchmark, Meta-Dataset, and demonstrate the benefits of POEM over other +meta-learning methods at representation learning from partial observations. We +further demonstrate the utility of POEM by meta-learning to represent an environ- +ment from partial views observed by an agent exploring the environment.1 +Minimise +Distance +Maximise +Consistency +(a) Standard Contrastive +(Meta-) Learners +(b) Partial Observation +Experts Model (POEM) +Figure 1: Standard contrastive (meta-) learners minimise a relative distance between representations. +This encourages the learning of features that are consistent in all views; in the above example this +corresponds to the pattern on the bird’s wing. To better handle partial observability, where features +may be disjoint between views, we propose Partial Observation Experts Modelling (POEM). POEM +instead maximises consistency between multiple views, by utilising representation uncertainty to +learn which features of the entity are captured by a view, and then combining these representations +together by weighting features by their uncertainty via a product of experts model (Hinton, 2002). +1 +INTRODUCTION +Modern contrastive learning methods (Radford et al., 2021; Chen et al., 2020; He et al., 2020; Oord +et al., 2019), and embedding-based meta-learning methods such as Prototypical Networks (Snell +et al., 2017; Vinyals et al., 2016; Sung et al., 2018; Edwards & Storkey, 2017), learn representations +by minimizing a relative distance between representations of related items compared with unrelated +1Implementation code is available at https://github.com/AdamJelley/POEM +1 +arXiv:2301.13136v1 [cs.LG] 30 Jan 2023 + +Accepted as a conference paper at ICLR 2023 +items (Ericsson et al., 2021). However, we argue that these approaches may learn to disregard po- +tentially relevant features from views that only inform part of the representation in order to achieve +better representational consistency, as demonstrated in Figure 1. We refer to such partially informa- +tive views as partial observations. The difficulty with partial observations occurs because distances +computed between representations must include contributions from all parts of the representation +vector. If the views provided are diverse, and therefore contain partially disjoint features, their rep- +resentations may appear different to a naive distance metric. For example, two puzzle pieces may +contain different information about the whole picture. We call this the problem of integrative repre- +sentation learning, where we wish to obtain a representation that integrates different but overlapping +information from each element of a set. +In this paper, we provide a probabilistic formalism for a few-shot objective that is able to learn to +capture representations in partially observable settings. It does so by building on a product of experts +(Hinton, 2002) to utilise representation uncertainty: a high variance in a representation component +indicates that the given view of the data poorly informs the given component, while low variance +indicates it informs it well. Given multiple views of the data, the product of experts component in +POEM combines the representations, weighting by the variance, to get a maximally informative and +consistent representation from the views. +To comprehensively evaluate our approach, we adapt a large-scale few-shot learning benchmark, +Meta-Dataset (Triantafillou et al., 2020), to evaluate representation learning from partial observa- +tions. We demonstrate that our approach, Partial Observation Experts Modelling (POEM), is able +to outperform standard few-shot baselines on our adapted benchmark, Partially Observed Meta- +Dataset (PO-Meta-Dataset), while still matching state-of-the-art on the standard benchmark. Fi- +nally, we demonstrate the potential for our approach to be applied to meta-learn representations of +environments from the partial views observed by an agent exploring that environment. +The main contributions of this work are: 1) A probabilistic formalism, POEM, that enables repre- +sentation learning under partial observability; 2) Comprehensive experimental evaluation of POEM +on an adaptation of Meta-Dataset designed to evaluate representation learning under partial observ- +ability, demonstrating that this approach outperforms standard baselines in this setting while still +matching state-of-the-art on the standard fully observed benchmark; 3) A demonstration of a poten- +tial application of POEM to meta-learn representations of environments from partial observations. +2 +RELATED WORK +2.1 +CONTRASTIVE LEARNING +Contrastive learning extracts features that are present in multiple views of a data item, by encour- +aging representations of related views to be close in an embedding space (Ericsson et al., 2021). In +computer vision and natural language applications these views typically consist of different augmen- +tations of data items, which are carefully crafted to preserve semantic features, and thereby act as an +inductive bias to encourage the contrastive learner to retain these consistent features (Le-Khac et al., +2020). A challenge in this approach is to prevent representational ‘collapse’, where all views are +mapped to the same representation. Standard contrastive approaches such as Contrastive Predictive +Coding (Oord et al., 2019), MoCo (He et al., 2020), and SimCLR (Chen et al., 2020) handle this +by computing feature space distance measures relative to the distances for negative views – pairs +of views that are encouraged to be distinct in the embedding space. In this work we take a similar +approach, where the negative views are partial observations of distinct items, but we aim to learn +to unify features from differing views, not just retain the consistent features. We learn to learn a +contrastive representation from partial views. We note that state-of-the-art representation learning +approaches such as CLIP (Radford et al., 2021), which leverage contrastive learning across modali- +ties, also suffer from extracting only a limited subset of features (F¨urst et al., 2022) due to using an +embedding-based approach (Vinyals et al., 2016) to match image and text representations. +2.2 +EMBEDDING-BASED META-LEARNING +Embedding-based meta-learners similarly learn representations of classes by extracting features that +are consistently present in the data samples (generally referred to as shots in the meta-learning liter- +ature) provided for each class, such that the class of new samples can be identified with a similarity +2 + +Accepted as a conference paper at ICLR 2023 +measure (Hospedales et al., 2020). These methods generally differ in terms of their approach to +combine features, and the distance metric used. Prototypical Networks (Snell et al., 2017) use a Eu- +clidian distance between the query representation and the average over the support representations +for a class (referred to as a prototype). Relation Networks (Sung et al., 2018) use the same proto- +type representation as Prototypical Networks, but use a parameterised relation module to learn to +compute the similarity between the query and the prototype rather than using a Euclidian distance. +Matching Networks (Vinyals et al., 2016) use a Cosine distance between the query sample and each +support sample as a weighting over the support labels, and so perform few-shot classification with- +out unifying the support representations. None of these approaches are designed to unify partially +informative support samples. The approach closest to that proposed in this paper is by Edwards & +Storkey (2017), where the authors map the different views to a statistic with an associated covariance +through a variational approach. However there is no control of the contribution of each view to the +variance, and the covariance is spherical, so the approach is also unsuitable for partial observation. +2.3 +OPTIMISATION-BASED META-LEARNING +The few-shot classification task can also be solved without learning embeddings. One sensible +baseline, fine-tuning of a previously pre-trained large model, simply treats each few-shot task as a +standard classification problem (Nakamura & Harada, 2019). For each task, one or more additional +output layers are added on top of a pre-trained embedding network and trained to predict the classes +of the support set (alongside optionally finetuning the embedding network). This can then be utilised +to predict the classes of the query set. +Taking this approach a step further, Model-Agnostic Meta-Learning (MAML) (Finn et al., 2017) +learns the initialisation of the embedding network, such that it can be rapidly fine-tuned on a new +few-shot task. Given the generality of this approach, many variants of this method now exist, such +as MAML++, Antoniou et al. (2018), Meta-SGD (Li et al., 2017), CAVIA (Zintgraf et al., 2019) +and fo-Proto-MAML (Triantafillou et al., 2020). One variant, LEO (Rusu et al., 2019), performs the +meta-optimisation on a latent representation of the embedding parameters, learned using a relational +network (Sung et al., 2018). However, none of these variants of this fundamental optimisation +based approach to few-shot learning (referred to as ’MAML’ for the remainder of this work) have +a mechanism for integrating partial information from the entire support set at inference time, or for +comparison with a partial query observation. +2.4 +OTHER META-LEARNING APPROACHES +Probabilisitic meta-learning methods, such as VERSA (Gordon et al., 2019), DKT (Patacchiola +et al., 2020) and Amortised Bayesian Prototype Meta-Learning (Sun et al., 2021), often unify both +embedding-based and optimisation based meta-learning by learning to output a posterior distribution +that captures uncertainty in predictions, but do not use uncertainty in features to optimally combine +support set information. Other recent work, such as DeepEMD (Zhang et al., 2022), has considered +the use of attention mechanisms or transformers with image patches (Hiller et al., 2022; Dong et al., +2020), or augmentations (Chen et al., 2021a). However, the purpose of these approaches is to iden- +tify informative patches or features within each support example, to improve fine-grained few-shot +learning performance or interpretability where relevant features may occupy only a small region of +the samples. As far as we know, there are no existing meta-learning methods that aim to integrate +partial information from across the support set for comparison with a partially informative query. +2.5 +PARTIAL OBSERVABILITY AND PRODUCT OF EXPERTS +Factor analysis is the linear counterpart to modern representation learners, but where partial observ- +ability is inherently expressed in the model. The inferential model for the latent space in factor +analysis is a product of each of the conditional Gaussian factors. In general, this form of inferential +model can be captured as a product of experts (Hinton, 2002). When those experts are Gaussian +distributions (Williams et al., 2001), this product of experts is fully tractable. By focusing on the +inferential components rather than the linear model, it is possible to generalise factor analysis in- +ference to nonlinear mappings (Tang et al., 2012). However, when only an inferential component +is required (as with representation learning), the product of experts can be used more flexibly, as in +our approach below. +3 + +Accepted as a conference paper at ICLR 2023 +3 +THEORETICAL FORMALISM +In this section, we introduce POEM, which incorporates a product of experts model for combining +different views with a prior representation, and then uses that representation to classify a query view. +3.1 +PRODUCT OF EXPERT PROTOTYPES +Let us consider data corresponding to partial observations, or views, of a set of items. In common +with most few-shot frameworks, we arrange the data into support sets and query sets. Each support +set consists of M data items: S = {Xm|m = 1, 2, . . . , M}, where the mth item Xm collects V m +views, where V may vary with m. Let xm +v denote the vth view of the mth data item, such that +Xm = {xm +1 , xm +2 , . . . , xm +V m}. The items in the support set are sampled randomly from the training +dataset. The query point, denoted x∗, here consists of a single different view corresponding to one +and only one of the M items in the support set (although in general we may consider N query points +simultaneously). We cast our representation learning problem as a meta-learning task. We must learn +a unified representation derived from the support set that can be compared with a representation of +the query view. We want that comparison to enable us to infer which support set item m = m∗ the +query view belongs to. +In this paper we are concerned with partial observability; that is, not every data view will inform +the whole representation. So instead of mapping each view to a deterministic point representation, +we map each view to a distributional representation where each component is a normalised density +that indicates the uncertainty in that component (called a factor). We denote this conditional density +φ, and on implementation parameterise the parameters of the distribution φ with a neural network. +We combine the corresponding factors for each view together using a product of experts, which +integrates a prior distribution along with the different views such that views with low variance in a +component strongly inform that component. +For a given support set, we compute a product of experts distribution for the representation zm: +p(zm|Xm) = +p(zm) �V m +v=1 φ(zm|xm +v ) +� +dz′ p(z′) �V m +v=1 φ(z′|xm +v ) +, +(1) +where p(z) is a prior density over the latent space. Now for a query point with a view that matches +other views from e.g. data item m, we can use Bayes rule to compute the probability that the query +point would be generated from the corresponding representation zm by +p(x∗|zm) = p(x∗)φ(zm|x∗) +p(zm) +, +(2) +where, again, p(z) = +� +dx p(x)φ(z|x) is the prior. +We put Eq.2 and Eq.1 together and marginalise over zm to get the marginal predictive distribution +p(x∗|Xm) = +� +dzm +� +p(zm) �V m +v=1 φ(zm|xm +v ) +� +dz′ p(z′) �V m +v=1 φ(z′|xm +v ) +� �p(x∗)φ(zm|x∗) +p(zm) +� +(3) += p(x∗) +�� +dzm φ(zm|x∗) �V m +v=1 φ(zm|xm +v ) +� +dz′ p(z′) �V m +v=1 φ(z′|xm +v ) +� += p(x∗)λ(x∗, Xm) +λ′(Xm) +(4) +where +λ(y, X) = +� +dz φ(z|y) +V� +v=1 +φ(z|xv), +and +(5) +λ′(X) = +� +dz p(z) +V� +v=1 +φ(z|xv). +(6) +The marginal predictive p(x∗|Xm) is used to form the training objective. In our few shot task, +we wish to maximize the likelihood for the correct match of query point to support set, accumu- +lated across all support/query selections indexed with t from the dataset. This provides a complete +4 + +Accepted as a conference paper at ICLR 2023 +negative log marginal likelihood objective to be minimized, as derived in appendix A.2: +L({St}, {x∗ +t }) = − +� +t +� +log λ(x∗, Xm∗) +λ′(Xm∗) +− log +� +m +λ(x∗, Xm) +λ′(Xm) +� +(7) +Full pseudocode for training POEM with this objective is provided in appendix A.3. +3.2 +INTERPRETATION OF OBJECTIVE +While the normalised factors φ can be chosen from any distribution class, we take φ to be Gaussian +with parameterised mean and precision for the remainder of this paper, rendering the integral in +Eq. 5 analytic. Approximating the prior p(z) by a Gaussian also renders Eq. 6 analytic. 2 We note +that other distributions with analytic products, such as Beta distributions, may also be of interest in +certain applications, but we leave an investigation of other distributional forms for φ to further work. +If the representations from each view for a support point are aligned with each other and the query +view (the means of all the Gaussians are similar), they will have a greater overlap and the integral of +the resulting product of Gaussians will be larger, leading to a greater value of λ(y, X). Furthermore, +increasing the precisions for aligned Gaussian components leads to greater λ(y, X), while, up to a +limit, decreasing the precisions for non-aligned Gaussian components leads to greater λ(y, X). +While the numerator in Eq. 4, λ(y, X), quantifies the overlap of the support set with the query, +the denominator λ′(X) contrasts this with the overlap of the support set representation with the +prior. Together, this factor is enhanced if it is beneficial in overlap terms to replace the prior with +the query representation, and reduced if such a replacement is detrimental. A greater consistency +between query and combined support set representations intuitively leads to a greater probability that +the query belongs to the class of the corresponding support set, effectively extending Prototypical +Networks to a probabilistic latent representation space (Snell et al., 2017). +As a result, this objective is a generalisation of a Prototypical Network that allows for (a) learnable +weighted averaging over support examples based on their informativeness to a given component; (b) +learnable combinations of features from subsets of support examples (via differing relative preci- +sions of components within support representations), and (c) partial comparison of the query sample +with the support samples (via differing relative precisions within the query). With all precisions +fixed to 1, this approach reproduces Prototypical Networks, neglecting small differences in scaling +factors that arise with varying numbers of views. This relationship is derived in Appendix A.4. +4 +EXPERIMENTAL EVALUATION +There is a lack of established benchmarks specifically targeted at the evaluation of representation +learning under partial observability. To design a comprehensive benchmark for few-shot represen- +tation learning under partial observability, we leverage Meta-Dataset (Triantafillou et al., 2020), a +recently proposed collection of few-shot learning benchmarks. We selected Meta-Dataset as the ba- +sis for our adapted benchmark as it consists of diverse datasets involving natural, human-made and +text-based visual concepts, with a variety of fine-grained classification tasks that require learning +from varying and unbalanced numbers of samples and classes. As a result, our derived benchmark +inherits these properties to provide a robust measure of the ability of a learning approach to learn +representations from partial observations. +To extend Meta-Dataset to incorporate partial observability, we take multiple views of each sample +and divide these views into support and query sets. Our adapted few-shot classification task is to +predict which sample a query view comes from, given a selection of support views of that sample, +as demonstrated in Figure 2. +In keeping with the spirit of Meta-Dataset, we vary the number of ways in the task (now the number +of images) from 5 to 25, taken from between 1 to 5 classes. Views are generated by applying +the standard augmentation operations used in SimCLR (Chen et al., 2020) and most other self- +supervised learning methods. However, to emphasise the focus on partial observability, the size of +2In reality, p(z) is typically flat over the region of non-negligible density of the product �V +v=1 φ(z|xv) so +does not affect the value of λ′ in Eq. 6 and can be neglected, as described in appendix A.1. +5 + +Accepted as a conference paper at ICLR 2023 +Meta-Dataset +Dataset +Classes +Samples +Views +VGG Flowers +010. Globe Thistle +Meta-Dataset +Meta-Dataset: +Few-Shot +Classification +of Classes +from Samples +PO-Meta-Dataset: +Few-Shot +Classification of +Samples from Views +Figure 2: Standard few-shot learning requires the prediction of an image class from a sample. Our +adapted task evaluates representation learning under partial observability by instead requiring pre- +diction of the underlying image from partial views. Views are generated with the standard contrastive +augmentations, with stronger cropping. We call the resulting benchmark Partially Observable Meta- +Dataset (PO-Meta-Dataset). +the random crops and the number of views was fixed, such that the entire support set for a sample +contains a maximum of 50% of the image. We also maintain a constant number of query views +per sample. Viewpoint information consisting of the coordinates of the view is provided to make +it possible for learners to understand where a view fits into a representation even in the absence of +overlapping views. Full details of the definition of the task are provided in appendix A.5. +We apply our proposed evaluation procedure to all datasets included in Meta-Dataset with a few +exceptions. ILSVRC (ImageNet, Russakovsky et al. (2015)) was not included since our network +backbones were pre-trained on this dataset, including the standard few-shot test classes (which is +also why this dataset was subsequently removed from the updated benchmark, MetaDataset-v2 (Du- +moulin et al., 2021)). Traffic Signs (Stallkamp et al., 2011) and MSCOCO (Lin et al., 2015) were not +included since these datasets are fully reserved for evaluation by Meta-Dataset and so do not have a +training set specified. Quick Draw (Fernandez-Fernandez et al., 2019) was also not included since +this dataset was found to be too large to use within the memory constraints of standard RTX2080 +GPUs. This leaves six diverse datasets: Aircraft (Maji et al., 2013), Birds (Wah et al., 2011), Flow- +ers (Nilsback & Zisserman, 2008), Fungi (Schroeder, Brigit, 2018), Omniglot (Lake et al., 2015) +and Textures (Cimpoi et al., 2014), on all of which our models were trained, validated and tested on +according to the data partitions specified by the Meta-Dataset benchmark. +The resulting benchmark, Partially Observed Meta-Dataset (PO-Meta-Dataset), therefore requires +that the learner coherently combine the information from the support views into a consistent repre- +sentation of the sample, such that the query view can be matched to the sample it originated from. +Since a maximum of 50% of each sample is seen in the support set, the task also requires generali- +sation to correctly match and classify query views. +4.1 +IMPLEMENTATION DETAILS +We utilise a re-implementation of Meta-Dataset benchmarking in PyTorch (Paszke et al., 2019) +which closely replicates the Meta-Dataset sampling procedure of uniformly sampling classes, fol- +lowed by a balanced query set (since all classes are considered equally important) and unbalanced +support sets (to mirror realistic variations in the appearances of classes). The experimental imple- +mentation, including full open-source code and data will be available on publication. +Following the MD-Transfer procedure used in Meta-Dataset, we leverage a single ResNet-18 (He +et al., 2015) classifier pre-trained on ImageNet (Russakovsky et al., 2015) at 126 × 126 resolution. +Since both a mean and precision must be learned to fully specify the model φv(z|xn +v), we add two +simple 3-layer MLP heads onto this backbone for POEM, each maintaining an embedding size of +6 + +0. +30 +40 +50 +8 - +0 +40Accepted as a conference paper at ICLR 2023 +512. For fair comparison, we also add the same 3-layer MLP head onto the backbone for the base- +lines. Using a larger embedding for the baselines was not found to be beneficial. During training, +gradients are backpropagated through the entire network such that both the randomly initialised +heads and pre-trained backbones are learned/fine-tuned. +We use a representative selection of meta-learning baselines utilised by Meta-Dataset for our re- +implementation. This includes a strong naive baseline (Finetuning, Nakamura & Harada (2019)), an +embedding-based approach (Prototypical Network, Snell et al. (2017)) and an optimisation-based +approach (MAML, Finn et al. (2017)), all modernised to use the ResNet-18 backbone as described +above. Recent competitions, such as the NeurIPS 2021 MetaDL Challenge (Baz et al., 2022; 2021), +have demonstrated that these fundamental approaches, updated to use modern pre-trained backbones +that are finetuned on the meta-task (exactly as in our experiments below) are still generally state- +of-the-art for novel datasets (Chen et al., 2021b), and so form strong baselines. In addition, our +re-implementation enables us to ensure that all learners are optimised for Meta-Dataset and that +comparisons between learners are fair, utilising the same benchmark parameters, model architectures +and where applicable, hyperparameters. Crucially, given the close connection between POEM and +Prototypical Networks, we ensure that all hyperparameters, including learning rates, scheduling and +architectures are identical for both methods. +4.2 +RESULTS +Our results on this novel representation learning benchmark, PO-Meta-Dataset, are given in table 1. +Test Source +Finetune +ProtoNet +MAML +POEM +Aircraft +46.5 ± 0.6 +48.5 ± 1.0 +37.5 ± 0.3 +55.3 ± 0.7 +Birds +62.6 ± 0.7 +67.4 ± 1.2 +52.5 ± 0.6 +71.1 ± 0.1 +Flowers +48.5 ± 0.4 +46.4 ± 0.7 +33.5 ± 0.3 +49.2 ± 1.5 +Fungi +61.0 ± 0.2 +61.4 ± 0.4 +46.1 ± 0.4 +64.8 ± 0.3 +Omniglot +71.3 ± 0.1 +87.8 ± 0.1 +47.4 ± 1.0 +89.2 ± 0.7 +Textures +83.2 ± 0.4 +76.7 ± 1.6 +73.1 ± 0.4 +81.4 ± 0.6 +Table 1: Few-shot classification accuracies on our adapted Meta-Dataset benchmark, PO-Meta- +Dataset. All learners use a ResNet-18 model pre-trained on ImageNet, with MLP heads to incorpo- +rate view information. POEM outperforms the baselines across the range of datasets, demonstrating +the benefits of the approach to learn and match representations from partial observations. +The results show that POEM outperforms the baselines at identifying views of images across a +diverse range of datasets, demonstrating the benefits of the approach to learn and match representa- +tions from partial observations. The only exception is the Textures dataset, for which the finetuning +baseline performs particularly strongly. We hypothesise that this is because the images in the Tex- +tures dataset are relatively uniform compared to the other datasets, so capturing the relative location +of views is less important than identifying very fine grained features that distinguish the samples, +which optimisation-based approaches are particularly effective at. +4.3 +ABLATION: META-DATASET +To demonstrate that the observed benefit of POEM over the baselines is due to the requirement of +the task to learn coherent representations from partial observations, we also evaluate our approach +against the baselines on the established Meta-Dataset benchmark. We now follow the standard few- +shot learning procedure as described in the paper (Triantafillou et al., 2020), but keep all learners +identical to those used in the evaluation above. +Our results on the standard Meta-Dataset benchmark are provided in table 2. As expected, we +find that POEM performs comparably with the baselines. Although Meta-Dataset provides realistic +few-shot learning tasks in terms of diversity of visual concepts, fine-grained classes and variable +shots and ways, each sample generally contains complete information including all relevant features +for the visual concept in question. Correctly classifying query samples does not generally require +any unification of views from support examples, but simply the identification of common features. +As a result, we see that the additional capacity of POEM to learn to weight support examples and +7 + +Accepted as a conference paper at ICLR 2023 +Test Source +Finetune +ProtoNet +MAML +POEM +Aircraft +56.2 ± 1.1 +47.2 ± 1.2 +35.9 ± 1.8 +46.5 ± 1.5 +Birds +52.6 ± 1.8 +78.3 ± 0.5 +65.2 ± 0.3 +79.4 ± 0.3 +Flowers +80.1 ± 2.0 +84.2 ± 0.7 +70.4 ± 0.4 +83.6 ± 1.3 +Fungi +33.6 ± 1.7 +84.7 ± 0.2 +18.9 ± 0.2 +81.0 ± 0.1 +Omniglot +89.6 ± 3.3 +98.7 ± 0.1 +94.7 ± 0.1 +98.6 ± 0.1 +Textures +60.4 ± 1.0 +65.3 ± 1.2 +56.1 ± 0.3 +65.7 ± 0.8 +Table 2: Few-shot classification accuracies on Meta-Dataset, all using a ResNet-18 backbone pre- +trained on ImageNet, with a 3 layer MLP head. POEM is comparable with the baselines. +combine partial features does not provide a significant performance improvement over the baselines +at few-shot classification in this fully observable benchmark. +In support of our hypothesis that feature uncertainty is not useful on this benchmark, we find that +the variance in the precisions relative to the means output by the POEM model generally decreases +during training and becomes negligible for all datasets, indicating that the precisions are not be- +ing utilised to improve performance and that the POEM objective is reducing to the Prototypical +Network objective, as discussed in section 3.2. This is further evidenced by the very similar per- +formances of POEM and the Prototypical Network across the entire set of datasets. However, on +PO-Meta-Dataset, we find that the relative variance in the precisions to the means is much larger +on convergence, which leads to the improved performance of POEM over the Prototypical Network +observed in Table 1. This is shown in appendix A.6. +5 +DEMONSTRATION OF LEARNING REPRESENTATIONS OF ENVIRONMENTS +We now apply POEM to the equivalent task of learning a representation of an environment from the +partial observations collected by an agent exploring that environment. +To do so, we utilise the 2D gridworld environment, MiniGrid (Chevalier-Boisvert et al., 2018). We +consider the 11 × 11 Simple Crossing environment, which consists of a procedurally generated +maze where the agent is required to traverse from the top left corner to the goal in the bottom right +corner. The MiniGrid environment provides an agent-centric viewpoint at each step in the trajectory, +consisting of a 7×7 window of the environment in front of the agent, taking into account the agent’s +current direction and where the line of sight is blocked by walls. +5.1 +META-LEARNING ENVIRONMENT REPRESENTATIONS VIA FEW-SHOT CLASSIFICATION +To generate few-shot episodes, we utilise two agents: an optimal agent that takes the optimal trajec- +tory from the start to the goal, and an exploratory agent that is incentivised to explore all possible +views in the environment. The support set for each environment is generated by running the optimal +agent in the environment and collecting the partial observations of this agent at each step during its +trajectory. The query set is similarly generated by running the exploratory agent in the environment, +filtering out any observations that are contained within the support set, and then randomly sampling +the desired number of queries from the remaining observations. +We generate these few-shot episodes dynamically, and train POEM to combine the support samples +(partial observations from the optimal trajectory) into a representation of the environment, such that +it can classify which environment a novel query observation has been collected from. A set of +sample environments and observations from those environments are shown in figures 3 and 4. +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 3: Sample environments. +3 +9 +5 +7 +7 +Figure 4: Sample queries labelled with targets correspond- +ing to the environment which they were observed in. +8 + +Accepted as a conference paper at ICLR 2023 +All observations are provided as pixels to a standard convolutional backbone, with the corresponding +agent location and direction appended to this representation and passed through an MLP head, equiv- +alent to the procedure utilised for the adapted Meta-Dataset experiments. As a baseline comparison, +we also train a Prototypical Network with an identical architecture on this task. Additionally, we +train an equivalent recurrent network architecture typically applied to POMDP tasks such as this +(Hausknecht & Stone, 2017), by adding a GRU layer (Cho et al., 2014; Chung et al., 2014) where +the hidden state of the GRU is updated at each timestep and then extracted as the unified represen- +tation of the agent. We find that POEM trains more quickly and reaches almost 10% higher final +environment recognition performance than both the Prototypical Network and GRU-based approach +over 100 test episodes (81.1% vs 72.4% and 72.1%), as shown in appendix A.8. This is a result of +POEM’s capacity to associate each observation with only part of the representation. +5.2 +RECONSTRUCTING ENVIRONMENTS FROM PARTIAL OBSERVATION TRAJECTORIES +Having learned an environment encoder using the few-shot learning procedure above, we now in- +vestigate the extent to which our representations can be used to reconstruct the environment. As +above, we generate trajectories with the optimal agent and feed these through the encoder to gener- +ate a representation of the environment. An MLP decoder is then trained to reconstruct the original +environment layout from the learned environment representation. The decoder attempts to predict a +one-hot representation of each possible grid cell, with a mean squared error loss. Given the trained +encoder and decoder, we are now able to generate a map of the environment the optimal agent has +traversed, solely from the agent’s partial observations, and without ever having seen the environment +as a whole. A sample of environments alongside their reconstructions are shown in figure 5. +Figure 5: Left: Ground truth environments explored by the agent. Right: Reconstructions of the +corresponding environments from POEM’s unified representation, encoded from the partial obser- +vations of the agent. +We see that the reconstructions clearly capture the approximate structure of each environment, +demonstrating that the agent has been able to integrate its observations from along its trajectory +into a single consistent representation. Since POEM enables the representation to be updated incre- +mentally with each partial observation of the environment at inference time, it would be possible +for an agent to update an internal environment representation at each step in its trajectory. There is +potential for utilising this approach for learning environment representations to be beneficial in the +context of exploration for reinforcement learning, but we leave such an investigation to future work. +6 +CONCLUSION +In this work, we have introduced Partial Observation Experts Modelling (POEM), a contrastive +meta-learning approach for few-shot learning in partially-observable settings. Unlike other stan- +dard contrastive and embedding-based meta-learning approaches, POEM utilises representational +uncertainty to enable observations to inform only part of a representation vector. This probabilis- +tic formalism enables consistent representation learning from multiple observations with a few-shot +learning objective. We have demonstrated that POEM is comparable to the state-of-the-art base- +lines on a comprehensive few-shot learning benchmark, and outperforms these baselines when this +benchmark is adapted to evaluate representation learning from partial observations. We have also +demonstrated a promising potential application for POEM to learn representations of an environ- +ment from an agent’s partial observations. We hope that this research inspires further work into +the challenging task of learning representations under partial observability and the creation of more +realistic partial observability benchmarks. +9 + +Accepted as a conference paper at ICLR 2023 +ACKNOWLEDGMENTS +Adam Jelley was kindly supported by Microsoft Research and EPSRC through Microsoft’s PhD +Scholarship Programme. Antreas Antoniou was supported by a Huawei DDMPLab Innovation Re- +search Grant. The Meta-Dataset experiments in this work were partly funded by Google Research +Compute Credits, and we thank Hugo Larochelle for his support in acquiring these compute credits. +REFERENCES +Antreas Antoniou, Harrison Edwards, and Amos Storkey. How to train your MAML. October 2018. +doi: 10.48550/arXiv.1810.09502. 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URL http: +//arxiv.org/abs/1810.03642. arXiv: 1810.03642. +A +APPENDIX +A.1 +GAUSSIAN PRODUCT RULES +Assuming the latent variable model φ(z|x) to be a diagonal covariance multivariate Gaussian, the +resulting integrals over latent variables become integrals over Gaussian products. This allows both +λ(y, X) (Equation 5) and λ′(X) (Equation 6) in the marginal predictive distribution (Equation 4) to +be evaluated analytically using the following univariate Gaussian product rules on each independent +dimension (Roweis, Sam, 1999). +Since a product of Gaussians � +i N(µi, τ −1 +i +) is itself a Gaussian, we have � +i N(µi, τ −1 +i +) = +SN(µ, τ −1), where +τ = +� +i +τi +(8) +µ = 1 +τ +� +i +τiµi +(9) +S = (2π) +(1−n) +2 +� +i τ 1/2 +i +τ 1/2 +exp +� +1 +2τµ2 − 1 +2 +� +i +τiµ2 +i +� +. +(10) +Therefore the integral of a Gaussian product is given by the resulting normalisation S. +In the case of evaluating the marginal predictive distribution p(x∗|Xm) (equation 4), this gives +S∗S +S′S = S∗ +S′ where S is the normalisation constant of the support product, S∗ is the normalisation +13 + +Accepted as a conference paper at ICLR 2023 +constant of the product of the query and normalised support product, and S′ is the normalisation +constant of the product of the prior p(z) and normalised support product. In reality, p(z) generally +has little impact as it is typically flat (τ → 0) over the region of non-negligible density of the +product �V +v=1 φ(z|xv) and so S′ ≈ 1 and we find S∗ +S′ ≈ S∗ so the ratios λ +λ′ in the objective can be +approximated by S∗, as in the simplified pseduocode in appendix A.3. +A.2 +DERIVATION OF OBJECTIVE FROM MARGINAL PREDICTIVE DISTRIBUTION +In section 3, we derived the marginal predictive distribution: +p(x∗|Xm) = p(x∗)λ(x∗, Xm) +λ′(Xm) +(11) +where +λ(y, X) = +� +dz φ(z|y) +dim(X) +� +v=1 +φ(z|xv), +and +(12) +λ′(X) = +� +dz p(z) +dim(X) +� +v=1 +φ(z|xv). +(13) +In our few shot task, the support data is chosen and then the query view is chosen uniformly at +random to match the views of one of the support data items. Let the hypothesis Hm indicate the +event that the query view x∗ comes from support point m. Then +P(Hm|S, x∗) = +P(Hm)P(S, x∗|Hm) +� +m′ P(Hm′)P(S, x∗|Hm′) = +(1/M)p(x∗|S, Hm) +� +m′(1/M)p(x∗|S, Hm′) +(14) += +(1/M)p(x∗|Xm) +� +m′(1/M)p(x∗|Xm′) = +p(x∗|Xm) +� +m′ p(x∗|Xm′). +(15) +From this we can formulate the training task: we wish to maximize the likelihood for the correct +match of query point to support set, accumulated across all support/query selections from the dataset. +Denote the tth support set by St, the tth query point by x∗ +t , and let mt denote the support point with +views that match the view of the query point. Then the complete negative log marginal likelihood +objective to be minimized is: +L({St}, {x∗ +t }) = − +� +t +log P(Hmt|St, x∗ +t ) +(16) += − +� +t +log +p(x∗|Xm) +� +m′ p(x∗|Xm′) +(17) += − +� +t +� +log λ(x∗, Xm∗) +λ′(Xm∗) +− log +� +m +λ(x∗, Xm) +λ′(Xm) +� +(18) +14 + +Accepted as a conference paper at ICLR 2023 +A.3 +PSEUDOCODE +Algorithm 1 Pytorch-Style Pseudocode: Gaussian Partial Observation Experts Modelling +# phi: dual-headed encoder network with shared backbone and output heads for mean and +precision of Gaussian embedding +# M: Number of items/classes in task +# V: Number of views of each item/class (in general can vary with m in range(M)) +# N: Number of query views +# D: Embedding dimension +# Load augmented partial views with view information +for (support_views, query_views, query_targets) in loader: +# support_views.shape = (M, V, ...) +# query_views.shape=(N, ...) +# query_targets.shape = (N,) +# Encode each support and query views +support_means, support_precisions = phi(support_views) # (M, V, D) +query_means, query_precisions = phi(query_views) # (N, D) +# Combine support views into unified representation of each item +# Gaussian products computed using equations in appendix A.1 +# Optionally include prior Gaussian here (neglected for simplified implementation) +environment_means, environment_precisions, log_environment_normalisation = +inner_gaussian_product(support_means, support_precisions) # Outputs: (M, D) +# Combine each query view with each unified support representation +env_query_mean, env_query_precisions, log_env_query_normalisation = +outer_gaussian_product(support_means, support_precisions, query_means, +query_precisions) # Outputs: (N, M, D) +# Predictions correspond to unified support with maximum overlap with query +_, predictions = log_env_query_normalisation.sum(2).max(1) # (N,) +# Cross entropy loss normalises with softmax and computes negative log-likelihood +loss = F.cross_entropy(log_env_query_normalisation, query_targets, reduction=’mean’) +# Optimization step +loss.backwards() +optimizer.step() +Algorithm 2 Language Agnostic Pseudocode: Gaussian Partial Observation Experts Modelling +Require: Training meta-set Dtrain ∈ T +Require: Learning rate α +1: Initialise dual-headed network φθ(z|x) +2: +▷ Heads correspond to mean µ and precision τ of Gaussian embedding z +3: while not converged do +4: +Sample task instance Ti = (X, x∗) ∼ Dtrain +5: +▷ Support set X consists of V m views of item m ∈ {1, ..., M}. +6: +▷ Query set x∗ consists of N queries, each one view from any one item. +7: +Encode each view in support set X into Gaussian z using φ(z|X) +8: +Encode each query view in x∗ into Gaussian z∗ using φ(z∗|x∗) +9: +for m ∈ {1, ..., M} do +10: +Compute Gaussian product over views �V m +v=1 φ(z|xm +v ) (using results in A.1) +11: +▷ This gives unified support representation (global environment representation) +12: +for n ∈ {1, ..., N} do +13: +Compute Gaussian product of query with support product φ(z∗ +n|x∗ +n) �V m +v=1 φ(z|xm +v ) +14: +end for +15: +end for +16: +Normalise resulting query-support normalisation constants Sm +n = +Sm +n +� +m Sm +n across items +17: +Compute negative log of Sm∗ +n +for correct support as loss L({Dt}, {x∗ +t }) (eq. 7) +18: +▷ Negative log likelihood for correct support +19: +Perform gradient step w.r.t. θ: θ ← φ − α∇θL({Dt}, {x∗ +t }) +20: end while +15 + +Accepted as a conference paper at ICLR 2023 +A.4 +EQUIVALENCE OF PROTOTYPICAL NETWORK OBJECTIVE TO POEM OBJECTIVE WITH +FIXED PRECISIONS +The probability of a query x∗ belonging to class n using the POEM objective is given by: +P(Hm|S, x∗) = λ(x∗; Xn) +λ′(Xn) +� � +m +λ(x∗; Xm) +λ′(Xm) +(19) +as defined in equation 15, where +λ(y, X) = +� +dz φ(z|y) +V� +v=1 +φ(z|xv), +and +(20) +λ′(X) = +� +dz p(z) +V� +v=1 +φ(z|xv). +(21) +. +Taking the precisions of the all Gaussian factors φ in λ and λ′ to be 1, we can apply the Gaussian +product rules given in appendix A.1 to calculate λ and λ′ analytically. We find that this gives: +pn = +Vn +Vn+1 +1 +2 exp +� +− +Vn +2(Vn+1) +� +µ − +� +i µni +Vn +�2� +� +m +Vm +Vm+1 +1 +2 exp +� +− +Vm +2(Vm+1) +� +µ − +� +i µmi +Vm +�2� +(22) +where µ is the representation mean of the query, and µni is the representation mean of support +sample i for class n, and Vn is the number of support samples for class n. +Equivalently, the probability of a query with representation vector µ belonging to a class n using a +Prototypical Network objective is given by: +pn = +exp +� +− +� +µ − +� +i µni +Vn +�2� +� +m exp +� +− +� +µ − +� +i µmi +Vm +�2� +(23) +We find that these are equivalent aside from the scaling factors +Vm +(2)(Vm+1) which only have a (sig- +nificant) effect when there are varying numbers of samples by class, and a greater effect when the +number of samples is smaller. Experimentally, we find that these scaling factors make little differ- +ence, as demonstrated in table 2 of section 4.3. +16 + +Accepted as a conference paper at ICLR 2023 +A.5 +PO-META-DATASET BENCHMARK ADDITIONAL DETAILS +Parameters used for adapted PO-Meta-Dataset are provided in Table A.5. All parameters not listed +chosen to match Meta-Dataset defaults. All augmentations are applied using Torchvision, with +parameters specified. +Table 3: PO-Meta-Dataset Parameters +PARAMETER +VALUE +Classes per Task +[1, 5] +Samples per Task +[5, 25] +Support Views per Sample +18 +Query Views per Sample +2 +Image Size +(84, 84) (except Omniglot, (28, 28)) +Crop Size +(14, 14) (1/6 in each dim, except Omniglot, random placement) +Color Jitter +(0.8, 0.8, 0.8, 0.2), p(apply) = 0.3 +Random Greyscale +0.2 +Random Horizontal Flip +0.5 +Gaussian Blur +((3, 3), (1, 0, 2.0)), p(apply) = 0.2 +All results computed over three runs. The Finetuning, Prototypical Network and POEM baselines +were run on on-premise RTX2080 GPUs. MAML required more memory and compute than avail- +able, so was run on cloud A100s. +A.6 +RELATIVE VARIANCE OF PRECISIONS DURING TRAINING ON META-DATASET AND +META-META-DATASET +The plot below shows the evolution of the variance in the representation precisions relative to the +variance in the representation means learned by POEM on two distinct datasets, Aircraft and VGG +Flowers. We see that for standard few-shot learning on Meta-Dataset, the variance in precisions is +negligible relative to the variance in the means, demonstrating that the representational uncertainty +is not useful in this task. Meanwhile, we see the variance in the precisions relative to the variance in +the means becoming large before converging to a value of O(100) on the Meta-Meta-Dataset task, +demonstrating that learning relative precisions is useful in this setting since each support sample +only informs part of the representation. +17 + +Relative Variance in Precisions to Variance in Means for POEM Representation During Training +Var(o) / +Var(μu) +300 +250 +200 +150 +100 +50 +10k +20k +30k +40k +Training Step + Meta-Dataset: +Aircraft +- Po-Meta-Dataset: Aircraft +- Po-Meta-Dataset: Flowers +- Meta-Dataset: FlowersAccepted as a conference paper at ICLR 2023 +A.7 +LEARNING REPRESENTATIONS OF ENVIRONMENTS ADDITIONAL DETAILS +Additional details about the parameters used for learning environment representations from agent +observations are provided in Table 4 +Table 4: Environment Representation Learning Parameters +PARAMETER +VALUE +Agent Training Algorithm +PPO (Schulman et al., 2017) (default hyperparameters) +Optimal Agent Reward +1 for reaching goal, -0.01 per timestep +Exploratory Agent Reward +1/N count exploration bonus (state defined by agent location and direction) +Encoder Conv Backbone Layers +5 +Encoder MLP Head Layers +3 +Encoder Embedding Dim +128 (corresponding ∼ 11 × 11 environment size) +Decoder MLP Layers +4 +A.8 +ENVIRONMENT RECOGNITION ACCURACY DURING TRAINING +POEM trains more quickly on the environment recognition task and reaches a higher final perfor- +mance than an equivalent Prototypical Network or Recurrent Network (GRU) (81.1% vs 72.4% and +72.1% ) over a subsequent 100 test episodes. +0 +1000 +2000 +3000 +4000 +5000 +Episodes +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Accuracy +Environment Recognition Accuracy +POEM +Prototypical Network +Recurrent Network (GRU) +18 + diff --git a/QtFPT4oBgHgl3EQfpDVG/content/tmp_files/load_file.txt b/QtFPT4oBgHgl3EQfpDVG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..43ebb3a322b6838c82fdd879a4380f2e2c80980d --- /dev/null +++ b/QtFPT4oBgHgl3EQfpDVG/content/tmp_files/load_file.txt @@ -0,0 +1,913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf,len=912 +page_content='Accepted as a conference paper at ICLR 2023 CONTRASTIVE META-LEARNING FOR PARTIALLY OBSERVABLE FEW-SHOT LEARNING Adam Jelley1, Amos Storkey1, Antreas Antoniou1, Sam Devlin2 1School of Informatics, University of Edinburgh, 2 Microsoft Research Cambridge ABSTRACT Many contrastive and meta-learning approaches learn representations by identify- ing common features in multiple views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, the formalism for these ap- proaches generally assumes features to be shared across views to be captured coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We consider the problem of learning a unified representation from partial observations, where useful features may be present in only some of the views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We approach this through a probabilistic formalism enabling views to map to representations with different levels of uncertainty in different components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' these views can then be integrated with one another through marginalisation over that uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Our approach, Partial Observation Experts Modelling (POEM), then enables us to meta-learn consistent representations from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We evaluate our approach on an adaptation of a comprehensive few-shot learn- ing benchmark, Meta-Dataset, and demonstrate the benefits of POEM over other meta-learning methods at representation learning from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We further demonstrate the utility of POEM by meta-learning to represent an environ- ment from partial views observed by an agent exploring the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 Minimise Distance Maximise Consistency (a) Standard Contrastive (Meta-) Learners (b) Partial Observation Experts Model (POEM) Figure 1: Standard contrastive (meta-) learners minimise a relative distance between representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This encourages the learning of features that are consistent in all views;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' in the above example this corresponds to the pattern on the bird’s wing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' To better handle partial observability, where features may be disjoint between views, we propose Partial Observation Experts Modelling (POEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' POEM instead maximises consistency between multiple views, by utilising representation uncertainty to learn which features of the entity are captured by a view, and then combining these representations together by weighting features by their uncertainty via a product of experts model (Hinton, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 1 INTRODUCTION Modern contrastive learning methods (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019), and embedding-based meta-learning methods such as Prototypical Networks (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Edwards & Storkey, 2017), learn representations by minimizing a relative distance between representations of related items compared with unrelated 1Implementation code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='com/AdamJelley/POEM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='13136v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='LG] 30 Jan 2023 Accepted as a conference paper at ICLR 2023 items (Ericsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, we argue that these approaches may learn to disregard po- tentially relevant features from views that only inform part of the representation in order to achieve better representational consistency, as demonstrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We refer to such partially informa- tive views as partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The difficulty with partial observations occurs because distances computed between representations must include contributions from all parts of the representation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' If the views provided are diverse, and therefore contain partially disjoint features, their rep- resentations may appear different to a naive distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' For example, two puzzle pieces may contain different information about the whole picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We call this the problem of integrative repre- sentation learning, where we wish to obtain a representation that integrates different but overlapping information from each element of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In this paper, we provide a probabilistic formalism for a few-shot objective that is able to learn to capture representations in partially observable settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' It does so by building on a product of experts (Hinton, 2002) to utilise representation uncertainty: a high variance in a representation component indicates that the given view of the data poorly informs the given component, while low variance indicates it informs it well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Given multiple views of the data, the product of experts component in POEM combines the representations, weighting by the variance, to get a maximally informative and consistent representation from the views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' To comprehensively evaluate our approach, we adapt a large-scale few-shot learning benchmark, Meta-Dataset (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020), to evaluate representation learning from partial observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We demonstrate that our approach, Partial Observation Experts Modelling (POEM), is able to outperform standard few-shot baselines on our adapted benchmark, Partially Observed Meta- Dataset (PO-Meta-Dataset), while still matching state-of-the-art on the standard benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Fi- nally, we demonstrate the potential for our approach to be applied to meta-learn representations of environments from the partial views observed by an agent exploring that environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The main contributions of this work are: 1) A probabilistic formalism, POEM, that enables repre- sentation learning under partial observability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2) Comprehensive experimental evaluation of POEM on an adaptation of Meta-Dataset designed to evaluate representation learning under partial observ- ability, demonstrating that this approach outperforms standard baselines in this setting while still matching state-of-the-art on the standard fully observed benchmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 3) A demonstration of a poten- tial application of POEM to meta-learn representations of environments from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 CONTRASTIVE LEARNING Contrastive learning extracts features that are present in multiple views of a data item, by encour- aging representations of related views to be close in an embedding space (Ericsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In computer vision and natural language applications these views typically consist of different augmen- tations of data items, which are carefully crafted to preserve semantic features, and thereby act as an inductive bias to encourage the contrastive learner to retain these consistent features (Le-Khac et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A challenge in this approach is to prevent representational ‘collapse’, where all views are mapped to the same representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Standard contrastive approaches such as Contrastive Predictive Coding (Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019), MoCo (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020), and SimCLR (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020) handle this by computing feature space distance measures relative to the distances for negative views – pairs of views that are encouraged to be distinct in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In this work we take a similar approach, where the negative views are partial observations of distinct items, but we aim to learn to unify features from differing views, not just retain the consistent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We learn to learn a contrastive representation from partial views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We note that state-of-the-art representation learning approaches such as CLIP (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021), which leverage contrastive learning across modali- ties, also suffer from extracting only a limited subset of features (F¨urst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2022) due to using an embedding-based approach (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2016) to match image and text representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 EMBEDDING-BASED META-LEARNING Embedding-based meta-learners similarly learn representations of classes by extracting features that are consistently present in the data samples (generally referred to as shots in the meta-learning liter- ature) provided for each class, such that the class of new samples can be identified with a similarity 2 Accepted as a conference paper at ICLR 2023 measure (Hospedales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' These methods generally differ in terms of their approach to combine features, and the distance metric used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Prototypical Networks (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017) use a Eu- clidian distance between the query representation and the average over the support representations for a class (referred to as a prototype).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Relation Networks (Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2018) use the same proto- type representation as Prototypical Networks, but use a parameterised relation module to learn to compute the similarity between the query and the prototype rather than using a Euclidian distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Matching Networks (Vinyals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2016) use a Cosine distance between the query sample and each support sample as a weighting over the support labels, and so perform few-shot classification with- out unifying the support representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' None of these approaches are designed to unify partially informative support samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The approach closest to that proposed in this paper is by Edwards & Storkey (2017), where the authors map the different views to a statistic with an associated covariance through a variational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However there is no control of the contribution of each view to the variance, and the covariance is spherical, so the approach is also unsuitable for partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 OPTIMISATION-BASED META-LEARNING The few-shot classification task can also be solved without learning embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' One sensible baseline, fine-tuning of a previously pre-trained large model, simply treats each few-shot task as a standard classification problem (Nakamura & Harada, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' For each task, one or more additional output layers are added on top of a pre-trained embedding network and trained to predict the classes of the support set (alongside optionally finetuning the embedding network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This can then be utilised to predict the classes of the query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Taking this approach a step further, Model-Agnostic Meta-Learning (MAML) (Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017) learns the initialisation of the embedding network, such that it can be rapidly fine-tuned on a new few-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Given the generality of this approach, many variants of this method now exist, such as MAML++, Antoniou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (2018), Meta-SGD (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017), CAVIA (Zintgraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019) and fo-Proto-MAML (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' One variant, LEO (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019), performs the meta-optimisation on a latent representation of the embedding parameters, learned using a relational network (Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, none of these variants of this fundamental optimisation based approach to few-shot learning (referred to as ’MAML’ for the remainder of this work) have a mechanism for integrating partial information from the entire support set at inference time, or for comparison with a partial query observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 OTHER META-LEARNING APPROACHES Probabilisitic meta-learning methods, such as VERSA (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019), DKT (Patacchiola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020) and Amortised Bayesian Prototype Meta-Learning (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021), often unify both embedding-based and optimisation based meta-learning by learning to output a posterior distribution that captures uncertainty in predictions, but do not use uncertainty in features to optimally combine support set information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Other recent work, such as DeepEMD (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2022), has considered the use of attention mechanisms or transformers with image patches (Hiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020), or augmentations (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, the purpose of these approaches is to iden- tify informative patches or features within each support example, to improve fine-grained few-shot learning performance or interpretability where relevant features may occupy only a small region of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As far as we know, there are no existing meta-learning methods that aim to integrate partial information from across the support set for comparison with a partially informative query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 PARTIAL OBSERVABILITY AND PRODUCT OF EXPERTS Factor analysis is the linear counterpart to modern representation learners, but where partial observ- ability is inherently expressed in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The inferential model for the latent space in factor analysis is a product of each of the conditional Gaussian factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In general, this form of inferential model can be captured as a product of experts (Hinton, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' When those experts are Gaussian distributions (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2001), this product of experts is fully tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' By focusing on the inferential components rather than the linear model, it is possible to generalise factor analysis in- ference to nonlinear mappings (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, when only an inferential component is required (as with representation learning), the product of experts can be used more flexibly, as in our approach below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 3 Accepted as a conference paper at ICLR 2023 3 THEORETICAL FORMALISM In this section, we introduce POEM, which incorporates a product of experts model for combining different views with a prior representation, and then uses that representation to classify a query view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 PRODUCT OF EXPERT PROTOTYPES Let us consider data corresponding to partial observations, or views, of a set of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In common with most few-shot frameworks, we arrange the data into support sets and query sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Each support set consists of M data items: S = {Xm|m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' , M}, where the mth item Xm collects V m views, where V may vary with m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Let xm v denote the vth view of the mth data item, such that Xm = {xm 1 , xm 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' , xm V m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The items in the support set are sampled randomly from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The query point, denoted x∗, here consists of a single different view corresponding to one and only one of the M items in the support set (although in general we may consider N query points simultaneously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We cast our representation learning problem as a meta-learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We must learn a unified representation derived from the support set that can be compared with a representation of the query view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We want that comparison to enable us to infer which support set item m = m∗ the query view belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In this paper we are concerned with partial observability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' that is, not every data view will inform the whole representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' So instead of mapping each view to a deterministic point representation, we map each view to a distributional representation where each component is a normalised density that indicates the uncertainty in that component (called a factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We denote this conditional density φ, and on implementation parameterise the parameters of the distribution φ with a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We combine the corresponding factors for each view together using a product of experts, which integrates a prior distribution along with the different views such that views with low variance in a component strongly inform that component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' For a given support set, we compute a product of experts distribution for the representation zm: p(zm|Xm) = p(zm) �V m v=1 φ(zm|xm v ) � dz′ p(z′) �V m v=1 φ(z′|xm v ) , (1) where p(z) is a prior density over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Now for a query point with a view that matches other views from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' data item m, we can use Bayes rule to compute the probability that the query point would be generated from the corresponding representation zm by p(x∗|zm) = p(x∗)φ(zm|x∗) p(zm) , (2) where, again, p(z) = � dx p(x)φ(z|x) is the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We put Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 together and marginalise over zm to get the marginal predictive distribution p(x∗|Xm) = � dzm � p(zm) �V m v=1 φ(zm|xm v ) � dz′ p(z′) �V m v=1 φ(z′|xm v ) � �p(x∗)φ(zm|x∗) p(zm) � (3) = p(x∗) �� dzm φ(zm|x∗) �V m v=1 φ(zm|xm v ) � dz′ p(z′) �V m v=1 φ(z′|xm v ) � = p(x∗)λ(x∗, Xm) λ′(Xm) (4) where λ(y, X) = � dz φ(z|y) V� v=1 φ(z|xv), and (5) λ′(X) = � dz p(z) V� v=1 φ(z|xv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (6) The marginal predictive p(x∗|Xm) is used to form the training objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In our few shot task, we wish to maximize the likelihood for the correct match of query point to support set, accumu- lated across all support/query selections indexed with t from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This provides a complete 4 Accepted as a conference paper at ICLR 2023 negative log marginal likelihood objective to be minimized, as derived in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2: L({St}, {x∗ t }) = − � t � log λ(x∗, Xm∗) λ′(Xm∗) − log � m λ(x∗, Xm) λ′(Xm) � (7) Full pseudocode for training POEM with this objective is provided in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 INTERPRETATION OF OBJECTIVE While the normalised factors φ can be chosen from any distribution class, we take φ to be Gaussian with parameterised mean and precision for the remainder of this paper, rendering the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 5 analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Approximating the prior p(z) by a Gaussian also renders Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 6 analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2 We note that other distributions with analytic products, such as Beta distributions, may also be of interest in certain applications, but we leave an investigation of other distributional forms for φ to further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' If the representations from each view for a support point are aligned with each other and the query view (the means of all the Gaussians are similar), they will have a greater overlap and the integral of the resulting product of Gaussians will be larger, leading to a greater value of λ(y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Furthermore, increasing the precisions for aligned Gaussian components leads to greater λ(y, X), while, up to a limit, decreasing the precisions for non-aligned Gaussian components leads to greater λ(y, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' While the numerator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 4, λ(y, X), quantifies the overlap of the support set with the query, the denominator λ′(X) contrasts this with the overlap of the support set representation with the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Together, this factor is enhanced if it is beneficial in overlap terms to replace the prior with the query representation, and reduced if such a replacement is detrimental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A greater consistency between query and combined support set representations intuitively leads to a greater probability that the query belongs to the class of the corresponding support set, effectively extending Prototypical Networks to a probabilistic latent representation space (Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As a result, this objective is a generalisation of a Prototypical Network that allows for (a) learnable weighted averaging over support examples based on their informativeness to a given component;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (b) learnable combinations of features from subsets of support examples (via differing relative preci- sions of components within support representations), and (c) partial comparison of the query sample with the support samples (via differing relative precisions within the query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' With all precisions fixed to 1, this approach reproduces Prototypical Networks, neglecting small differences in scaling factors that arise with varying numbers of views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This relationship is derived in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 4 EXPERIMENTAL EVALUATION There is a lack of established benchmarks specifically targeted at the evaluation of representation learning under partial observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' To design a comprehensive benchmark for few-shot represen- tation learning under partial observability, we leverage Meta-Dataset (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020), a recently proposed collection of few-shot learning benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We selected Meta-Dataset as the ba- sis for our adapted benchmark as it consists of diverse datasets involving natural, human-made and text-based visual concepts, with a variety of fine-grained classification tasks that require learning from varying and unbalanced numbers of samples and classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As a result, our derived benchmark inherits these properties to provide a robust measure of the ability of a learning approach to learn representations from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' To extend Meta-Dataset to incorporate partial observability, we take multiple views of each sample and divide these views into support and query sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Our adapted few-shot classification task is to predict which sample a query view comes from, given a selection of support views of that sample, as demonstrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In keeping with the spirit of Meta-Dataset, we vary the number of ways in the task (now the number of images) from 5 to 25, taken from between 1 to 5 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Views are generated by applying the standard augmentation operations used in SimCLR (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020) and most other self- supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, to emphasise the focus on partial observability, the size of 2In reality, p(z) is typically flat over the region of non-negligible density of the product �V v=1 φ(z|xv) so does not affect the value of λ′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 6 and can be neglected, as described in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 5 Accepted as a conference paper at ICLR 2023 Meta-Dataset Dataset Classes Samples Views VGG Flowers 010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Globe Thistle Meta-Dataset Meta-Dataset: Few-Shot Classification of Classes from Samples PO-Meta-Dataset: Few-Shot Classification of Samples from Views Figure 2: Standard few-shot learning requires the prediction of an image class from a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Our adapted task evaluates representation learning under partial observability by instead requiring pre- diction of the underlying image from partial views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Views are generated with the standard contrastive augmentations, with stronger cropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We call the resulting benchmark Partially Observable Meta- Dataset (PO-Meta-Dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' the random crops and the number of views was fixed, such that the entire support set for a sample contains a maximum of 50% of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We also maintain a constant number of query views per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Viewpoint information consisting of the coordinates of the view is provided to make it possible for learners to understand where a view fits into a representation even in the absence of overlapping views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Full details of the definition of the task are provided in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We apply our proposed evaluation procedure to all datasets included in Meta-Dataset with a few exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' ILSVRC (ImageNet, Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (2015)) was not included since our network backbones were pre-trained on this dataset, including the standard few-shot test classes (which is also why this dataset was subsequently removed from the updated benchmark, MetaDataset-v2 (Du- moulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Traffic Signs (Stallkamp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2011) and MSCOCO (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2015) were not included since these datasets are fully reserved for evaluation by Meta-Dataset and so do not have a training set specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Quick Draw (Fernandez-Fernandez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019) was also not included since this dataset was found to be too large to use within the memory constraints of standard RTX2080 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This leaves six diverse datasets: Aircraft (Maji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2013), Birds (Wah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2011), Flow- ers (Nilsback & Zisserman, 2008), Fungi (Schroeder, Brigit, 2018), Omniglot (Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2015) and Textures (Cimpoi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2014), on all of which our models were trained, validated and tested on according to the data partitions specified by the Meta-Dataset benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The resulting benchmark, Partially Observed Meta-Dataset (PO-Meta-Dataset), therefore requires that the learner coherently combine the information from the support views into a consistent repre- sentation of the sample, such that the query view can be matched to the sample it originated from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Since a maximum of 50% of each sample is seen in the support set, the task also requires generali- sation to correctly match and classify query views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 IMPLEMENTATION DETAILS We utilise a re-implementation of Meta-Dataset benchmarking in PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2019) which closely replicates the Meta-Dataset sampling procedure of uniformly sampling classes, fol- lowed by a balanced query set (since all classes are considered equally important) and unbalanced support sets (to mirror realistic variations in the appearances of classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The experimental imple- mentation, including full open-source code and data will be available on publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Following the MD-Transfer procedure used in Meta-Dataset, we leverage a single ResNet-18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2015) classifier pre-trained on ImageNet (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2015) at 126 × 126 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Since both a mean and precision must be learned to fully specify the model φv(z|xn v), we add two simple 3-layer MLP heads onto this backbone for POEM, each maintaining an embedding size of 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 30 40 50 8 - 0 40Accepted as a conference paper at ICLR 2023 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' For fair comparison, we also add the same 3-layer MLP head onto the backbone for the base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Using a larger embedding for the baselines was not found to be beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' During training, gradients are backpropagated through the entire network such that both the randomly initialised heads and pre-trained backbones are learned/fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We use a representative selection of meta-learning baselines utilised by Meta-Dataset for our re- implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This includes a strong naive baseline (Finetuning, Nakamura & Harada (2019)), an embedding-based approach (Prototypical Network, Snell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (2017)) and an optimisation-based approach (MAML, Finn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (2017)), all modernised to use the ResNet-18 backbone as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Recent competitions, such as the NeurIPS 2021 MetaDL Challenge (Baz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2021), have demonstrated that these fundamental approaches, updated to use modern pre-trained backbones that are finetuned on the meta-task (exactly as in our experiments below) are still generally state- of-the-art for novel datasets (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2021b), and so form strong baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In addition, our re-implementation enables us to ensure that all learners are optimised for Meta-Dataset and that comparisons between learners are fair, utilising the same benchmark parameters, model architectures and where applicable, hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Crucially, given the close connection between POEM and Prototypical Networks, we ensure that all hyperparameters, including learning rates, scheduling and architectures are identical for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 RESULTS Our results on this novel representation learning benchmark, PO-Meta-Dataset, are given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Test Source Finetune ProtoNet MAML POEM Aircraft 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 Birds 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 Flowers 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 Fungi 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 Omniglot 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 Textures 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 Table 1: Few-shot classification accuracies on our adapted Meta-Dataset benchmark, PO-Meta- Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' All learners use a ResNet-18 model pre-trained on ImageNet, with MLP heads to incorpo- rate view information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' POEM outperforms the baselines across the range of datasets, demonstrating the benefits of the approach to learn and match representations from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The results show that POEM outperforms the baselines at identifying views of images across a diverse range of datasets, demonstrating the benefits of the approach to learn and match representa- tions from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The only exception is the Textures dataset, for which the finetuning baseline performs particularly strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We hypothesise that this is because the images in the Tex- tures dataset are relatively uniform compared to the other datasets, so capturing the relative location of views is less important than identifying very fine grained features that distinguish the samples, which optimisation-based approaches are particularly effective at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 ABLATION: META-DATASET To demonstrate that the observed benefit of POEM over the baselines is due to the requirement of the task to learn coherent representations from partial observations, we also evaluate our approach against the baselines on the established Meta-Dataset benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We now follow the standard few- shot learning procedure as described in the paper (Triantafillou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2020), but keep all learners identical to those used in the evaluation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Our results on the standard Meta-Dataset benchmark are provided in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As expected, we find that POEM performs comparably with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Although Meta-Dataset provides realistic few-shot learning tasks in terms of diversity of visual concepts, fine-grained classes and variable shots and ways, each sample generally contains complete information including all relevant features for the visual concept in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Correctly classifying query samples does not generally require any unification of views from support examples, but simply the identification of common features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As a result, we see that the additional capacity of POEM to learn to weight support examples and 7 Accepted as a conference paper at ICLR 2023 Test Source Finetune ProtoNet MAML POEM Aircraft 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 Birds 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 Flowers 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 Fungi 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 Omniglot 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 Textures 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 Table 2: Few-shot classification accuracies on Meta-Dataset, all using a ResNet-18 backbone pre- trained on ImageNet, with a 3 layer MLP head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' POEM is comparable with the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' combine partial features does not provide a significant performance improvement over the baselines at few-shot classification in this fully observable benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In support of our hypothesis that feature uncertainty is not useful on this benchmark, we find that the variance in the precisions relative to the means output by the POEM model generally decreases during training and becomes negligible for all datasets, indicating that the precisions are not be- ing utilised to improve performance and that the POEM objective is reducing to the Prototypical Network objective, as discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This is further evidenced by the very similar per- formances of POEM and the Prototypical Network across the entire set of datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' However, on PO-Meta-Dataset, we find that the relative variance in the precisions to the means is much larger on convergence, which leads to the improved performance of POEM over the Prototypical Network observed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This is shown in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 5 DEMONSTRATION OF LEARNING REPRESENTATIONS OF ENVIRONMENTS We now apply POEM to the equivalent task of learning a representation of an environment from the partial observations collected by an agent exploring that environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' To do so, we utilise the 2D gridworld environment, MiniGrid (Chevalier-Boisvert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We consider the 11 × 11 Simple Crossing environment, which consists of a procedurally generated maze where the agent is required to traverse from the top left corner to the goal in the bottom right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The MiniGrid environment provides an agent-centric viewpoint at each step in the trajectory, consisting of a 7×7 window of the environment in front of the agent, taking into account the agent’s current direction and where the line of sight is blocked by walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 META-LEARNING ENVIRONMENT REPRESENTATIONS VIA FEW-SHOT CLASSIFICATION To generate few-shot episodes, we utilise two agents: an optimal agent that takes the optimal trajec- tory from the start to the goal, and an exploratory agent that is incentivised to explore all possible views in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The support set for each environment is generated by running the optimal agent in the environment and collecting the partial observations of this agent at each step during its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The query set is similarly generated by running the exploratory agent in the environment, filtering out any observations that are contained within the support set, and then randomly sampling the desired number of queries from the remaining observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We generate these few-shot episodes dynamically, and train POEM to combine the support samples (partial observations from the optimal trajectory) into a representation of the environment, such that it can classify which environment a novel query observation has been collected from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A set of sample environments and observations from those environments are shown in figures 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 1 2 3 4 5 6 7 8 9 10 Figure 3: Sample environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 3 9 5 7 7 Figure 4: Sample queries labelled with targets correspond- ing to the environment which they were observed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 8 Accepted as a conference paper at ICLR 2023 All observations are provided as pixels to a standard convolutional backbone, with the corresponding agent location and direction appended to this representation and passed through an MLP head, equiv- alent to the procedure utilised for the adapted Meta-Dataset experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As a baseline comparison, we also train a Prototypical Network with an identical architecture on this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Additionally, we train an equivalent recurrent network architecture typically applied to POMDP tasks such as this (Hausknecht & Stone, 2017), by adding a GRU layer (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2014) where the hidden state of the GRU is updated at each timestep and then extracted as the unified represen- tation of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We find that POEM trains more quickly and reaches almost 10% higher final environment recognition performance than both the Prototypical Network and GRU-based approach over 100 test episodes (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1% vs 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4% and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1%), as shown in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This is a result of POEM’s capacity to associate each observation with only part of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 RECONSTRUCTING ENVIRONMENTS FROM PARTIAL OBSERVATION TRAJECTORIES Having learned an environment encoder using the few-shot learning procedure above, we now in- vestigate the extent to which our representations can be used to reconstruct the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' As above, we generate trajectories with the optimal agent and feed these through the encoder to gener- ate a representation of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' An MLP decoder is then trained to reconstruct the original environment layout from the learned environment representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The decoder attempts to predict a one-hot representation of each possible grid cell, with a mean squared error loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Given the trained encoder and decoder, we are now able to generate a map of the environment the optimal agent has traversed, solely from the agent’s partial observations, and without ever having seen the environment as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A sample of environments alongside their reconstructions are shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Figure 5: Left: Ground truth environments explored by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Right: Reconstructions of the corresponding environments from POEM’s unified representation, encoded from the partial obser- vations of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We see that the reconstructions clearly capture the approximate structure of each environment, demonstrating that the agent has been able to integrate its observations from along its trajectory into a single consistent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Since POEM enables the representation to be updated incre- mentally with each partial observation of the environment at inference time, it would be possible for an agent to update an internal environment representation at each step in its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' There is potential for utilising this approach for learning environment representations to be beneficial in the context of exploration for reinforcement learning, but we leave such an investigation to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 6 CONCLUSION In this work, we have introduced Partial Observation Experts Modelling (POEM), a contrastive meta-learning approach for few-shot learning in partially-observable settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Unlike other stan- dard contrastive and embedding-based meta-learning approaches, POEM utilises representational uncertainty to enable observations to inform only part of a representation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This probabilis- tic formalism enables consistent representation learning from multiple observations with a few-shot learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We have demonstrated that POEM is comparable to the state-of-the-art base- lines on a comprehensive few-shot learning benchmark, and outperforms these baselines when this benchmark is adapted to evaluate representation learning from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We have also demonstrated a promising potential application for POEM to learn representations of an environ- ment from an agent’s partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We hope that this research inspires further work into the challenging task of learning representations under partial observability and the creation of more realistic partial observability benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 9 Accepted as a conference paper at ICLR 2023 ACKNOWLEDGMENTS Adam Jelley was kindly supported by Microsoft Research and EPSRC through Microsoft’s PhD Scholarship Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Antreas Antoniou was supported by a Huawei DDMPLab Innovation Re- search Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The 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+page_content='06347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Jake Snell, Kevin Swersky, and Richard S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Prototypical Networks for Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='05175 [cs, stat], June 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='05175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv: 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='05175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Johannes Stallkamp, Marc Schlipsing, Jan Salmen, and Christian Igel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The German traffic sign recognition benchmark: a multi-class classification competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In The 2011 international joint conference on neural networks, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 1453–1460, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='organization: IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Zhuo Sun, Jijie Wu, Xiaoxu Li, Wenming Yang, and Jing-Hao Xue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Amortized Bayesian Proto- type Meta-learning: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Hospedales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Learning to Compare: Relation Network for Few-Shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Technical Re- port arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='06025, arXiv, March 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='06025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='06025 [cs] version: 2 type: article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Yichuan Tang, Ruslan Salakhutdinov, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Deep Mixtures of Factor Analysers, June 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4635 [cs, stat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Utku Evci, Kelvin Xu, Ross Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Meta- Dataset: A Dataset of Datasets for Learning to Learn from Few Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Technical Re- port arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03096, arXiv, April 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03096 [cs, stat] type: article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Oriol Vinyals, Charles Blundell, Timothy Lillicrap, koray kavukcuoglu, and Daan Wierstra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Match- ing Networks for One Shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, volume 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='cc/ paper/2016/hash/90e1357833654983612fb05e3ec9148c-Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The caltech-ucsd birds-200-2011 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Publisher: California Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Christopher Williams, Felix Agakov, and Stephen Felderhof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Products of gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Dietterich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Becker, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Ghahramani (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' ), Advances in Neural Information Processing Systems, vol- ume 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' MIT Press, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='cc/paper/2001/ file/8232e119d8f59aa83050a741631803a6-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Chi Zhang, Yujun Cai, Guosheng Lin, and Chunhua Shen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' DeepEMD: Differentiable Earth Mover’s Distance for Few-Shot Learning, January 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 06777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='06777 [cs, eess].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Luisa M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Zintgraf, Kyriacos Shiarlis, Vitaly Kurin, Katja Hofmann, and Shimon Whiteson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Fast Context Adaptation via Meta-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03642 [cs, stat], June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' URL http: //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='org/abs/1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' arXiv: 1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='03642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 GAUSSIAN PRODUCT RULES Assuming the latent variable model φ(z|x) to be a diagonal covariance multivariate Gaussian, the resulting integrals over latent variables become integrals over Gaussian products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' This allows both λ(y, X) (Equation 5) and λ′(X) (Equation 6) in the marginal predictive distribution (Equation 4) to be evaluated analytically using the following univariate Gaussian product rules on each independent dimension (Roweis, Sam, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Since a product of Gaussians � i N(µi, τ −1 i ) is itself a Gaussian, we have � i N(µi, τ −1 i ) = SN(µ, τ −1), where τ = � i τi (8) µ = 1 τ � i τiµi (9) S = (2π) (1−n) 2 � i τ 1/2 i τ 1/2 exp � 1 2τµ2 − 1 2 � i τiµ2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (10) Therefore the integral of a Gaussian product is given by the resulting normalisation S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In the case of evaluating the marginal predictive distribution p(x∗|Xm) (equation 4), this gives S∗S S′S = S∗ S′ where S is the normalisation constant of the support product, S∗ is the normalisation 13 Accepted as a conference paper at ICLR 2023 constant of the product of the query and normalised support product, and S′ is the normalisation constant of the product of the prior p(z) and normalised support product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' In reality, p(z) generally has little impact as it is typically flat (τ → 0) over the region of non-negligible density of the product �V v=1 φ(z|xv) and so S′ ≈ 1 and we find S∗ S′ ≈ S∗ so the ratios λ λ′ in the objective can be approximated by S∗, as in the simplified pseduocode in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 DERIVATION OF OBJECTIVE FROM MARGINAL PREDICTIVE DISTRIBUTION In section 3, we derived the marginal predictive distribution: p(x∗|Xm) = p(x∗)λ(x∗, Xm) λ′(Xm) (11) where λ(y, X) = � dz φ(z|y) dim(X) � v=1 φ(z|xv), and (12) λ′(X) = � dz p(z) dim(X) � v=1 φ(z|xv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (13) In our few shot task, the support data is chosen and then the query view is chosen uniformly at random to match the views of one of the support data items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Let the hypothesis Hm indicate the event that the query view x∗ comes from support point m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Then P(Hm|S, x∗) = P(Hm)P(S, x∗|Hm) � m′ P(Hm′)P(S, x∗|Hm′) = (1/M)p(x∗|S, Hm) � m′(1/M)p(x∗|S, Hm′) (14) = (1/M)p(x∗|Xm) � m′(1/M)p(x∗|Xm′) = p(x∗|Xm) � m′ p(x∗|Xm′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (15) From this we can formulate the training task: we wish to maximize the likelihood for the correct match of query point to support set, accumulated across all support/query selections from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Denote the tth support set by St, the tth query point by x∗ t , and let mt denote the support point with views that match the view of the query point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Then the complete negative log marginal likelihood objective to be minimized is: L({St}, {x∗ t }) = − � t log P(Hmt|St, x∗ t ) (16) = − � t log p(x∗|Xm) � m′ p(x∗|Xm′) (17) = − � t � log λ(x∗, Xm∗) λ′(Xm∗) − log � m λ(x∗, Xm) λ′(Xm) � (18) 14 Accepted as a conference paper at ICLR 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 PSEUDOCODE Algorithm 1 Pytorch-Style Pseudocode: Gaussian Partial Observation Experts Modelling # phi: dual-headed encoder network with shared backbone and output heads for mean and precision of Gaussian embedding # M: Number of items/classes in task # V: Number of views of each item/class (in general can vary with m in range(M)) # N: Number of query views # D: Embedding dimension # Load augmented partial views with view information for (support_views,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' query_views,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' query_targets) in loader: # support_views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='shape = (M, V, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=') # query_views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='shape=(N, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=') # query_targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='shape = (N,) # Encode each support and query views support_means, support_precisions = phi(support_views) # (M, V, D) query_means, query_precisions = phi(query_views) # (N, D) # Combine support views into unified representation of each item # Gaussian products computed using equations in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 # Optionally include prior Gaussian here (neglected for simplified implementation) environment_means,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' environment_precisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' log_environment_normalisation = inner_gaussian_product(support_means,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' support_precisions) # Outputs: (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' D) # Combine each query view with each unified support representation env_query_mean,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' env_query_precisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' log_env_query_normalisation = outer_gaussian_product(support_means,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' support_precisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' query_means,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' query_precisions) # Outputs: (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' D) # Predictions correspond to unified support with maximum overlap with query _,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' predictions = log_env_query_normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='sum(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='max(1) # (N,) # Cross entropy loss normalises with softmax and computes negative log-likelihood loss = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='cross_entropy(log_env_query_normalisation, query_targets, reduction=’mean’) # Optimization step loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='backwards() optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='step() Algorithm 2 Language Agnostic Pseudocode: Gaussian Partial Observation Experts Modelling Require: Training meta-set Dtrain ∈ T Require: Learning rate α 1: Initialise dual-headed network φθ(z|x) 2: ▷ Heads correspond to mean µ and precision τ of Gaussian embedding z 3: while not converged do 4: Sample task instance Ti = (X, x∗) ∼ Dtrain 5: ▷ Support set X consists of V m views of item m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 6: ▷ Query set x∗ consists of N queries, each one view from any one item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 7: Encode each view in support set X into Gaussian z using φ(z|X) 8: Encode each query view in x∗ into Gaussian z∗ using φ(z∗|x∗) 9: for m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', M} do 10: Compute Gaussian product over views �V m v=1 φ(z|xm v ) (using results in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1) 11: ▷ This gives unified support representation (global environment representation) 12: for n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', N} do 13: Compute Gaussian product of query with support product φ(z∗ n|x∗ n) �V m v=1 φ(z|xm v ) 14: end for 15: end for 16: Normalise resulting query-support normalisation constants Sm n = Sm n � m Sm n across items 17: Compute negative log of Sm∗ n for correct support as loss L({Dt}, {x∗ t }) (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 7) 18: ▷ Negative log likelihood for correct support 19: Perform gradient step w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' θ: θ ← φ − α∇θL({Dt}, {x∗ t }) 20: end while 15 Accepted as a conference paper at ICLR 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 EQUIVALENCE OF PROTOTYPICAL NETWORK OBJECTIVE TO POEM OBJECTIVE WITH FIXED PRECISIONS The probability of a query x∗ belonging to class n using the POEM objective is given by: P(Hm|S, x∗) = λ(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Xn) λ′(Xn) � � m λ(x∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Xm) λ′(Xm) (19) as defined in equation 15, where λ(y, X) = � dz φ(z|y) V� v=1 φ(z|xv), and (20) λ′(X) = � dz p(z) V� v=1 φ(z|xv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' (21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Taking the precisions of the all Gaussian factors φ in λ and λ′ to be 1, we can apply the Gaussian product rules given in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 to calculate λ and λ′ analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We find that this gives: pn = Vn Vn+1 1 2 exp � − Vn 2(Vn+1) � µ − � i µni Vn �2� � m Vm Vm+1 1 2 exp � − Vm 2(Vm+1) � µ − � i µmi Vm �2� (22) where µ is the representation mean of the query, and µni is the representation mean of support sample i for class n, and Vn is the number of support samples for class n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Equivalently, the probability of a query with representation vector µ belonging to a class n using a Prototypical Network objective is given by: pn = exp � − � µ − � i µni Vn �2� � m exp � − � µ − � i µmi Vm �2� (23) We find that these are equivalent aside from the scaling factors Vm (2)(Vm+1) which only have a (sig- nificant) effect when there are varying numbers of samples by class, and a greater effect when the number of samples is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Experimentally, we find that these scaling factors make little differ- ence, as demonstrated in table 2 of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 16 Accepted as a conference paper at ICLR 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 PO-META-DATASET BENCHMARK ADDITIONAL DETAILS Parameters used for adapted PO-Meta-Dataset are provided in Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' All parameters not listed chosen to match Meta-Dataset defaults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' All augmentations are applied using Torchvision, with parameters specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Table 3: PO-Meta-Dataset Parameters PARAMETER VALUE Classes per Task [1, 5] Samples per Task [5, 25] Support Views per Sample 18 Query Views per Sample 2 Image Size (84, 84) (except Omniglot, (28, 28)) Crop Size (14, 14) (1/6 in each dim, except Omniglot, random placement) Color Jitter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2), p(apply) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 Random Greyscale 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 Random Horizontal Flip 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 Gaussian Blur ((3, 3), (1, 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='0)), p(apply) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 All results computed over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' The Finetuning, Prototypical Network and POEM baselines were run on on-premise RTX2080 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' MAML required more memory and compute than avail- able, so was run on cloud A100s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 RELATIVE VARIANCE OF PRECISIONS DURING TRAINING ON META-DATASET AND META-META-DATASET The plot below shows the evolution of the variance in the representation precisions relative to the variance in the representation means learned by POEM on two distinct datasets, Aircraft and VGG Flowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' We see that for standard few-shot learning on Meta-Dataset, the variance in precisions is negligible relative to the variance in the means, demonstrating that the representational uncertainty is not useful in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' Meanwhile, we see the variance in the precisions relative to the variance in the means becoming large before converging to a value of O(100) on the Meta-Meta-Dataset task, demonstrating that learning relative precisions is useful in this setting since each support sample only informs part of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 17 Relative Variance in Precisions to Variance in Means for POEM Representation During Training Var(o) / Var(μu) 300 250 200 150 100 50 10k 20k 30k 40k Training Step Meta-Dataset: Aircraft Po-Meta-Dataset: Aircraft Po-Meta-Dataset: Flowers Meta-Dataset: FlowersAccepted as a conference paper at ICLR 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 LEARNING REPRESENTATIONS OF ENVIRONMENTS ADDITIONAL DETAILS Additional details about the parameters used for learning environment representations from agent observations are provided in Table 4 Table 4: Environment Representation Learning Parameters PARAMETER VALUE Agent Training Algorithm PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=', 2017) (default hyperparameters) Optimal Agent Reward 1 for reaching goal, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='01 per timestep Exploratory Agent Reward 1/N count exploration bonus (state defined by agent location and direction) Encoder Conv Backbone Layers 5 Encoder MLP Head Layers 3 Encoder Embedding Dim 128 (corresponding ∼ 11 × 11 environment size) Decoder MLP Layers 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 ENVIRONMENT RECOGNITION ACCURACY DURING TRAINING POEM trains more quickly on the environment recognition task and reaches a higher final perfor- mance than an equivalent Prototypical Network or Recurrent Network (GRU) (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1% vs 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4% and 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1% ) over a subsequent 100 test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content=' 0 1000 2000 3000 4000 5000 Episodes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} +page_content='8 Accuracy Environment Recognition Accuracy POEM Prototypical Network Recurrent Network (GRU) 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFPT4oBgHgl3EQfpDVG/content/2301.13136v1.pdf'} diff --git a/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/2301.05585v1.pdf.txt b/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/2301.05585v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8132e64c1327f8496eb65df8ca11040cff797388 --- /dev/null +++ b/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/2301.05585v1.pdf.txt @@ -0,0 +1,2468 @@ +arXiv:2301.05585v1 [stat.ME] 13 Jan 2023 +Bivariate Distributions on the Unit Square: Theoretical +Properties and Applications +Roberto Vila ∗1, 2, Narayanaswamy Balakrishnan † 2, Helton Saulo ‡ 1, and +Peter Zörnig §1 +1Department of Statistics, University of Brasília, Brasília, Brazil +2 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada +January 16, 2023 +Abstract +We present the bivariate unit-log-symmetric model which is based on the bivariate log-symmetric +distribution (BLS) defined in Vila et al. (2022). It is a flexible family of bivariate distributions over +the unit square. We study mathematical properties like stochastic representations, quantiles, condi- +tional distributions, independence of the marginal distributions and moments. Maximum likelihood +estimators, simulation results and applications to soccer data are also presented. +Keywords. Bivariate unit-log-symmetric distribution · Bivariate log-symmetric distribution · Bivariate model · +MCMC · Proportion data · Soccer data. +Mathematics Subject Classification (2010). MSC 60E05 · MSC 62Exx · MSC 62Fxx. +1 +Introduction +Bivariate distributions over the unit-square have been intensively studied in the literature. Many of +them are based on the beta distribution and its generalizations. Models of this type have been studies since +the 1980s. Other distributions on the unit square are based on the generalized arcsine distribution and the +inverse Gaussian distribution. A very recent model, the bivariate unit-sinh-normal distributions, is based +on the bivariate Birnbaum-Saunders distribution, see Martínez-Flórez et al. (2022). Bivariate distributions +over the unit square arise naturally in comparing indices, rates or proportions in the interval (0, 1). +∗rovig161@gmail.com +†bala@mcmaster.ca +‡heltonsaulo@gmail.com +§peter@unb.br +1 + +In the present paper we study the bivariate unit-log-symmetric (BULS) distribution defined over +the unit-square which is obtained as a modification of the bivariate log-symmetric (BLS) distribution +introduced in Vila et al. (2022). The definitions of BLS and BULS are given in Section 2, indicating +some special cases of the BULS. In Section 3 we study some properties of the new model. Subsection +3.1 presents a stochastic representation. Making use of this representation we derive formulas for the +marginal quantiles in Subsection 3.2. In Subsection 3.3 we study the conditional distributions of the +BULS. We derive more compact formulas for the conditional densities, using the distribution functions +of the normal, the 푡-Student, the hyperbolic, the Laplace and the slash distribution. One of the uses of +having closed formulas for the conditional densities (of the BULS model), for example, is for studying +Heckman-type selection models (Heckman, 1979) when the selection variables have bounded support. +In Subsection 3.4 we determine the distribution of the squared Mahalanobis distance of a random vector +W = (푊1,푊2)⊤ with BULS distribution. A necessary condition for the independence of the components +of W is represented in Subsection 3.5. Section 3 ends with formulas for the moments of 푊1 and 푊2 based +on the stochastic representation. In Section 4 the log-likelihood function and the likelihood equations for +the BULS distribution are determined. In Section 5 we carry out a Monte Carlo simulation study. We +evaluate the performance of the ML-estimators by means of bias, root mean square error and coverage +probability. In Section 6 we present two applications to soccer data. In Subsection 6.1 we model the +vector W = (푊1, 푊2)⊤, where 푊1 represents the time elapsed until a first kick goal (of any team) and the +time elapsed until a goal of any type of the home team. It turns out that specific BULS distributions are +adequate to model the vector W . In Subsection 6.2 we study the very actual data of the 2022 FIFA World +Cup which has been realized in December of 2022. Now the components of the vector W represent the +pass completion proportions of medium passes (14 to 18 meters) and long passes (longer than 37 meters). +These data can also be well fitted BULS distributions. +2 +The bivariate unit-log-symmetric model +In this section (specifically, in Subsection 2.2), we define the model of interest in this paper, the bivariate +unit-log-symmetric model (BULS). To define this model we first need to establish the bivariate log- +symmetric distribution (BLS) which was naturally defined in Vila et al. (2022). +2.1 +The BLS distribution +Following Vila et al. (2022), a continuous random vector T = (푇1, 푇2)⊤ has a bivariate log-symmetric +(BLS) distribution if its joint probability density function (PDF) is given by +푓푇1,푇2(푡1, 푡2; θ) = +1 +푡1푡2휎1휎2 +� +1 − 휌2 푍푔푐 +푔푐 +� +�푡1 +2 − 2휌�푡1�푡2 + �푡2 +2 +1 − 휌2 +� +, +푡1, 푡2 > 0, +(2.1) +where +�푡푖 = log +�� 푡푖 +휂푖 +�1/휎푖� +, 휂푖 = exp(휇푖), 푖 = 1, 2, +2 + +with θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤ being the parameter vector, 휇푖 ∈ R, 휎푖 > 0, 푖 = 1, 2 and 휌 ∈ (−1, 1). +Furthermore, 푍푔푐 > 0 is the partition function, that is, +푍푔푐 = +∫ ∞ +0 +∫ ∞ +0 +1 +푡1푡2휎1휎2 +� +1 − 휌2 푔푐 +� +�푡1 +2 − 2휌�푡1�푡2 + �푡2 +2 +1 − 휌2 +� +d푡1d푡2 = 휋 +∫ ∞ +0 +푔푐(푢) d푢, +(2.2) +and 푔푐 is a scalar function referred to as the density generator (see Fang et al., 1990). The second integral +in (2.2) is consequence of a change of variables, for more details see Proposition 3.1 of Vila et al. (2022). +When a random vector T is BLS distributed, with parameter vector θ, we write T ∼ BLS(θ, 푔푐). +2.2 +The BULS distribution +We say that a continuous random vector W = (푊1,푊2)⊤ has a bivariate unit-log-symmetric (BULS) +distribution with parameter vector θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤, denoted by W ∼ BULS(θ, 푔푐), if its PDF is +as follows (for 0 < 푤1, 푤2 < 1) +푓푊1,푊2(푤1, 푤2; θ) = +1 +(1 − 푤1)푡1휎1(1 − 푤2)푡2휎2 +� +1 − 휌2 푍푔푐 +푔푐 +� +�푤2 +1 − 2휌�푤1�푤2 + �푤2 +2 +1 − 휌2 +� +, +(2.3) +where +�푤푖 = log +�� 푡푖 +휂푖 +�1/휎푖� +, 푡푖 = − log(1 − 푤푖), 휂푖 = exp(휇푖), 푖 = 1, 2, +with 휎푖 > 0, 푖 = 1, 2, 휌 ∈ (−1, 1), and 푍푔푐 and 푔푐 are as given in (2.1). In Subsection 3.1 we prove that the +BULS PDF (2.3) is obtained by considering 푊푖 = 1 − exp(−푇푖), 푖 = 1, 2, with = (푇1,푇2)⊤ ∼ BLS(θ, 푔푐). +Table 1 presents some examples of bivariate unit-log-symmetric distributions. +Table 1: Partition functions (푍푔푐) and density generators (푔푐) for some distributions. +Distribution +푍푔푐 +푔푐 +Parameter +Bivariate unit-log-normal +2휋 +exp(−푥/2) +− +Bivariate unit-log-Student-푡 +Γ(휈/2)휈휋 +Γ((휈+2)/2) +(1 + 푥 +휈)−(휈+2)/2 +휈 > 0 +Bivariate unit-log-hyperbolic +2휋(휈+1) exp(−휈) +휈2 +exp(−휈 +√ +1 + 푥 ) +휈 > 0 +Bivariate unit-log-Laplace +휋 +퐾0( +√ +2푥 ) +− +Bivariate unit-log-slash +휋 +푞 2 +2−푞 +2 +푥− 푞+2 +2 훾( 푞+2 +2 , 푥 +2) +푞 > 0 +In +Table +1, +Γ(푡) += +∫ ∞ +0 푥푡−1 exp(−푥) d푥, +푡 +> +0, +is +the +gamma +function, +퐾휆(푢) += +(1/2)(푢/2)휆 ∫ ∞ +0 푡−휆−1 exp(−푡 − 푢2/4푡) d푡, 푢 > 0, is the modified Bessel function of the third kind with +index 휆 (see appendix of Kotz et al., 2001); and 훾(푠, 푥) = +∫ 푥 +0 푡푠−1 exp(−푡) d푡 is the lower incomplete +gamma function. +3 + +Let T = (푇1,푇2)⊤ ∼ BLS(θ, 푔푐). By using (2.3) it is clear that the random vector X = (푋1, 푋2)⊤, +where +푋푖 = log(푇푖) = log +� +− log(1 − 푊푖) +� +, +푖 = 1, 2, +(2.4) +has a bivariate elliptically symmetric (BSY) distribution (see p. 592 in Balakrishnan and Lai, 2009). That +is, the PDF of X is as follows +푓푋1,푋2(푥1, 푥2; θ∗) = +1 +휎1휎2 +� +1 − 휌2 푍푔푐 +푔푐 +� +�푥1 +2 − 2휌�푥1�푥2 + �푥2 +2 +1 − 휌2 +� +, +−∞ < 푥1, 푥2 < ∞, +(2.5) +where +�푥푖 = 푥푖 − 휇푖 +휎푖 +, 푖 = 1, 2, +with θ∗ = (휇1, 휇2, 휎1, 휎2, 휌) being the parameter vector and 푍푔푐 is the partition function stated in (2.2). In +this case, the notation X ∼ BSY(θ∗, 푔푐) is used. +It is a simple task to observe that the joint cumulative distribution function (CDF) of W ∼ BULS(θ, 푔푐), +denoted by 퐹푊1,푊2(푤1, 푤2; θ), is written as +퐹푊1,푊2(푤1, 푤2; θ) = 퐹푇1,푇2 +� − log(1 − 푤1), − log(1 − 푤2); θ� += 퐹푋1,푋2 +� log[− log(1 − 푤1)], log[− log(1 − 푤2)]; θ∗ +�, +wherein 퐹푇1,푇2(푡1, 푡2; θ) and 퐹푋1,푋2(푥1, 푥2; θ∗) denote the CDFs of T ∼ BLS(θ, 푔푐) and X ∼ BES(θ∗, 푔푐), +respectively. Notice that there is no single closed form for the CDF of X with the exception of the bivariate +normal case. +3 +Some basic properties of model +In this section, some mathematical properties of proposed bivariate unit-log-symmetric distribution +are discussed. +3.1 +Stochastic representation +Proposition 3.1. The random vector W = (푊1,푊2)⊤ has a BULS distribution if +푊1 = 1 − exp +� +− 휂1 exp(휎1푍1) +� +, +푊2 = 1 − exp +� +− 휂2 exp �휎2휌푍1 + 휎2 +� +1 − 휌2 푍2 +�� +, +where 푍1 = 푅퐷푈1 and 푍2 = 푅 +√ +1 − 퐷2 푈2; 푈1, 푈2,푅, and 퐷 are mutually independent random variables, +휌 ∈ (−1, 1), 휂푖 = exp(휇푖), and P(푈푖 = −1) = P(푈푖 = 1) = 1/2, 푖 = 1, 2. The random variable 퐷 is positive +and has PDF 푓퐷(푑) = 2/(휋 +√ +1 − 푑2 ), 푑 ∈ (0, 1). Further, the positive random variable 푅 has PDF given +by 푓푅(푟) = 2푟푔푐(푟2)/ +∫ ∞ +0 +푔푐(푢) d푢, 푟 > 0. +4 + +Proof. It is well-known that (see Proposition 3.2 of Vila et al., 2022), the random vector T = (푇1, 푇2)⊤ has +a BLS distribution if +푇1 = 휂1 exp(휎1푍1), +푇2 = 휂2 exp �휎2휌푍1 + 휎2 +� +1 − 휌2 푍2 +�. +(3.1) +Moreover, by (2.4), 푊푖 = 1 − exp(−푇푖), 푖 = 1, 2. Hence the proof follows. +□ +3.2 +Marginal Quantiles +Given 푝 ∈ (0, 1), let 푄푊푖(푝) be the 푝-quantile of 푊푖, for 푖 = 1, 2. By using the stochastic representation +of Proposition 3.1, for W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐) we have +푝 = P(푊1 ⩽ 푄푊1(푝)) = P�1 − exp +� +− 휂1 exp(휎1푍1) +� +⩽ 푄푊1(푝)� += P +� +푍1 ⩽ log +�� +−log(1 − 푄푊1(푝)) +휂1 +�1/휎1 �� +and +푝 = P(푊2 ⩽ 푄푊2(푝)) = P +� +1 − exp +� +−휂2 exp +� +휎2휌푍1 + 휎2 +� +1 − 휌2 푍2 +�� +⩽ 푄푊2(푝) +� += P +� +휌푍1 + +� +1 − 휌2 푍2 ⩽ log +�� +−log(1 − 푄푊2(푝)) +휂2 +�1/휎2 �� +. +Hence, the 푝-quantiles 푄푍1(푝) and 푄푍2(푝) of 푍1 and 푍2, respectively, satisfy +log +�� +−log(1 − 푄푊1(푝)) +휂1 +�1/휎1 � += 푄푍1(푝) +and +log +�� +−log(1 − 푄푊2(푝)) +휂2 +�1/휎2 � += 푄휌푍1+√ +1−휌2 푍2(푝). +Therefore, the 푝-quantiles 푄푊1(푝) and 푄푊2(푝) are given by +푄푊1(푝) = 1 − exp +� +− 휂1 exp(휎1푄푍1(푝)) +� +, +푄푊2(푝) = 1 − exp +� +− 휂2 exp �휎2푄휌푍1+√ +1−휌2 푍2(푝)�� +, +respectively. +5 + +3.3 +Conditional distributions +Lemma 3.2. If W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐), then the PDF of 푊2 | 푊1 = 푤1 is written as +푓푊2(푤2 | 푊1 = 푤1) = +1 +(1 − 푤2)푡2휎2 +� +1 − 휌2 푓푍2 +� +1 +� +1 − 휌2 (�푤2 − 휌�푤1) +���� 푍1 = �푤1 +� +, +(3.2) +where �푤푖, 푖 = 1, 2, and 푡2 are defined in (2.3), and 푍1 and 푍2 are as in Proposition 3.1. +Proof. If 푊1 = 푤1, then 푍1 = log +� +(−log(1 − 푤1)/휂1)1/휎1 � += �푤1. Thus, the conditional distribution of +푊2 given 푊1 = 푤1 is the same as the distribution of +1 − exp +� +−휂2 exp +� +휎2휌�푤1 + 휎2 +� +1 − 휌2 푍2 +�� ����푊1 = 푤1. +Consequently, +퐹푊2(푤2 | 푊1 = 푤1) = P +� +1 − exp +� +−휂2 exp +� +휎2휌�푤1 + 휎2 +� +1 − 휌2 푍2 +�� +⩽ 푤2 +����푊1 = 푤1 +� += P +� +푍2 ⩽ +1 +� +1 − 휌2 (�푤2 − 휌�푤1) +���� 푍1 = �푤1 +� +. +Then, by differentiating 퐹푊2(푤2 | 푊1 = 푤1) with respect to 푤2, (3.2) is obtained. +□ +Lemma 3.3. For a Borelian subset 퐵 of (0, 1), the following identity is satisfied +P +� +휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵 +� += P(푍2 ∈ 퐵). +In other words, 휌푍1 + +� +1 − 휌2 푍2 and 푍2 have the same distribution. +Proof. It is clear that the density of the sum 휌푍1 + +� +1 − 휌2 푍2 are related through their joint density 푓푍1,푍2 +as follows +푓휌푍1+√ +1−휌2 푍2(푠2) = +1 +� +1 − 휌2 +∫ ∞ +−∞ +푓푍1,푍2 +� +푧, 푠2 − 휌푧 +� +1 − 휌2 +� +d푧. +(3.3) +From Item (13) of Saulo et al. (2022), the joint PDF of 푍1 and 푍2 is given by +푓푍1,푍2(푥, 푦) = +1 +푍푔푐 +푔푐(푥2 + 푦2), +−∞ < 푥, 푦 < ∞. +(3.4) +Then the integral in (3.3) is += +1 +� +1 − 휌2 푍푔푐 +∫ ∞ +−∞ +푔푐 +� +푧2 + +� 푠2 − 휌푧 +� +1 − 휌2 +�2� +d푧. +(3.5) +6 + +Using the identity +푧2 + +� 푠2 − 휌푧 +� +1 − 휌2 +�2 += +푧2 − 2휌푧푠2 + 푠2 +2 +1 − 휌2 += +� 푧 − 휌푠2 +� +1 − 휌2 +�2 ++ 푠2 +2 +the integral in (3.5) is written as += +1 +� +1 − 휌2 푍푔푐 +∫ ∞ +−∞ +푔푐 +�� 푧 − 휌푠2 +� +1 − 휌2 +�2 ++ 푠2 +2 +� +d푧. +Taking the change of variables 푠1 = (푧 − 휌푠2)/ +� +1 − 휌2 , the above integral is += +1 +푍푔푐 +∫ ∞ +−∞ +푔푐(푠2 +1 + 푠2 +2) d푠1 = +∫ ∞ +−∞ +푓푍1,푍2(푠1, 푠2) d푠1, +where in the last line we used (3.4). Therefore +푓휌푍1+√ +1−휌2 푍2(푠2) = +∫ ∞ +−∞ +푓푍1,푍2(푠1, 푠2) d푠1 = 푓푍2(푠2). +(3.6) +Hence, by (3.6), it is clear that 휌푍1 + +� +1 − 휌2 푍2 and 푍2 are equal in distribution. +□ +Lemma 3.3 plays a fundamental role in the proof of the following theorem. This result provides a +simple formula for determining the conditional distribution of 푊1 given 푊2 ∈ 퐵 whenever the marginal +and conditional distributions of W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) are known. The usefulness of this result +is essential for studying Heckman-type selection models (Heckman, 1979) when the selection variables +have unitary support. +Theorem 3.4. For a Borelian subset 퐵 of (0, 1), we define the following Borelian set: +퐵푟 = +1 +√ +1 − 푟2 log +�� +−log(1 − 퐵) +휂2 +�1/휎2� +− +푟 +√ +1 − 푟2 �푤1, +−1 < 푟 < 1, +(3.7) +where �푤1 is as in (2.3). If W ∼ BULS(θ, 푔푐), then the PDF of 푊1 | 푊2 ∈ 퐵 is given by +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓푍1(�푤1) P(푍2 ∈ 퐵휌 | 푍1 = �푤1) +P(푍2 ∈ 퐵0) +, +in which 푡1 is as in (2.3), 퐵푟 is as in (3.7), and 푍1 and 푍2 are as in Proposition 3.1. +Proof. Let 퐵 be a Borelian subset of (0, 1). Observe that +푓푊1(푤1 | 푊2 ∈ 퐵) = 푓푊1(푤1) +∫ +퐵 푓푊2(푤2 | 푊1 = 푤1) d푤2 +P(푊2 ∈ 퐵) +. +7 + +Since 푓푊1(푤1) = 푓푍1(�푤1)/[(1 − 푤1)푡1휎1] and P(푊2 ∈ 퐵) = P�휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵0 +�, where 퐵0 is given +in (3.7) with 푟 = 0, the term on the right-hand side of the above identity is += +1 +(1 − 푤1)푡1휎1 +푓푍1(�푤1) +∫ +퐵 푓푊2(푤2 | 푊1 = 푤1) d푤2 +P�휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵0 +� . +By using the formula for 푓푊2(푤2|푊1 = 푤1) provided by Lemma 3.2, the previous expression is += +1 +(1 − 푤1)푡1휎1휎2 +� +1 − 휌2 푓푍1(�푤1) +∫ +퐵 +1 +(1−푤2)푡2 푓푍2 +� +1 +√ +1−휌2 �푤2 − +휌 +√ +1−휌2 �푤1 +��� 푍1 = �푤1 +� +d푤2 +P�휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵0 +� +, +where �푤푖 and 푡푖, 푖 = 1, 2, are as in (2.3). +Finally, by applying the change of variable 푧 = (�푤2 − +휌 �푤1)/ +� +1 − 휌2 , the above expression is += +1 +(1 − 푤1)푡1휎1 +푓푍1(�푤1) +∫ +퐵휌 푓푍2(푧 | 푍1 = �푤1) d푧 +P�휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵0 +� . +In other words, we have proved that +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓푍1(�푤1) +∫ +퐵휌 푓푍2(푧 | 푍1 = �푤1) d푧 +P�휌푍1 + +� +1 − 휌2 푍2 ∈ 퐵0 +� . +Finally, by combining the above identity with Lemma 3.3, the proof follows. +□ +Using Theorem 3.4, in what remains of this subsection, for each generator (푔푐) of Table 1, we present +closed formulas of the conditional densities of 푊1 | 푊2 ∈ 퐵 corresponding to bivariate unit-log-normal +(Corollary 3.5), bivariate unit-log-Student-푡 (Corollary 3.6), bivariate unit-log-hyperbolic (Corollary 3.7), +bivariate unit-log-Laplace (Corollary 3.8) and bivariate unit-log-slash (Corollary 3.9). +Corollary 3.5 (Gaussian generator). Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = exp(−푥/2) be the +generator of the bivariate unit-log-normal distribution. Then, for each Borelian subset 퐵 of (0, 1), the PDF +of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +휙(�푤1) Φ(퐵휌) +Φ(퐵0) , +where Φ(퐶) = +∫ +퐶 휙(푥)d푥 and 휙(푥) is the standard normal PDF. Further, �푤1 and 푡1 are as in (2.3), and 퐵푟 +is as in (3.7). +Proof. It is well-known that the bivariate log-normal distribution has a stochastic representation as in +(3.1), where 푍1 ∼ 푁(0, 1) and 푍2 ∼ 푁(0, 1), and 푍2 | 푍1 = 푥 ∼ 푁(0, 1) (Abdous et al., 2005). Hence, +P(푍2 ∈ 퐵0) = Φ(퐵0) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = Φ(퐵휌). Then, by applying Theorem 3.4, the proof +follows. +□ +8 + +Corollary 3.6 (Student-푡 generator). Let W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = (1+ (푥/휈))−(휈+2)/2, +휈 > 0, be the generator of the bivariate unit-log-Student-푡 distribution with 휈 degrees of freedom. Then, +for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓휈(�푤1) +퐹휈+1 +�� +휈+1 +휈+�푤2 +1 퐵휌 +� +퐹휈(퐵0) +, +where 퐹휈(퐶) = +∫ +퐶 푓휈(푥)d푥 and 푓휈(푥) is the standard Student-푡 PDF with 휈 degrees of freedom. +Proof. It is well-known that the bivariate log-Student-푡 distribution has a stochastic representation as in +(3.1), where 푍1 ∼ 푡휈 and 푍2 ∼ 푡휈 (Student-푡 with 휈 degrees of freedom), and (see Corollary 3.7 of Vila et al., +2022) +푍2 | 푍1 = 푥 ∼ +� +휈 + 푥2 +휈 + 1 푡휈+1. +Hence, P(푍2 ∈ 퐵0) = 퐹휈(퐵0) and +P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹휈+1 +�� +휈 + 1 +휈 + �푤2 +1 +퐵휌 +� +. +By applying Theorem 3.4, the proof of corollary follows. +□ +Corollary 3.7 (Hyperbolic generator). Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = exp(−휈 +√ +1 + 푥 ), +be the generator of the bivariate unit-log-hyperbolic distribution. Then, for each Borelian subset 퐵 of +(0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓GH(�푤1; 3/2, 휈, 1) +퐹GH +� +퐵휌; 1, 휈, +� +1 + �푤2 +1 +� +퐹GH(퐵0; 3/2, 휈, 1) +, +where 퐹GH(퐶; 휆, 훼, 훿) = +∫ +퐶 푓GH(푥; 휆, 훼, 훿)d푥 and 푓GH(푥; 휆, 훼, 훿) is the generalised hyperbolic (GH) PDF +(see Definition A.1 in the Appendix). +Proof. It is well-known that the bivariate log-hyperbolic distribution has a stochastic representation as in +(3.1), where 푍1 ∼ GH(3/2, 휈, 1) and 푍2 ∼ GH(3/2, 휈, 1) (Deng and Yao, 2018, Subsection 2.1, p. 3). +Moreover, the distribution of 푍2 given 푍1 = 푥 is GH(1/2, +√ +2 , |푥|) (Proposition A.1). Then P(푍2 ∈ 퐵0) = +퐹GH(퐵0; 3/2, 휈, 1) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹GH +�퐵휌; 1, 휈, +� +1 + �푤2 +1 +�. By applying Theorem 3.4, the +proof follows. +□ +Corollary 3.8 (Laplace generator). Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = 퐾0( +√ +2푥 ) be the +generator of the bivariate unit-log-Laplace distribution. Then, for each Borelian subset 퐵 of (0, 1), the +PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓L(�푤1) 퐹GH(퐵휌; 1 +2, +√ +2 , |�푤1|) +퐹L(퐵0) +, +where 퐹L(퐶) = +∫ +퐶 푓L(푥)d푥 and 푓L(푥) = exp(− +√ +2 |푥|)/ +√ +2 is the Laplace PDF with scale parameter 1/ +√ +2 , +and 퐹GH is defined in Corollary 3.7. +9 + +Proof. It is well-known that the bivariate log-Laplace distribution has a stochastic representation as in +(3.1), where 푍1 ∼ Laplace(0, 1/ +√ +2 ) and 푍2 ∼ Laplace(0, 1/ +√ +2 ) (Kotz et al., 2001, Subsection 5.1.4, +p. 234). Further, the distribution of 푍2 given 푍1 = 푥 is GH(1/2, +√ +2 , |푥|) (Proposition A.2). Hence, +P(푍2 ∈ 퐵0) = 퐹L(퐵0) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹GH(퐵휌; 1/2, +√ +2 , |�푤1|). By applying Theorem 3.4, the +proof follows. +□ +Corollary +3.9 +(Slash +generator). Let +W += +(푊1,푊2)⊤ +∼ +BULS(θ, 푔푐) +and +푔푐(푥) += +푥−(푞+2)/2훾((푞 + 2)/2, 푥/2), be the generator of the bivariate unit-log-slash distribution. Then, for each +Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) +푓푊1(푤1 | 푊2 ∈ 퐵) = +1 +(1 − 푤1)푡1휎1 +푓SL(�푤1; 푞) 퐹ESL +�퐵휌; �푤1, 푞 + 1� +퐹SL(퐵0; 푞) +, +where 퐹SL(퐶; 푞) = +∫ +퐶 푓SL(푥; 푞)d푥 and 푓SL(푥; 푞) = 푞 +∫ 1 +0 푡푞휙(푡푥)d푡 is the classical slash PDF, and +퐹ESL(퐶; 푎, 푞) = +∫ +퐶 푓ESL(푥; 푎, 푞)d푥, where 푓ESL(푥; 푎, 푞) is the generalised hyperbolic (ESL) PDF (see +Definition A.2 in the Appendix). +Proof. It is well-known that the bivariate log-slash distribution has a stochastic representation as in (3.1), +where 푍1 ∼ SL(푞) and 푍2 ∼ SL(푞) (Wang and Genton, 2006, Section 2, p. +211). +Moreover, the +distribution of 푍2 given 푍1 = 푥 is ESL(푥, 푞 + 1) (Proposition A.3). Hence, P(푍2 ∈ 퐵0) = 퐹SL(퐵0; 푞) and +P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹ESL +�퐵휌; �푤1, 푞 + 1� . By applying Theorem 3.4, the proof follows. +□ +In summary, Table 2 presents some examples of conditional PDFs corresponding to the bivariate +unit-log-symmetric distributions of Table 1. +Table 2: Conditional densities of 푊1| 푊2 ∈ 퐵 and density generators (푔푐) for some distributions. +Distribution +푔푐 +푓푊1(푤1 | 푊2 ∈ 퐵) +Bivariate unit-log-normal +exp(−푥/2) +1 +(1−푤1)푡1휎1 휙(�푤1) Φ(퐵휌) +Φ(퐵0) +Bivariate unit-log-Student-푡 +(1 + 푥 +휈)−(휈+2)/2 +1 +(1−푤1)푡1휎1 푓휈(�푤1) +퐹휈+1 +�� +휈+1 +휈+ � +푤2 +1 +퐵휌 +� +퐹휈(퐵0) +Bivariate unit-log-hyperbolic +exp(−휈 +√ +1 + 푥 ) +1 +(1−푤1)푡1휎1 푓GH(�푤1; 3/2, 휈, 1) +퐹GH +� +퐵휌;1,휈,√ +1+�푤2 +1 +� +퐹GH(퐵0;3/2,휈,1) +Bivariate unit-log-Laplace +퐾0( +√ +2푥 ) +1 +(1−푤1)푡1휎1 푓L(�푤1) +퐹GH(퐵휌; 1 +2, +√ +2 ,|�푤1|) +퐹L(퐵0) +Bivariate unit-log-slash +푥− 푞+2 +2 훾( 푞+2 +2 , 푥 +2) +1 +(1−푤1)푡1휎1 푓SL(�푤1; 푞) 퐹ESL(퐵휌;�푤1,푞+1) +퐹SL(퐵0;푞) +10 + +3.4 +The squared Mahalanobis Distance +The squared Mahalanobis distance of a random vector W = (푊1, 푊2)⊤ and the vector log(η) = +(log(휂1), log(휂2))⊤ of a BULS distribution is defined as +푑2(W , log(η)) = +� +푊2 +1 − 2휌 � +푊1 � +푊2 + � +푊2 +2 +1 − 휌2 +, +where +� +푊푖 = log +��푇푖 +휂푖 +�1/휎푖� +, 푇푖 = − log(1 − 푊푖), 휂푖 = exp(휇푖), 푖 = 1, 2. +Analogously to Propositions 3.8 and 3.9 of Vila et al. (2022) we have the following formulas for the +CDF and PDF of the random variable 푑2(W , log(η)): +퐹푑2(W ,log(η))(푥) = +4 +푍푔푐 +∫ √푥 +0 +�∫ √ +푥−푧2 +1 +0 +푔푐(푧2 +1 + 푧2 +2) d푧2 +� +d푧1, +푥 > 0, +푓푑2(W ,log(η))(푥) = +휋 +푍푔푐 +푔푐(푥), +푥 > 0, +where 푍푔푐 is as in (2.2). +For example, by taking 푔푐(푥) = exp(−푥/2) and 푍푔푐 = 2휋 (see Table 1), we get 푑2(W , log(η)) ∼ +휒2 +2 (chi-square with 2 degrees of freedom). +By taking 푔푐(푥) = (1 + (푥/휈))−(휈+2)/2 and 푍푔푐 += +Γ(휈/2)휈휋/Γ((휈 + 2)/2) (see Table 1), we have 푑2(W , log(η)) ∼ 2퐹2,휈, where 퐹2,휈 denotes the F- +distribution with 2 and 휈 degrees of freedom. +3.5 +Independence +Proposition 3.10. Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐). If 휌 = 0 and the density generator 푔푐 in (2.3) +satisfies +푔푐 +�푥2 + 푦2� = 푔푐1 +�푥2�푔푐2 +�푦2�, +∀(푥, 푦) ∈ R2, +(3.8) +for some density generators 푔푐1 and 푔푐2, then 푊1 and 푊2 are independent. +Proof. The proof follows the same steps as the proof of Proposition 3.11 of Vila et al. (2022). For the +convenience of the reader, we put the details of this one here. +Let 휌 = 0. By using (3.8), the joint density (2.3) of (푊1, 푊2) satisfies +푓푊1,푊2(푤1, 푤2; θ) = +푍푔푐1 푍푔푐2 +푍푔푐 +푓1(푤1; 휇1, 휎1) 푓2(푤2; 휇2, 휎2), +∀(푤1, 푤2) ∈ (0, 1) × (0, 1), +(3.9) +where 푓푖(푤푖; 휇푖, 휎푖) = 푔푐푖 +� � +푤푖 +2�/[(1 − 푤푖)푡푖휎푖푍푔푐푖 ], 0 < 푤푖 < 1; 푍푔푐푖 = +∫ ∞ +−∞ 푔푐푖 +�푧푖2� d푧푖, 푖 = 1, 2; and � +푤푖 +and 푡푖 are as in (2.3). Integrating (3.9) in terms of 푤1 and 푤2 we obtain +푍푔푐1 푍푔푐2 +푍푔푐 += 1, +11 + +and consequently, 푍푔푐 = 푍푔푐1 푍푔푐2. Therefore, +푓푊1,푊2(푤1, 푤2; θ) = 푓1(푤1; 휇1, 휎1) 푓2(푤2; 휇2, 휎2), +∀(푤1, 푤2) ∈ (0, 1) × (0, 1). +Moreover, it is simple to verify that 푓1 and 푓2 are PDFs corresponding to univariate symmetric random +variables (Vanegas and Paula, 2016). +Then 푊1 and 푊2 are statistically independent, and even more, +푓푖 = 푓푊푖, for 푖 = 1, 2 (see Proposition 2.5 of James, 2004). +□ +Remark 3.11. In Table 1, the density generator of the bivariate unit-log-normal is the unique one that +fulfills (3.8). +3.6 +Moments +Since for W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐), 0 < 푊푖 < 1, its is clear that 0 ⩽ E(푊푟 +푖 ) ⩽ 1, for any 푟 > 0 and +푖 = 1, 2. Therefore, the positive moments of 푊푖 always exist. +In general, for any 푟 ∈ R, the moments of 푊푖 admit the following representation: +E(푊푟 +1) += E +� +1 − exp +� +− 휂1 exp(휎1푍1) +��푟 , +E(푊푟 +2) += E�1 − exp +� +− 휂2 exp �휎2 +� +휌푍1 + +� +1 − 휌2 푍2 +����푟, +where 푍1 and 푍2 are given in Proposition 3.1. +4 +Maximum likelihood estimation +Let {(푊1푖, 푊2푖)⊤ : 푖 = 1, . . . , 푛} be a bivariate random sample of size 푛 from the BULS(θ, 푔푐) distribution +with PDF as given in (2.3), and let (푤1푖, 푤2푖)⊤ be the correspondent observations of (푊1푖,푊2푖)⊤. Then, +the log-likelihood function for θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤, without the additive constant, is expressed as +ℓ(θ) = −푛 +2 +� +푖=1 +log(휎푖) − 푛 +2 log �1 − 휌2� + +푛 +� +푖=1 +log 푔푐 +� +�푤2 +1푖 − 2휌�푤1푖�푤2푖 + �푤2 +2푖 +1 − 휌2 +� +, +0 < 푤1푖, 푤2푖 < 1, +where +�푤푘푖 = log +��푡푘푖 +휂푘 +�1/휎푘� +, 푡푘푖 = − log(1 − 푤푘푖), 휂푘 = exp(휇푘), 푘 = 1, 2; 푖 = 1, . . . , 푛. +In the case that a supremum �θ = ( �휂1, �휂2, � +휎1, � +휎2, �휌)⊤ exists, it must satisfy the following likelihood +equations: +휕ℓ(θ) +휕휂1 +���� +θ=�θ += 0, +휕ℓ(θ) +휕휂2 += 0, +휕ℓ(θ) +휕휎1 +���� +θ=�θ += 0, +휕ℓ(θ) +휕휎2 +���� +θ=�θ += 0, +휕ℓ(θ) +휕휌 +���� +θ=�θ += 0, +(4.1) +12 + +with +휕ℓ(θ) +휕휂1 += +2 +휎1휂1(1 − 휌2) +푛 +� +푖=1 +�휌�푤2푖 − �푤1푖 +�퐺(�푤1푖, �푤2푖), +휕ℓ(θ) +휕휂2 += +2 +휎2휂2(1 − 휌2) +푛 +� +푖=1 +�휌�푤1푖 − �푤2푖 +�퐺(�푤1푖, �푤2푖), +휕ℓ(θ) +휕휎1 += − 푛 +휎1 ++ +2 +휎1(1 − 휌2) +푛 +� +푖=1 +�푤1푖 +�휌�푤2푖 − �푤1푖 +�퐺(�푤1푖, �푤2푖), +휕ℓ(θ) +휕휎2 += − 푛 +휎2 ++ +2 +휎2(1 − 휌2) +푛 +� +푖=1 +�푤2푖 +�휌�푤1푖 − �푤2푖 +�퐺(�푤1푖, �푤2푖), +휕ℓ(θ) +휕휌 += +푛휌 +1 − 휌2 − +2 +(1 − 휌2)2 +푛 +� +푖=1 +�휌�푤1푖 − �푤2푖 +� �휌�푤2푖 − �푤1푖 +�퐺(�푤1푖, �푤2푖), +(4.2) +where we are adopting the notation: +퐺(�푤1푖, �푤2푖) = 푔′ +푐(푥휌,푖) +푔푐(푥휌,푖), +(4.3) +with 푥휌,푖 = (�푤2 +1푖 − 2휌�푤1푖�푤2푖 + �푤2 +2푖)/(1 − 휌2), 푖 = 1, . . . , 푛. +Observe that the likelihood equations (4.1) can be written as follows +푛 +� +푖=1 +�푤1푖 퐺(�푤1푖, �푤2푖) +���� +θ=�θ += 0, +푛 +� +푖=1 +��푤2 +1푖 − �푤2 +2푖 +� 퐺(�푤1푖, �푤2푖) +���� +θ=�θ += 0, +푛 +� +푖=1 +�푤2푖 +� +2휌�푤2푖 − (1 + 휌2)�푤1푖 +� +퐺(�푤1푖, �푤2푖) +���� +θ=�θ += −푛�휌(1 − �휌2) +2 +. +Any nontrivial root �θ of the above likelihood equations is known as an ML estimator in the loose sense. +When the parameter value provides the absolute maximum of the log-likelihood function, it is called an +ML estimator in the strict sense. +In the following proposition we study the existence of the ML estimator �휌 when the other parameters +are known. +Proposition 4.1. Let 푔푐 be a density generator satisfying the following condition: +푔′ +푐(푥) = 푟(푥)푔푐(푥), +−∞ < 푥 < ∞, +(4.4) +for some real-valued function 푟(푥) so that lim휌→±1 푟(푥휌,푖) = 푐 ∈ (−∞, 0), where 푥휌,푖, 푖 = 1, . . . , 푛, are as +in (4.3). If the parameters 휂1, 휂2, 휎1 and 휎2 are known, then the equation (4.2) has at least one root on the +interval (−1, 1). +13 + +Proof. The proof of this result follows by direct application of Intermediate value theorem, so details of +the proof are omitted. For more details of the proof, see Proposition 5.1. of Vila et al. (2022). +□ +For the BULS model no closed-form solution to the maximization problem is known or available, and an +MLE can only be found via numerical optimization. Under mild regularity conditions (Cox and Hinkley, +1974; Davison, 2008), the asymptotic distribution of ML estimator �θ of θ is easily determined by the +convergence in law: (�θ − θ) +풟 +−→ 푁(0, 퐼−1(θ)), where 0 is the zero mean vector and 퐼−1(θ) is the inverse +expected Fisher information matrix. The main use of the last convergence is to construct confidence regions +and to perform hypothesis testing for θ (Davison, 2008). +5 +Simulation study +In this section, we carry out a Monte Carlo simulation study to evaluate the performance of the ML +estimators for the BULS distributions. For illustration purposes we only present results for the bivariate +unit-log-normal model. +The simulation scenario considers the following setting: 1,000 Monte Carlo +replications, sample size 푛 ∈ (50, 50, 100, 150), vector of true parameters (휂1, 휂2, 휎1, 휎2) = (1, 1, 0.5, 0.5), +휌 ∈ {0, 0.25, 0.5, 0.75, 0.95} (negative values of 휌 produce the same results and then are omitted). To +study the ML estimators, we compute the bias, root mean square error (RMSE), and coverage probability +(CP), defined by +� +Bias(�휃) += +1 +푁 +푁 +� +푖=1 +�휃(푖) − 휃, +� +RMSE(�휃) += +� +� +� +1 +푁 +푁 +� +푖=1 +(�휃(푖) − 휃)2 , +� +CP(�휃) += +1 +푁 +푁 +� +푖=1 +I(휃 ∈ [퐿(푖) +�휃 ,푈(푖) +�휃 ]), +where 휃 and �휃(푖) are the true parameter value and its respective 푖-th ML estimate, 푁 is the number of Monte +Carlo replicas, I is an indicator function taking the value 1 if 휃 ∈ [퐿(푖) +�휃 , 푈(푖) +�휃 ], and 0 otherwise, where 퐿(푖) +�휃 +and 푈(푖) +�휃 +are the 푖-th upper and lower limit estimates of the 95% confidence interval. We expect that, as the +sample size increases, the bias and RMSE reduce, and the CP approaches the 95% nominal level. +The simulation results are presented in Figure 1. We observe that the results obtained for the chosen +bivariate unit-log-normal distribution are as expected. As the sample size increases, the bias and RMSE +tend to decrease. Moreover, the CP approaches the 95% nominal level. Finally, in general, the results do +not seem to depend on the parameter 휌. +14 + +0 +100 +300 +500 +700 +0.000 +0.005 +0.010 +0.015 +n +Bias +η^1(ρ = 0) +η^1(ρ = 0.25) +η^1(ρ = 0.5) +η^1(ρ = 0.75) +η^1(ρ = 0.95) +0 +100 +300 +500 +700 +0.000 +0.005 +0.010 +0.015 +n +Bias +η^2(ρ = 0) +η^2(ρ = 0.25) +η^2(ρ = 0.5) +η^2(ρ = 0.75) +η^2(ρ = 0.95) +0 +100 +300 +500 +700 +−0.06 +−0.02 +0.00 +0.02 +n +Bias +σ^1(ρ = 0) +σ^1(ρ = 0.25) +σ^1(ρ = 0.5) +σ^1(ρ = 0.75) +σ^1(ρ = 0.95) +0 +100 +300 +500 +700 +−0.06 +−0.02 +0.00 +0.02 +n +Bias +σ^2(ρ = 0) +σ^2(ρ = 0.25) +σ^2(ρ = 0.5) +σ^2(ρ = 0.75) +σ^2(ρ = 0.95) +0 +100 +300 +500 +700 +−0.008 +−0.004 +0.000 +0.004 +n +Bias +ρ^(ρ = 0) +ρ^(ρ = 0.25) +ρ^(ρ = 0.5) +ρ^(ρ = 0.75) +ρ^(ρ = 0.95) +0 +100 +300 +500 +700 +0.02 +0.04 +0.06 +0.08 +0.10 +n +RMSE +η^1(ρ = 0) +η^1(ρ = 0.25) +η^1(ρ = 0.5) +η^1(ρ = 0.75) +η^1(ρ = 0.95) +0 +100 +300 +500 +700 +0.02 +0.04 +0.06 +0.08 +0.10 +n +RMSE +η^2(ρ = 0) +η^2(ρ = 0.25) +η^2(ρ = 0.5) +η^2(ρ = 0.75) +η^2(ρ = 0.95) +0 +100 +300 +500 +700 +0.0 +0.1 +0.2 +0.3 +0.4 +n +RMSE +σ^1(ρ = 0) +σ^1(ρ = 0.25) +σ^1(ρ = 0.5) +σ^1(ρ = 0.75) +σ^1(ρ = 0.95) +0 +100 +300 +500 +700 +0.0 +0.1 +0.2 +0.3 +0.4 +n +RMSE +σ^2(ρ = 0) +σ^2(ρ = 0.25) +σ^2(ρ = 0.5) +σ^2(ρ = 0.75) +σ^2(ρ = 0.95) +0 +100 +300 +500 +700 +0.00 +0.05 +0.10 +0.15 +0.20 +n +RMSE +ρ^(ρ = 0) +ρ^(ρ = 0.25) +ρ^(ρ = 0.5) +ρ^(ρ = 0.75) +ρ^(ρ = 0.95) +0 +100 +300 +500 +700 +0.88 +0.90 +0.92 +0.94 +0.96 +n +CP +η^1(ρ = 0) +η^1(ρ = 0.25) +η^1(ρ = 0.5) +η^1(ρ = 0.75) +η^1(ρ = 0.95) +0 +100 +300 +500 +700 +0.88 +0.90 +0.92 +0.94 +0.96 +n +CP +η^2(ρ = 0) +η^2(ρ = 0.25) +η^2(ρ = 0.5) +η^2(ρ = 0.75) +η^2(ρ = 0.95) +0 +100 +300 +500 +700 +0.84 +0.88 +0.92 +0.96 +n +CP +σ^1(ρ = 0) +σ^1(ρ = 0.25) +σ^1(ρ = 0.5) +σ^1(ρ = 0.75) +σ^1(ρ = 0.95) +0 +100 +300 +500 +700 +0.84 +0.88 +0.92 +0.96 +n +CP +σ^2(ρ = 0) +σ^2(ρ = 0.25) +σ^2(ρ = 0.5) +σ^2(ρ = 0.75) +σ^2(ρ = 0.95) +0 +100 +300 +500 +700 +0.86 +0.90 +0.94 +n +CP +ρ^(ρ = 0) +ρ^(ρ = 0.25) +ρ^(ρ = 0.5) +ρ^(ρ = 0.75) +ρ^(ρ = 0.95) +Figure 1: Monte Carlo simulation results for the bivariate unit-log-normal model. +6 +Application to soccer data +In this section, two real soccer data sets, corresponding to times elapsed until scored goals of UEFA +Champions League and pass completions of 2022 FIFA World Cup, are analyzed. The UEFA Champions +League data set was extracted from Meintanis (2007), whereas the 2022 FIFA World Cup data set is new +and is analyzed for the first time here. +6.1 +UEFA Champions League +We consider a bivariate data set on the group stage of the UEFA Champions League for the seasons 2004/05 +and 2005/06. Only matches with at least one goal scored directly from a kick by any team, and with at +least one goal scored by the home team, are considered; see Meintanis (2007). The first variable (푊1) is +the time (in minutes) elapsed until a first kick goal is scored by any team, and the second one 푊2 is the +time (in minutes) elapsed until a first goal of any type is scored by the home team. The times are divided +by 90 minutes (full game time) to obtain data on the unit square (0, 1) × (0, 1); see Table 7. +Table 3 provides descriptive statistics for the variables 푊1 and 푊2, including the minimum, median, +mean, maximum, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS), and +coefficient of kurtosis (CK). We observe in the variable 푊1, the mean and median to be, respectively, 0.454 +15 + +and 0.456, i.e., the mean is almost equal to the median, which indicates symmetry in the data. The CV +is 49.274%, which means a moderate level of dispersion around the mean. Furthermore, the CS value +also confirms the symmetry nature. The variable 푊1 has mean equal to 0.365 and median equal to 0.311, +which indicates the small positively skewed feature in the distribution of the data. Moreover, the CV value +is 69.475%, showing a moderate level of dispersion around the mean. The CS confirms the small skewed +nature and the CK value indicates the small kurtosis feature in the data. +Table 3: Summary statistics for the UEFA Champions League data set. +Variables +푛 +Minimum +Median +Mean +Maximum +SD +CV +CS +CK +푊1 +37 +0.022 +0.456 +0.454 +0.911 +0.224 +49.274 +0.164 +-0.930 +푊2 +37 +0.022 +0.311 +0.365 +0.944 +0.254 +69.475 +0.522 +-0.839 +The ML estimates and the standard errors (in parentheses) for the bivariate unit-log-symmetric model +parameters are presented in Table 4. The extra parameters, associated with log-Student-푡, log-hyperbolic +and log-slash models, were estimated using the profile log-likelihood; see Saulo et al. (2022). Table 4 +also presents the log-likelihood value, and the values of the Akaike (AIC) and Bayesian (BIC) information +criteria. We observe that the log-hyperbolic model provides better fit than other models based on the +values of log-likelihood, AIC and BIC. Note, however, that the values of log-likelihood, AIC and BIC of +all bivariate unit-log-symmetric models are quite close to each other. +Table 4: ML estimates (with standard errors in parentheses), log-likelihood, AIC and BIC values for the +indicated bivariate unit-log-symmetric models. +Distribuiton +�휂1 +�휂2 +�휎1 +�휎2 +�휌 +�휈 +Log-likelihood +AIC +BIC +Log-normal +0.5288* +0.3414* +0.8865* +1.1355* +0.4956* +– +-36.693 +83.386 +91.441 +(0.0771) +(0.0637) +(0.1031) +(0.1320) +(0.1240) +Log-Student-푡 +0.5541* +0.3783* +0.7431* +0.9734* +0.4723* +7 +-35.487 +80.974 +89.029 +(0.0751) +(0.0672) +(0.1033) +(0.1308) +(0.1463) +Log-hyperbolic +0.5458* +0.3816* +0.8456* +1.0950* +0.4893* +2 +-35.470 +80.940 +88.996 +(0.0752) +(0.0677) +(0.1162) +(0.1462) +(0.1428) +Log-Laplace +0.5680* +0.5679* +0.9928* +1.3231* +0.5281* +– +-36.009 +82.019 +90.073 +(0.0020) +(0.0021) +(0.1692) +(0.2164) +(0.1639) +Log-slash +0.5629* +0.3715* +0.6203* +0.8302* +0.4472* +5 +-35.560 +81.120 +89.174 +(0.0749) +(0.0666) +(0.0847) +(0.1096) +(0.1472) +∗ significant at 5% level. +Figure 2 shows the QQ plots of the Mahalanobis distance for the bivariate unit-log-symmetric models +considered in Table 4. The QQ plot is a plot of the empirical quantiles of the Mahalanobis distance against +the theoretical quantiles of the respective reference distribution (see Subsection 3.4). Therefore, points +falling along a straight line indicate a good fit. From Figure 2, we see clearly that, with the exception of +the log-Student-푡 case, the Mahalanobis distances in the considered models conform relatively well with +their reference distributions. We also see that, in all the cases, there is a point away from the reference line, +which may be an outlier. +16 + +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +10 +12 +theoretical quantile +empirical quantile +(a) Log-normal +0 +2 +4 +6 +8 +0 +5 +10 +15 +20 +theoretical quantile +empirical quantile +(b) Log-Student-푡 +0 +2 +4 +6 +8 +10 +12 +0 +5 +10 +15 +theoretical quantile +empirical quantile +(c) Log-hyperbolic +0 +5 +10 +15 +0 +2 +4 +6 +8 +10 +theoretical quantile +empirical quantile +(d) Log-Laplace +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +theoretical quantile +empirical quantile +(e) Log-slash +Figure 2: QQ plot of the Mahalanobis distances for the indicated models. +6.2 +2022 FIFA World Cup +In this subsection, data on the 2022 FIFA World Cup are used to illustrate the proposed methodology. The +data are available at https://www.kaggle.com/. The first variable (푊1) is the medium pass completion +proportion, that is, successful passes between 14 and 18 meters. The second variable (푊2) is the long pass +completion proportion, namely, passes longer than 37 meters; see Table 7. +Table 5 provides descriptive statistics for the variables 푊1 and 푊2. We observe in the variable 푊1, the +mean and median to be, respectively, 0.454 and 0.456, i.e., the mean is almost equal to the median, which +indicates symmetry in the data. The CV is 49.274%, which means a moderate level of dispersion around +the mean. Furthermore, the CS value also confirms the symmetry nature. The variable 푊2 has mean equal +to 0.365 and median equal to 0.311, which indicates the small positively skewed feature in the distribution +of the data. Moreover, the CV value is 69.475%, showing a moderate level of dispersion around the mean. +The CS confirms the small skewed nature and the CK value indicates the small kurtosis feature in the data. +17 + +Table 5: Summary statistics for the 2022 FIFA World Cup data set. +Variables +푛 +Minimum +Median +Mean +Maximum +SD +CV +CS +CK +푊1 +32 +0.769 +0.860 +0.860 +0.931 +0.038 +4.376 +-0.373 +-0.194 +푊2 +32 +0.427 +0.556 +0.550 +0.751 +0.075 +13.713 +0.308 +-0.425 +Table 6 presents the estimation results for the bivariate unit-log-symmetric models. The results of this +table reveal that the log-normal model provides better adjustment than the other models based on the values +of log-likelihood, AIC and BIC. +Figure 3 shows the QQ plots of the Mahalanobis distances (see Subsection 3.4) for the bivariate unit- +log-symmetric models considered in Table 6. We see clearly that the log-normal model provides better fit +than the other bivariate unit-log-symmetric models. +Table 6: ML estimates (with standard errors in parentheses), log-likelihood, AIC and BIC values for the +indicated bivariate unit-log-symmetric models. +Distribuiton +�휂1 +�휂2 +�휎1 +�휎2 +�휌 +�휈 +Log-likelihood +AIC +BIC +Log-normal +1.9872* +0.7953* +0.1364* +0.2089* +0.7343* +– +20.791 +-31.581 +-24.252 +(0.0479) +(0.0294) +(0.0171) +(0.0261) +(0.0815) +Log-Student-푡 +1.9954* +0.7936* +-0.1257* +-0.1949* +0.7423* +9 +20.130 +-30.260 +-22.931 +(0.0485) +(0.0299) +(0.0178) +(0.0271) +(0.0868) +Log-hyperbolic +1.9908* +0.7942* +0.3956* +0.6088* +0.7378* +10 +20.618 +-31.236 +-23.907 +(0.0482) +(0.0296) +(0.0523) +(0.0800) +(0.0841) +Log-Laplace +1.9908* +0.8089* +0.1563* +0.2425* +0.7471* +– +16.915 +-23.830 +-16.501 +(0.0023) +(0.0021) +(0.0278) +(0.0415) +(0.0938) +Log-slash +1.9897* +0.7935* +0.1173* +0.1802* +0.7392* +8 +20.613 +-28.919 +-23.898 +(0.0482) +(0.0295) +(0.0154) +(0.0237) +(0.0844) +∗ significant at 5% level. +7 +Conclusions +In the present paper we have proposed a family of bivariate distributions over the unit square. By adequately +defining the density generator we can transform any distribution over the real line into a bivariate distribution +over the region (0, 1) × (0, 1). Such a model has several potential applications, since the simultaneous +modeling of quantities like proportions, rates or indices frequently arises in applied sciences like economy, +medicine engineering and social sciences. We have addressed diverse theoretical properties like stochastic +representation, quantiles, conditional distributions, independence and moments and practical questions like +estimation, a Monte Carlo Simulation Study and applications to soccer data. The present research can be +extended in several possible directions. By changing the density generator, numerous special forms of the +BULS distribution can be constructed. Furthermore, generalizations to higher dimensions can be studied. +18 + +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +theoretical quantile +empirical quantile +(a) Log-normal +0 +2 +4 +6 +8 +0 +2 +4 +6 +8 +10 +theoretical quantile +empirical quantile +(b) Log-Student-푡 +0 +2 +4 +6 +8 +10 +12 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +theoretical quantile +empirical quantile +(c) Log-hyperbolic +0 +2 +4 +6 +8 +10 +12 +14 +0 +1 +2 +3 +4 +5 +6 +theoretical quantile +empirical quantile +(d) Log-Laplace +0 +5 +10 +15 +20 +0 +2 +4 +6 +8 +10 +12 +theoretical quantile +empirical quantile +(e) Log-slash +Figure 3: QQ plot of the Mahalanobis distances for the indicated models. +Acknowledgements +Roberto Vilaand Helton Saulo gratefully acknowledgefinancial support from CNPq, +CAPES and FAP-DF, Brazil. +Disclosure statement +There are no conflicts of interest to disclose. +References +Abdous, B., Fougères and A.-L., Ghoudi, K., “Extreme behaviour for bivariate elliptical distributions", +Canadian Journal of Statistics, pp. 317–334, 2005. +Balakrishnan, N. and Lai, C-D., Continuous Bivariate Distributions, Springer-Verlag, New York, 2009. +Cox, D. R. and Hinkley, D. V., Theoretical statistics, London, UK: Chapman and Hall, 1974. +Davison, A. C., Statistical Models (First ed.). Cambridge Series in Statistical and Probabilistic 408 Math- +ematics, Cambridge University Press, 2008. +19 + +Deng, X. and Yao, J. (2018). On the property of multivariate generalized hyperbolic distribution and the +Stein-type inequality. Communications in Statistics-Theory and Methods, 47:5346–5356. +Fang, K. T., Kotz, S. and Ng, K. W., Symmetric multivariate and related distributions, Chapman and Hall, +London, 1990. +Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press, San Diego, +2000. +Heckman, J. J., “Sample selection bias as a specification error", Econometrica, pp. 153-161, 1979. +James, B. R., Probabilidade: um curso em nível intermediário, Projeto Euclides, Brazil, 2004. +Kotz, S., Kozubowski, T. J. and Podgórski, K., The Laplace Distribution and Generalizations, 2001. +Martínez-Flórez, G., Lemonte, A. J., Moreno-Arenas, G. and Tovar-Falón, R., “The Bivariate Unit-Sinh- +Normal Distribution and Its Related Regression Model”, Mathematics, 2022. +Meintanis, S. G., “Test of fit for Marshall-Olkin distributions with applications", Journal of Statistical +Planning and Inference, 137:3954-3963, 2007. +Rohatgi V. K. and Saleh, A. K. Md. E., An introduction to probability theory and mathematical statistics, +John Wiley & Sons, 3rd edition, 2015. +Saulo, H., Balakrishnan, N., Zhu, X., Gonzales, J. F. B. and Leão, J., “Estimation in generalized bivariate +Birnbaum-Saunders models", Metrika, pp. 427-453, 2017. +Saulo, H., Dasilva, A., Leiva, V., Sánchez, L., and Fuente-Mella, H. L. “Log-symmetric quantile regression +models”, Statistica Neerlandica, 76:124–163, 2022. +Saulo, H., Vila, R., Cordeiro, S. S. and Leiva, V. “Bivariate symmetric Heckman models and their +characterization”, Journal of Multivariate Analysis, 105097, 2022. +Vanegas, L. H. and Paula, G. A., “Log-symmetric distributions: Statistical properties and parameter +estimation", Brazilian Journal of Probability and Statistics, pp. 196-220, 2016. +Vila, R., Balakrishnan, N., Saulo, H. and Protazio, A. “Bivariate Log-symmetric Models: Theoretical Prop- +erties and Parameter Estimation”, Preprint, Avaliable at https://arxiv.org/pdf/2211.13839.pdf, +2022. +Wang, J. and Genton, M. G. (2006). The multivariate skew-slash distribution. Journal of Statistical +Planning and Inference, 136:209–220. +20 + +Appendix A +Some additional results +For the convenience of the reader, in this section, some complementary results related to the Subsection +3.3 are presented in detail. +Definition A.1. We say that a random variable 푋 follows a univariate generalised hyperbolic (GH) distri- +bution, denoted by 푋 ∼ GH(휆, 훼, 훿), if its PDF is given by +푓GH(푥; 휆, 훼, 훿) = +( +√ +훼2 /훿)휆 +√ +2휋 퐾휆(훿 +√ +훼2 ) +퐾휆−1/2(훼 +√ +훿2 + 푥2 ) +( +√ +훿2 + 푥2 /훼)1/2−휆 , +−∞ < 푥 < ∞. +Here, 퐾푟 is the modified Bessel function of the third kind with index 푟, 휆 ∈ R, 훼 ∈ R and 훿 > 0 is a scale +parameter. +The following result has appeared in a multivariated version in Deng and Yao (2018, Proposition 3, p. +5). +Proposition A.1 (Hyperbolic generator). Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.1. If +푔푐(푥) = exp(−휈 +√ +1 + 푥 ) then the conditional distribution of 푍2 given 푍1 = 푥 is GH(1, 휈, +√ +1 + 푥2 ) and its +unconditional distributions both are GH(3/2, 휈, 1). +Proof. By using (3.4), the joint PDF of 푍1 and 푍2 is (see Table 1) +푓푍1,푍2(푥, 푦) = 휈2 exp(휈) +2휋(휈 + 1) exp +� +−휈 +� +1 + 푥2 + 푦2 +� +. +(A.1) +The marginal PDF of 푍1 is given by +푓푍1(푥) = +∫ ∞ +−∞ +푓푍1,푍2(푥, 푦) d푦 = 휈2 exp(휈) +2휋(휈 + 1) +∫ ∞ +−∞ +exp +� +−휈 +� +1 + 푥2 + 푦2 +� +d푦 += 2휈2 exp(휈) +2휋(휈 + 1) +∫ ∞ +0 +exp +� +−휈 +� +1 + 푥2 + 푦2 +� +d푦. +By using the Formula 6 of Section 3.46-3.48 of reference Gradshteyn and Ryzhik (2000, p. +364): +∫ ∞ +0 exp(−푎 +√ +푏2 + 푥2 )d푥 = 푏퐾1(푎푏), the above integral is += 2휈2 exp(휈) +2휋(휈 + 1) +� +1 + 푥2 퐾1(휈 +� +1 + 푥2 ). +(A.2) +Since 퐾3/2(휈) = 2√휋 exp(−휈)(휈 + 1)(휈/2)3/2/휈3, the above espression is written as += +휈3/2 +√ +2휋 퐾3/2(휈) +퐾1(휈 +√ +1 + 푥2 ) +( +√ +1 + 푥2 /휈 )−1 = 푓GH(푥; 3/2, 휈, 1). +This proves that 푍1 ∼ GH(3/2, 휈, 1). Analogously, we prove that 푍2 ∼ GH(3/2, 휈, 1). +21 + +On the other hand, employing (A.1) and (A.2), and by using the the well-known identity 퐾1/2(푧) = +� +휋/(2푧) exp(−푧), the conditional PDF of 푍2 given 푍1 = 푥 is +푓푍2 | 푍1(푦 | 푥) = +exp(−휈 +� +1 + 푥2 + 푦2 ) +2 +√ +1 + 푥2 퐾1(휈 +√ +1 + 푥2 ) += +휈/ +√ +1 + 푥2 +√ +2휋 퐾1(휈 +√ +1 + 푥2 ) +퐾1/2(휈 +� +1 + 푥2 + 푦2 ) +( +� +1 + 푥2 + 푦2 /휈)−1/2 += 푓GH(푥; 1, 휈, +� +1 + 푥2 ). +We thus complete the proof. +□ +The following result has also appeared in a multivariated version in the reference Kotz et al. (2001, +Theorem 6.7.1, p. 253). +Proposition A.2 (Laplace generator). Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.1. If +푔푐(푥) = 퐾0( +√ +2푥 ) then the conditional distribution of 푍2 given 푍1 = 푥 is GH(1/2, +√ +2 , |푥|) and its +unconditional distributions both are Laplace(0, 1/ +√ +2 ). +Proof. By (3.4) and by using the definitions of 퐾0 and 푍푔푐 in Table 1, +푓푍1,푍2(푥, 푦) = 1 +휋 퐾0 +�� +2(푥2 + 푦2) +� += 1 +2휋 +∫ ∞ +0 +1 +푡 exp +� +−푡 − 푥2 + 푦2 +2푡 +� +d푡. +(A.3) +An algebraic calculation shows that the marginal of 푍1 is +푓푍1(푥) = +∫ ∞ +−∞ +푓푍1,푍2(푥, 푦) d푦 = +1 +√ +2 +exp(− +√ +2 |푥|) = 푓L(푥), +(A.4) +where 푓L(푥) = exp(− +√ +2 |푥|)/ +√ +2 is the Laplace PDF with scale parameter 1/ +√ +2 . +That is, 푍1 ∼ +Laplace(0, 1/ +√ +2 ). Analogously we verify that 푍2 ∼ Laplace(0, 1/ +√ +2 ). +On the other hand, by using (A.3), (A.4) and the well-known identity 퐾1/2(푧) = +� +휋/(2푧) exp(−푧), the +conditional PDF of 푍2 given 푍1 = 푥 can be written as +푓푍2 | 푍1(푦 | 푥) = +1 +휋 퐾0 +�� +2(푥2 + 푦2) +� +1 +√ +2 +exp(− +√ +2 |푥|) += +( +√ +2 /|푥|)1/2 +√ +2휋 퐾1/2( +√ +2 |푥|) +퐾0 +�√ +2 +� +푥2 + 푦2 +� += 푓GH(푥; 1/2, +√ +2 , |푥|). +Following the Definition A.1 we get that 푍2 | 푍1 = 푥 ∼ GH(1/2, +√ +2 , |푥|). +□ +Definition A.2. We say that a random variable 푋 follows a univariate extended slash (ESL) distribution, +denoted by 푋 ∼ ESL(푎, 푞), if its PDF is given by +푓ESL(푥; 푎, 푞) = +∫ 1 +0 +푡푞휙(푡푎)휙(푡푥) d푡 +∫ 1 +0 +푢푞−1휙(푢푎) d푢 +, +−∞ < 푥 < ∞, +22 + +where 휙 denotes the PDF of the standard normal distribution. +Choosing 푎 = 0 the classical slash (SL) PDF is obtained. The resulting density takes the form +푓SL(푥; 푞) = 푞 +∫ 1 +0 +푡푞휙(푡푥) d푡, +−∞ < 푥 < ∞ += 푞 2 +푞 +2 −1 +√휋 +|푥|−(푞+1) 훾 +�푞 + 1 +2 +, 푥2 +2 +� +. +In this case we denote 푋 ∼ SL(푞). +Proposition A.3 (Slash generator). Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.1. +If +푔푐(푥) = 푥−(푞+2)/2훾((푞 + 2)/2, 푥/2) then the conditional distribution of 푍2 given 푍1 = 푥 is ESL(푥, 푞 + 1) +and its unconditional distributions both are SL(푞). +Proof. By (3.4) and by definition of 푍푔푐 in Table 1, the joint PDF of 푍1 and 푍2 is given by +푓푍1,푍2(푥, 푦) = 푞 2 +푞 +2 −1 +휋 +(푥2 + 푦2)− 푞+2 +2 훾 +�푞 + 2 +2 +, 푥2 + 푦2 +2 +� += 푞 +∫ 1 +0 +푡푞+1휙(푡푥)휙(푡푦) d푡. +(A.5) +Then the marginal PDF of 푍1 is +푓푍1(푥) = +∫ ∞ +−∞ +푓푍1,푍2(푥, 푦) d푦 = 푞 +∫ 1 +0 +푡푞+1휙(푡푥) +�∫ ∞ +−∞ +휙(푡푦) d푦 +� +d푡 += 푞 +∫ 1 +0 +푡푞휙(푡푥) d푡 = 푓SL(푥; 푞). +(A.6) +This proves that (see Definition A.2), 푍1 ∼ SL(푞). Analogously it is proved that 푍2 ∼ SL(푞). +On the other hand, by (A.5) and (A.6), the conditional PDF of 푍2 given 푍1 = 푥, is +푓푍2 | 푍1(푦 | 푥) = +∫ 1 +0 +푡푞+1휙(푡푥)휙(푡푦) d푡 +∫ 1 +0 +푢푞휙(푢푥) d푢 += 푓ESL(푦; 푥, 푞 + 1). +Following the Definition A.2 we have that 푍2 | 푍1 = 푥 ∼ ESL(푥, 푞 + 1). +□ +Appendix B +Data sets +23 + +Table 7: UEFA Champions League and 2022 FIFA World Cup data sets. +UEFA +FIFA +W1 +W2 +W1 +W2 +1 +0.289 +0.222 +0.888 +0.541 +2 +0.700 +0.200 +0.815 +0.474 +3 +0.211 +0.211 +0.907 +0.624 +4 +0.733 +0.944 +0.891 +0.606 +5 +0.444 +0.444 +0.827 +0.517 +6 +0.544 +0.544 +0.898 +0.557 +7 +0.089 +0.089 +0.856 +0.462 +8 +0.767 +0.789 +0.861 +0.618 +9 +0.433 +0.433 +0.890 +0.603 +10 +0.911 +0.533 +0.860 +0.477 +11 +0.800 +0.800 +0.920 +0.646 +12 +0.733 +0.689 +0.894 +0.587 +13 +0.278 +0.100 +0.913 +0.648 +14 +0.456 +0.033 +0.849 +0.471 +15 +0.178 +0.833 +0.781 +0.427 +16 +0.200 +0.200 +0.828 +0.442 +17 +0.244 +0.156 +0.864 +0.581 +18 +0.467 +0.467 +0.820 +0.527 +19 +0.022 +0.022 +0.846 +0.526 +20 +0.400 +0.578 +0.879 +0.601 +21 +0.378 +0.378 +0.860 +0.481 +22 +0.589 +0.433 +0.885 +0.616 +23 +0.600 +0.078 +0.862 +0.592 +24 +0.567 +0.311 +0.769 +0.463 +25 +0.844 +0.711 +0.845 +0.495 +26 +0.711 +0.167 +0.846 +0.489 +27 +0.289 +0.533 +0.931 +0.751 +28 +0.178 +0.178 +0.863 +0.555 +29 +0.489 +0.144 +0.856 +0.447 +30 +0.278 +0.156 +0.879 +0.569 +31 +0.611 +0.122 +0.812 +0.613 +32 +0.544 +0.544 +0.841 +0.594 +33 +0.267 +0.267 +– +– +34 +0.489 +0.333 +– +– +35 +0.467 +0.033 +– +– +36 +0.300 +0.522 +– +– +37 +0.311 +0.311 +– +– +24 + diff --git a/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/load_file.txt b/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5e39fea678facaf8be75ea98814b9ee306ced8a --- /dev/null +++ b/R9E5T4oBgHgl3EQfaA9_/content/tmp_files/load_file.txt @@ -0,0 +1,1210 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf,len=1209 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='05585v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='ME] 13 Jan 2023 Bivariate Distributions on the Unit Square: Theoretical Properties and Applications Roberto Vila ∗1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Narayanaswamy Balakrishnan † 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Helton Saulo ‡ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' and Peter Zörnig §1 1Department of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' University of Brasília,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Brasília,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Brazil 2 Department of Mathematics and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' McMaster University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hamilton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Canada January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 2023 Abstract We present the bivariate unit-log-symmetric model which is based on the bivariate log-symmetric distribution (BLS) defined in Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is a flexible family of bivariate distributions over the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We study mathematical properties like stochastic representations, quantiles, condi- tional distributions, independence of the marginal distributions and moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Maximum likelihood estimators, simulation results and applications to soccer data are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Bivariate unit-log-symmetric distribution · Bivariate log-symmetric distribution · Bivariate model · MCMC · Proportion data · Soccer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' MSC 60E05 · MSC 62Exx · MSC 62Fxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1 Introduction Bivariate distributions over the unit-square have been intensively studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Many of them are based on the beta distribution and its generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Models of this type have been studies since the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Other distributions on the unit square are based on the generalized arcsine distribution and the inverse Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' A very recent model, the bivariate unit-sinh-normal distributions, is based on the bivariate Birnbaum-Saunders distribution, see Martínez-Flórez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Bivariate distributions over the unit square arise naturally in comparing indices, rates or proportions in the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' ∗rovig161@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='com †bala@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='ca ‡heltonsaulo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='com §peter@unb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='br 1 In the present paper we study the bivariate unit-log-symmetric (BULS) distribution defined over the unit-square which is obtained as a modification of the bivariate log-symmetric (BLS) distribution introduced in Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The definitions of BLS and BULS are given in Section 2, indicating some special cases of the BULS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Section 3 we study some properties of the new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 presents a stochastic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Making use of this representation we derive formulas for the marginal quantiles in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 we study the conditional distributions of the BULS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We derive more compact formulas for the conditional densities, using the distribution functions of the normal, the 푡-Student, the hyperbolic, the Laplace and the slash distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' One of the uses of having closed formulas for the conditional densities (of the BULS model), for example, is for studying Heckman-type selection models (Heckman, 1979) when the selection variables have bounded support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4 we determine the distribution of the squared Mahalanobis distance of a random vector W = (푊1,푊2)⊤ with BULS distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' A necessary condition for the independence of the components of W is represented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Section 3 ends with formulas for the moments of 푊1 and 푊2 based on the stochastic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Section 4 the log-likelihood function and the likelihood equations for the BULS distribution are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Section 5 we carry out a Monte Carlo simulation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We evaluate the performance of the ML-estimators by means of bias, root mean square error and coverage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Section 6 we present two applications to soccer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 we model the vector W = (푊1, 푊2)⊤, where 푊1 represents the time elapsed until a first kick goal (of any team) and the time elapsed until a goal of any type of the home team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It turns out that specific BULS distributions are adequate to model the vector W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 we study the very actual data of the 2022 FIFA World Cup which has been realized in December of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Now the components of the vector W represent the pass completion proportions of medium passes (14 to 18 meters) and long passes (longer than 37 meters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' These data can also be well fitted BULS distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 2 The bivariate unit-log-symmetric model In this section (specifically, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2), we define the model of interest in this paper, the bivariate unit-log-symmetric model (BULS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' To define this model we first need to establish the bivariate log- symmetric distribution (BLS) which was naturally defined in Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 The BLS distribution Following Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022), a continuous random vector T = (푇1, 푇2)⊤ has a bivariate log-symmetric (BLS) distribution if its joint probability density function (PDF) is given by 푓푇1,푇2(푡1, 푡2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) = 1 푡1푡2휎1휎2 � 1 − 휌2 푍푔푐 푔푐 � �푡1 2 − 2휌�푡1�푡2 + �푡2 2 1 − 휌2 � , 푡1, 푡2 > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) where �푡푖 = log �� 푡푖 휂푖 �1/휎푖� , 휂푖 = exp(휇푖), 푖 = 1, 2, 2 with θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤ being the parameter vector, 휇푖 ∈ R, 휎푖 > 0, 푖 = 1, 2 and 휌 ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Furthermore, 푍푔푐 > 0 is the partition function, that is, 푍푔푐 = ∫ ∞ 0 ∫ ∞ 0 1 푡1푡2휎1휎2 � 1 − 휌2 푔푐 � �푡1 2 − 2휌�푡1�푡2 + �푡2 2 1 − 휌2 � d푡1d푡2 = 휋 ∫ ∞ 0 푔푐(푢) d푢, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) and 푔푐 is a scalar function referred to as the density generator (see Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The second integral in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) is consequence of a change of variables, for more details see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' When a random vector T is BLS distributed, with parameter vector θ, we write T ∼ BLS(θ, 푔푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 The BULS distribution We say that a continuous random vector W = (푊1,푊2)⊤ has a bivariate unit-log-symmetric (BULS) distribution with parameter vector θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤, denoted by W ∼ BULS(θ, 푔푐), if its PDF is as follows (for 0 < 푤1, 푤2 < 1) 푓푊1,푊2(푤1, 푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) = 1 (1 − 푤1)푡1휎1(1 − 푤2)푡2휎2 � 1 − 휌2 푍푔푐 푔푐 � �푤2 1 − 2휌�푤1�푤2 + �푤2 2 1 − 휌2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) where �푤푖 = log �� 푡푖 휂푖 �1/휎푖� , 푡푖 = − log(1 − 푤푖), 휂푖 = exp(휇푖), 푖 = 1, 2, with 휎푖 > 0, 푖 = 1, 2, 휌 ∈ (−1, 1), and 푍푔푐 and 푔푐 are as given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 we prove that the BULS PDF (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) is obtained by considering 푊푖 = 1 − exp(−푇푖), 푖 = 1, 2, with = (푇1,푇2)⊤ ∼ BLS(θ, 푔푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 1 presents some examples of bivariate unit-log-symmetric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 1: Partition functions (푍푔푐) and density generators (푔푐) for some distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Distribution 푍푔푐 푔푐 Parameter Bivariate unit-log-normal 2휋 exp(−푥/2) − Bivariate unit-log-Student-푡 Γ(휈/2)휈휋 Γ((휈+2)/2) (1 + 푥 휈)−(휈+2)/2 휈 > 0 Bivariate unit-log-hyperbolic 2휋(휈+1) exp(−휈) 휈2 exp(−휈 √ 1 + 푥 ) 휈 > 0 Bivariate unit-log-Laplace 휋 퐾0( √ 2푥 ) − Bivariate unit-log-slash 휋 푞 2 2−푞 2 푥− 푞+2 2 훾( 푞+2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푥 2) 푞 > 0 In Table 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Γ(푡) = ∫ ∞ 0 푥푡−1 exp(−푥) d푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푡 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' is the gamma function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 퐾휆(푢) = (1/2)(푢/2)휆 ∫ ∞ 0 푡−휆−1 exp(−푡 − 푢2/4푡) d푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푢 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' is the modified Bessel function of the third kind with index 휆 (see appendix of Kotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' and 훾(푠, 푥) = ∫ 푥 0 푡푠−1 exp(−푡) d푡 is the lower incomplete gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3 Let T = (푇1,푇2)⊤ ∼ BLS(θ, 푔푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) it is clear that the random vector X = (푋1, 푋2)⊤, where 푋푖 = log(푇푖) = log � − log(1 − 푊푖) � , 푖 = 1, 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) has a bivariate elliptically symmetric (BSY) distribution (see p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 592 in Balakrishnan and Lai, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' That is, the PDF of X is as follows 푓푋1,푋2(푥1, 푥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ∗) = 1 휎1휎2 � 1 − 휌2 푍푔푐 푔푐 � �푥1 2 − 2휌�푥1�푥2 + �푥2 2 1 − 휌2 � , −∞ < 푥1, 푥2 < ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) where �푥푖 = 푥푖 − 휇푖 휎푖 , 푖 = 1, 2, with θ∗ = (휇1, 휇2, 휎1, 휎2, 휌) being the parameter vector and 푍푔푐 is the partition function stated in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In this case, the notation X ∼ BSY(θ∗, 푔푐) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is a simple task to observe that the joint cumulative distribution function (CDF) of W ∼ BULS(θ, 푔푐), denoted by 퐹푊1,푊2(푤1, 푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ), is written as 퐹푊1,푊2(푤1, 푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) = 퐹푇1,푇2 � − log(1 − 푤1), − log(1 − 푤2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ� = 퐹푋1,푋2 � log[− log(1 − 푤1)], log[− log(1 − 푤2)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ∗ �, wherein 퐹푇1,푇2(푡1, 푡2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) and 퐹푋1,푋2(푥1, 푥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ∗) denote the CDFs of T ∼ BLS(θ, 푔푐) and X ∼ BES(θ∗, 푔푐), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Notice that there is no single closed form for the CDF of X with the exception of the bivariate normal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3 Some basic properties of model In this section, some mathematical properties of proposed bivariate unit-log-symmetric distribution are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 Stochastic representation Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The random vector W = (푊1,푊2)⊤ has a BULS distribution if 푊1 = 1 − exp � − 휂1 exp(휎1푍1) � , 푊2 = 1 − exp � − 휂2 exp �휎2휌푍1 + 휎2 � 1 − 휌2 푍2 �� , where 푍1 = 푅퐷푈1 and 푍2 = 푅 √ 1 − 퐷2 푈2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푈1, 푈2,푅, and 퐷 are mutually independent random variables, 휌 ∈ (−1, 1), 휂푖 = exp(휇푖), and P(푈푖 = −1) = P(푈푖 = 1) = 1/2, 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The random variable 퐷 is positive and has PDF 푓퐷(푑) = 2/(휋 √ 1 − 푑2 ), 푑 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Further, the positive random variable 푅 has PDF given by 푓푅(푟) = 2푟푔푐(푟2)/ ∫ ∞ 0 푔푐(푢) d푢, 푟 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 2022), the random vector T = (푇1, 푇2)⊤ has a BLS distribution if 푇1 = 휂1 exp(휎1푍1), 푇2 = 휂2 exp �휎2휌푍1 + 휎2 � 1 − 휌2 푍2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) Moreover, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4), 푊푖 = 1 − exp(−푇푖), 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 Marginal Quantiles Given 푝 ∈ (0, 1), let 푄푊푖(푝) be the 푝-quantile of 푊푖, for 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using the stochastic representation of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1, for W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐) we have 푝 = P(푊1 ⩽ 푄푊1(푝)) = P�1 − exp � − 휂1 exp(휎1푍1) � ⩽ 푄푊1(푝)� = P � 푍1 ⩽ log �� −log(1 − 푄푊1(푝)) 휂1 �1/휎1 �� and 푝 = P(푊2 ⩽ 푄푊2(푝)) = P � 1 − exp � −휂2 exp � 휎2휌푍1 + 휎2 � 1 − 휌2 푍2 �� ⩽ 푄푊2(푝) � = P � 휌푍1 + � 1 − 휌2 푍2 ⩽ log �� −log(1 − 푄푊2(푝)) 휂2 �1/휎2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence, the 푝-quantiles 푄푍1(푝) and 푄푍2(푝) of 푍1 and 푍2, respectively, satisfy log �� −log(1 − 푄푊1(푝)) 휂1 �1/휎1 � = 푄푍1(푝) and log �� −log(1 − 푄푊2(푝)) 휂2 �1/휎2 � = 푄휌푍1+√ 1−휌2 푍2(푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Therefore, the 푝-quantiles 푄푊1(푝) and 푄푊2(푝) are given by 푄푊1(푝) = 1 − exp � − 휂1 exp(휎1푄푍1(푝)) � , 푄푊2(푝) = 1 − exp � − 휂2 exp �휎2푄휌푍1+√ 1−휌2 푍2(푝)�� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 Conditional distributions Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐), then the PDF of 푊2 | 푊1 = 푤1 is written as 푓푊2(푤2 | 푊1 = 푤1) = 1 (1 − 푤2)푡2휎2 � 1 − 휌2 푓푍2 � 1 � 1 − 휌2 (�푤2 − 휌�푤1) ���� 푍1 = �푤1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) where �푤푖, 푖 = 1, 2, and 푡2 are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3), and 푍1 and 푍2 are as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If 푊1 = 푤1, then 푍1 = log � (−log(1 − 푤1)/휂1)1/휎1 � = �푤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Thus, the conditional distribution of 푊2 given 푊1 = 푤1 is the same as the distribution of 1 − exp � −휂2 exp � 휎2휌�푤1 + 휎2 � 1 − 휌2 푍2 �� ����푊1 = 푤1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Consequently, 퐹푊2(푤2 | 푊1 = 푤1) = P � 1 − exp � −휂2 exp � 휎2휌�푤1 + 휎2 � 1 − 휌2 푍2 �� ⩽ 푤2 ����푊1 = 푤1 � = P � 푍2 ⩽ 1 � 1 − 휌2 (�푤2 − 휌�푤1) ���� 푍1 = �푤1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, by differentiating 퐹푊2(푤2 | 푊1 = 푤1) with respect to 푤2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For a Borelian subset 퐵 of (0, 1), the following identity is satisfied P � 휌푍1 + � 1 − 휌2 푍2 ∈ 퐵 � = P(푍2 ∈ 퐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In other words, 휌푍1 + � 1 − 휌2 푍2 and 푍2 have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is clear that the density of the sum 휌푍1 + � 1 − 휌2 푍2 are related through their joint density 푓푍1,푍2 as follows 푓휌푍1+√ 1−휌2 푍2(푠2) = 1 � 1 − 휌2 ∫ ∞ −∞ 푓푍1,푍2 � 푧, 푠2 − 휌푧 � 1 − 휌2 � d푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) From Item (13) of Saulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022), the joint PDF of 푍1 and 푍2 is given by 푓푍1,푍2(푥, 푦) = 1 푍푔푐 푔푐(푥2 + 푦2), −∞ < 푥, 푦 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) Then the integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) is = 1 � 1 − 휌2 푍푔푐 ∫ ∞ −∞ 푔푐 � 푧2 + � 푠2 − 휌푧 � 1 − 휌2 �2� d푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) 6 Using the identity 푧2 + � 푠2 − 휌푧 � 1 − 휌2 �2 = 푧2 − 2휌푧푠2 + 푠2 2 1 − 휌2 = � 푧 − 휌푠2 � 1 − 휌2 �2 + 푠2 2 the integral in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) is written as = 1 � 1 − 휌2 푍푔푐 ∫ ∞ −∞ 푔푐 �� 푧 − 휌푠2 � 1 − 휌2 �2 + 푠2 2 � d푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Taking the change of variables 푠1 = (푧 − 휌푠2)/ � 1 − 휌2 , the above integral is = 1 푍푔푐 ∫ ∞ −∞ 푔푐(푠2 1 + 푠2 2) d푠1 = ∫ ∞ −∞ 푓푍1,푍2(푠1, 푠2) d푠1, where in the last line we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Therefore 푓휌푍1+√ 1−휌2 푍2(푠2) = ∫ ∞ −∞ 푓푍1,푍2(푠1, 푠2) d푠1 = 푓푍2(푠2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6) Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6), it is clear that 휌푍1 + � 1 − 휌2 푍2 and 푍2 are equal in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 plays a fundamental role in the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' This result provides a simple formula for determining the conditional distribution of 푊1 given 푊2 ∈ 퐵 whenever the marginal and conditional distributions of W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The usefulness of this result is essential for studying Heckman-type selection models (Heckman, 1979) when the selection variables have unitary support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For a Borelian subset 퐵 of (0, 1), we define the following Borelian set: 퐵푟 = 1 √ 1 − 푟2 log �� −log(1 − 퐵) 휂2 �1/휎2� − 푟 √ 1 − 푟2 �푤1, −1 < 푟 < 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7) where �푤1 is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If W ∼ BULS(θ, 푔푐), then the PDF of 푊1 | 푊2 ∈ 퐵 is given by 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓푍1(�푤1) P(푍2 ∈ 퐵휌 | 푍1 = �푤1) P(푍2 ∈ 퐵0) , in which 푡1 is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3), 퐵푟 is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7), and 푍1 and 푍2 are as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let 퐵 be a Borelian subset of (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Observe that 푓푊1(푤1 | 푊2 ∈ 퐵) = 푓푊1(푤1) ∫ 퐵 푓푊2(푤2 | 푊1 = 푤1) d푤2 P(푊2 ∈ 퐵) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 7 Since 푓푊1(푤1) = 푓푍1(�푤1)/[(1 − 푤1)푡1휎1] and P(푊2 ∈ 퐵) = P�휌푍1 + � 1 − 휌2 푍2 ∈ 퐵0 �, where 퐵0 is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7) with 푟 = 0, the term on the right-hand side of the above identity is = 1 (1 − 푤1)푡1휎1 푓푍1(�푤1) ∫ 퐵 푓푊2(푤2 | 푊1 = 푤1) d푤2 P�휌푍1 + � 1 − 휌2 푍2 ∈ 퐵0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using the formula for 푓푊2(푤2|푊1 = 푤1) provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2, the previous expression is = 1 (1 − 푤1)푡1휎1휎2 � 1 − 휌2 푓푍1(�푤1) ∫ 퐵 1 (1−푤2)푡2 푓푍2 � 1 √ 1−휌2 �푤2 − 휌 √ 1−휌2 �푤1 ��� 푍1 = �푤1 � d푤2 P�휌푍1 + � 1 − 휌2 푍2 ∈ 퐵0 � , where �푤푖 and 푡푖, 푖 = 1, 2, are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Finally, by applying the change of variable 푧 = (�푤2 − 휌 �푤1)/ � 1 − 휌2 , the above expression is = 1 (1 − 푤1)푡1휎1 푓푍1(�푤1) ∫ 퐵휌 푓푍2(푧 | 푍1 = �푤1) d푧 P�휌푍1 + � 1 − 휌2 푍2 ∈ 퐵0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In other words, we have proved that 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓푍1(�푤1) ∫ 퐵휌 푓푍2(푧 | 푍1 = �푤1) d푧 P�휌푍1 + � 1 − 휌2 푍2 ∈ 퐵0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Finally, by combining the above identity with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3, the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, in what remains of this subsection, for each generator (푔푐) of Table 1, we present closed formulas of the conditional densities of 푊1 | 푊2 ∈ 퐵 corresponding to bivariate unit-log-normal (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5), bivariate unit-log-Student-푡 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6), bivariate unit-log-hyperbolic (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7), bivariate unit-log-Laplace (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8) and bivariate unit-log-slash (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5 (Gaussian generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = exp(−푥/2) be the generator of the bivariate unit-log-normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 휙(�푤1) Φ(퐵휌) Φ(퐵0) , where Φ(퐶) = ∫ 퐶 휙(푥)d푥 and 휙(푥) is the standard normal PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Further, �푤1 and 푡1 are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3), and 퐵푟 is as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that the bivariate log-normal distribution has a stochastic representation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1), where 푍1 ∼ 푁(0, 1) and 푍2 ∼ 푁(0, 1), and 푍2 | 푍1 = 푥 ∼ 푁(0, 1) (Abdous et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence, P(푍2 ∈ 퐵0) = Φ(퐵0) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = Φ(퐵휌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, by applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ 8 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6 (Student-푡 generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1, 푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = (1+ (푥/휈))−(휈+2)/2, 휈 > 0, be the generator of the bivariate unit-log-Student-푡 distribution with 휈 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓휈(�푤1) 퐹휈+1 �� 휈+1 휈+�푤2 1 퐵휌 � 퐹휈(퐵0) , where 퐹휈(퐶) = ∫ 퐶 푓휈(푥)d푥 and 푓휈(푥) is the standard Student-푡 PDF with 휈 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that the bivariate log-Student-푡 distribution has a stochastic representation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1), where 푍1 ∼ 푡휈 and 푍2 ∼ 푡휈 (Student-푡 with 휈 degrees of freedom), and (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7 of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 2022) 푍2 | 푍1 = 푥 ∼ � 휈 + 푥2 휈 + 1 푡휈+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence, P(푍2 ∈ 퐵0) = 퐹휈(퐵0) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹휈+1 �� 휈 + 1 휈 + �푤2 1 퐵휌 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, the proof of corollary follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7 (Hyperbolic generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = exp(−휈 √ 1 + 푥 ), be the generator of the bivariate unit-log-hyperbolic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓GH(�푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3/2, 휈, 1) 퐹GH � 퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1, 휈, � 1 + �푤2 1 � 퐹GH(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3/2, 휈, 1) , where 퐹GH(퐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휆, 훼, 훿) = ∫ 퐶 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휆, 훼, 훿)d푥 and 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휆, 훼, 훿) is the generalised hyperbolic (GH) PDF (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that the bivariate log-hyperbolic distribution has a stochastic representation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1), where 푍1 ∼ GH(3/2, 휈, 1) and 푍2 ∼ GH(3/2, 휈, 1) (Deng and Yao, 2018, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, the distribution of 푍2 given 푍1 = 푥 is GH(1/2, √ 2 , |푥|) (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then P(푍2 ∈ 퐵0) = 퐹GH(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3/2, 휈, 1) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹GH �퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1, 휈, � 1 + �푤2 1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8 (Laplace generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = 퐾0( √ 2푥 ) be the generator of the bivariate unit-log-Laplace distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓L(�푤1) 퐹GH(퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1 2, √ 2 , |�푤1|) 퐹L(퐵0) , where 퐹L(퐶) = ∫ 퐶 푓L(푥)d푥 and 푓L(푥) = exp(− √ 2 |푥|)/ √ 2 is the Laplace PDF with scale parameter 1/ √ 2 , and 퐹GH is defined in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that the bivariate log-Laplace distribution has a stochastic representation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1), where 푍1 ∼ Laplace(0, 1/ √ 2 ) and 푍2 ∼ Laplace(0, 1/ √ 2 ) (Kotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', 2001, Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 234).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Further, the distribution of 푍2 given 푍1 = 푥 is GH(1/2, √ 2 , |푥|) (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence, P(푍2 ∈ 퐵0) = 퐹L(퐵0) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹GH(퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1/2, √ 2 , |�푤1|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9 (Slash generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐) and 푔푐(푥) = 푥−(푞+2)/2훾((푞 + 2)/2, 푥/2), be the generator of the bivariate unit-log-slash distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, for each Borelian subset 퐵 of (0, 1), the PDF of 푊1 | 푊2 ∈ 퐵 is given by (for 0 < 푤1 < 1) 푓푊1(푤1 | 푊2 ∈ 퐵) = 1 (1 − 푤1)푡1휎1 푓SL(�푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) 퐹ESL �퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' �푤1, 푞 + 1� 퐹SL(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) , where 퐹SL(퐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) = ∫ 퐶 푓SL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞)d푥 and 푓SL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) = 푞 ∫ 1 0 푡푞휙(푡푥)d푡 is the classical slash PDF, and 퐹ESL(퐶;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푎, 푞) = ∫ 퐶 푓ESL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푎, 푞)d푥, where 푓ESL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푎, 푞) is the generalised hyperbolic (ESL) PDF (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' It is well-known that the bivariate log-slash distribution has a stochastic representation as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1), where 푍1 ∼ SL(푞) and 푍2 ∼ SL(푞) (Wang and Genton, 2006, Section 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, the distribution of 푍2 given 푍1 = 푥 is ESL(푥, 푞 + 1) (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Hence, P(푍2 ∈ 퐵0) = 퐹SL(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) and P(푍2 ∈ 퐵휌 | 푍1 = �푤1) = 퐹ESL �퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' �푤1, 푞 + 1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4, the proof follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ In summary, Table 2 presents some examples of conditional PDFs corresponding to the bivariate unit-log-symmetric distributions of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 2: Conditional densities of 푊1| 푊2 ∈ 퐵 and density generators (푔푐) for some distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Distribution 푔푐 푓푊1(푤1 | 푊2 ∈ 퐵) Bivariate unit-log-normal exp(−푥/2) 1 (1−푤1)푡1휎1 휙(�푤1) Φ(퐵휌) Φ(퐵0) Bivariate unit-log-Student-푡 (1 + 푥 휈)−(휈+2)/2 1 (1−푤1)푡1휎1 푓휈(�푤1) 퐹휈+1 �� 휈+1 휈+ � 푤2 1 퐵휌 � 퐹휈(퐵0) Bivariate unit-log-hyperbolic exp(−휈 √ 1 + 푥 ) 1 (1−푤1)푡1휎1 푓GH(�푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3/2, 휈, 1) 퐹GH � 퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1,휈,√ 1+�푤2 1 � 퐹GH(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3/2,휈,1) Bivariate unit-log-Laplace 퐾0( √ 2푥 ) 1 (1−푤1)푡1휎1 푓L(�푤1) 퐹GH(퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1 2, √ 2 ,|�푤1|) 퐹L(퐵0) Bivariate unit-log-slash 푥− 푞+2 2 훾( 푞+2 2 , 푥 2) 1 (1−푤1)푡1휎1 푓SL(�푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) 퐹ESL(퐵휌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='�푤1,푞+1) 퐹SL(퐵0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='푞) 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4 The squared Mahalanobis Distance The squared Mahalanobis distance of a random vector W = (푊1, 푊2)⊤ and the vector log(η) = (log(휂1), log(휂2))⊤ of a BULS distribution is defined as 푑2(W , log(η)) = � 푊2 1 − 2휌 � 푊1 � 푊2 + � 푊2 2 1 − 휌2 , where � 푊푖 = log ��푇푖 휂푖 �1/휎푖� , 푇푖 = − log(1 − 푊푖), 휂푖 = exp(휇푖), 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Analogously to Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9 of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022) we have the following formulas for the CDF and PDF of the random variable 푑2(W , log(η)): 퐹푑2(W ,log(η))(푥) = 4 푍푔푐 ∫ √푥 0 �∫ √ 푥−푧2 1 0 푔푐(푧2 1 + 푧2 2) d푧2 � d푧1, 푥 > 0, 푓푑2(W ,log(η))(푥) = 휋 푍푔푐 푔푐(푥), 푥 > 0, where 푍푔푐 is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For example, by taking 푔푐(푥) = exp(−푥/2) and 푍푔푐 = 2휋 (see Table 1), we get 푑2(W , log(η)) ∼ 휒2 2 (chi-square with 2 degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By taking 푔푐(푥) = (1 + (푥/휈))−(휈+2)/2 and 푍푔푐 = Γ(휈/2)휈휋/Γ((휈 + 2)/2) (see Table 1), we have 푑2(W , log(η)) ∼ 2퐹2,휈, where 퐹2,휈 denotes the F- distribution with 2 and 휈 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5 Independence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If 휌 = 0 and the density generator 푔푐 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) satisfies 푔푐 �푥2 + 푦2� = 푔푐1 �푥2�푔푐2 �푦2�, ∀(푥, 푦) ∈ R2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8) for some density generators 푔푐1 and 푔푐2, then 푊1 and 푊2 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The proof follows the same steps as the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='11 of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For the convenience of the reader, we put the details of this one here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let 휌 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8), the joint density (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) of (푊1, 푊2) satisfies 푓푊1,푊2(푤1, 푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) = 푍푔푐1 푍푔푐2 푍푔푐 푓1(푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휇1, 휎1) 푓2(푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휇2, 휎2), ∀(푤1, 푤2) ∈ (0, 1) × (0, 1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9) where 푓푖(푤푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휇푖, 휎푖) = 푔푐푖 � � 푤푖 2�/[(1 − 푤푖)푡푖휎푖푍푔푐푖 ], 0 < 푤푖 < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푍푔푐푖 = ∫ ∞ −∞ 푔푐푖 �푧푖2� d푧푖, 푖 = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' and � 푤푖 and 푡푖 are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9) in terms of 푤1 and 푤2 we obtain 푍푔푐1 푍푔푐2 푍푔푐 = 1, 11 and consequently, 푍푔푐 = 푍푔푐1 푍푔푐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Therefore, 푓푊1,푊2(푤1, 푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' θ) = 푓1(푤1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휇1, 휎1) 푓2(푤2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휇2, 휎2), ∀(푤1, 푤2) ∈ (0, 1) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, it is simple to verify that 푓1 and 푓2 are PDFs corresponding to univariate symmetric random variables (Vanegas and Paula, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then 푊1 and 푊2 are statistically independent, and even more, 푓푖 = 푓푊푖, for 푖 = 1, 2 (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5 of James, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In Table 1, the density generator of the bivariate unit-log-normal is the unique one that fulfills (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6 Moments Since for W = (푊1,푊2)⊤ ∼ BULS(θ, 푔푐), 0 < 푊푖 < 1, its is clear that 0 ⩽ E(푊푟 푖 ) ⩽ 1, for any 푟 > 0 and 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Therefore, the positive moments of 푊푖 always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In general, for any 푟 ∈ R, the moments of 푊푖 admit the following representation: E(푊푟 1) = E � 1 − exp � − 휂1 exp(휎1푍1) ��푟 , E(푊푟 2) = E�1 − exp � − 휂2 exp �휎2 � 휌푍1 + � 1 − 휌2 푍2 ����푟, where 푍1 and 푍2 are given in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 4 Maximum likelihood estimation Let {(푊1푖, 푊2푖)⊤ : 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' , 푛} be a bivariate random sample of size 푛 from the BULS(θ, 푔푐) distribution with PDF as given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3), and let (푤1푖, 푤2푖)⊤ be the correspondent observations of (푊1푖,푊2푖)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Then, the log-likelihood function for θ = (휂1, 휂2, 휎1, 휎2, 휌)⊤, without the additive constant, is expressed as ℓ(θ) = −푛 2 � 푖=1 log(휎푖) − 푛 2 log �1 − 휌2� + 푛 � 푖=1 log 푔푐 � �푤2 1푖 − 2휌�푤1푖�푤2푖 + �푤2 2푖 1 − 휌2 � , 0 < 푤1푖, 푤2푖 < 1, where �푤푘푖 = log ��푡푘푖 휂푘 �1/휎푘� , 푡푘푖 = − log(1 − 푤푘푖), 휂푘 = exp(휇푘), 푘 = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' , 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In the case that a supremum �θ = ( �휂1, �휂2, � 휎1, � 휎2, �휌)⊤ exists, it must satisfy the following likelihood equations: 휕ℓ(θ) 휕휂1 ���� θ=�θ = 0, 휕ℓ(θ) 휕휂2 = 0, 휕ℓ(θ) 휕휎1 ���� θ=�θ = 0, 휕ℓ(θ) 휕휎2 ���� θ=�θ = 0, 휕ℓ(θ) 휕휌 ���� θ=�θ = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) 12 with 휕ℓ(θ) 휕휂1 = 2 휎1휂1(1 − 휌2) 푛 � 푖=1 �휌�푤2푖 − �푤1푖 �퐺(�푤1푖, �푤2푖), 휕ℓ(θ) 휕휂2 = 2 휎2휂2(1 − 휌2) 푛 � 푖=1 �휌�푤1푖 − �푤2푖 �퐺(�푤1푖, �푤2푖), 휕ℓ(θ) 휕휎1 = − 푛 휎1 + 2 휎1(1 − 휌2) 푛 � 푖=1 �푤1푖 �휌�푤2푖 − �푤1푖 �퐺(�푤1푖, �푤2푖), 휕ℓ(θ) 휕휎2 = − 푛 휎2 + 2 휎2(1 − 휌2) 푛 � 푖=1 �푤2푖 �휌�푤1푖 − �푤2푖 �퐺(�푤1푖, �푤2푖), 휕ℓ(θ) 휕휌 = 푛휌 1 − 휌2 − 2 (1 − 휌2)2 푛 � 푖=1 �휌�푤1푖 − �푤2푖 � �휌�푤2푖 − �푤1푖 �퐺(�푤1푖, �푤2푖), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) where we are adopting the notation: 퐺(�푤1푖, �푤2푖) = 푔′ 푐(푥휌,푖) 푔푐(푥휌,푖), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) with 푥휌,푖 = (�푤2 1푖 − 2휌�푤1푖�푤2푖 + �푤2 2푖)/(1 − 휌2), 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' , 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Observe that the likelihood equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) can be written as follows 푛 � 푖=1 �푤1푖 퐺(�푤1푖, �푤2푖) ���� θ=�θ = 0, 푛 � 푖=1 ��푤2 1푖 − �푤2 2푖 � 퐺(�푤1푖, �푤2푖) ���� θ=�θ = 0, 푛 � 푖=1 �푤2푖 � 2휌�푤2푖 − (1 + 휌2)�푤1푖 � 퐺(�푤1푖, �푤2푖) ���� θ=�θ = −푛�휌(1 − �휌2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Any nontrivial root �θ of the above likelihood equations is known as an ML estimator in the loose sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' When the parameter value provides the absolute maximum of the log-likelihood function, it is called an ML estimator in the strict sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In the following proposition we study the existence of the ML estimator �휌 when the other parameters are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let 푔푐 be a density generator satisfying the following condition: 푔′ 푐(푥) = 푟(푥)푔푐(푥), −∞ < 푥 < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) for some real-valued function 푟(푥) so that lim휌→±1 푟(푥휌,푖) = 푐 ∈ (−∞, 0), where 푥휌,푖, 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' , 푛, are as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If the parameters 휂1, 휂2, 휎1 and 휎2 are known, then the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) has at least one root on the interval (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The proof of this result follows by direct application of Intermediate value theorem, so details of the proof are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For more details of the proof, see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' of Vila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ For the BULS model no closed-form solution to the maximization problem is known or available, and an MLE can only be found via numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Under mild regularity conditions (Cox and Hinkley, 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Davison, 2008), the asymptotic distribution of ML estimator �θ of θ is easily determined by the convergence in law: (�θ − θ) 풟 −→ 푁(0, 퐼−1(θ)), where 0 is the zero mean vector and 퐼−1(θ) is the inverse expected Fisher information matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The main use of the last convergence is to construct confidence regions and to perform hypothesis testing for θ (Davison, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 5 Simulation study In this section, we carry out a Monte Carlo simulation study to evaluate the performance of the ML estimators for the BULS distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' For illustration purposes we only present results for the bivariate unit-log-normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The simulation scenario considers the following setting: 1,000 Monte Carlo replications, sample size 푛 ∈ (50, 50, 100, 150), vector of true parameters (휂1, 휂2, 휎1, 휎2) = (1, 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5), 휌 ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95} (negative values of 휌 produce the same results and then are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' To study the ML estimators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' we compute the bias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' root mean square error (RMSE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' and coverage probability (CP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' defined by � Bias(�휃) = 1 푁 푁 � 푖=1 �휃(푖) − 휃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' � RMSE(�휃) = � � � 1 푁 푁 � 푖=1 (�휃(푖) − 휃)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' � CP(�휃) = 1 푁 푁 � 푖=1 I(휃 ∈ [퐿(푖) �휃 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='푈(푖) �휃 ]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' where 휃 and �휃(푖) are the true parameter value and its respective 푖-th ML estimate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푁 is the number of Monte Carlo replicas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' I is an indicator function taking the value 1 if 휃 ∈ [퐿(푖) �휃 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푈(푖) �휃 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' and 0 otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' where 퐿(푖) �휃 and 푈(푖) �휃 are the 푖-th upper and lower limit estimates of the 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We expect that, as the sample size increases, the bias and RMSE reduce, and the CP approaches the 95% nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The simulation results are presented in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We observe that the results obtained for the chosen bivariate unit-log-normal distribution are as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' As the sample size increases, the bias and RMSE tend to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, the CP approaches the 95% nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Finally, in general, the results do not seem to depend on the parameter 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 14 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='015 n Bias η^1(ρ = 0) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='015 n Bias η^2(ρ = 0) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 n Bias σ^1(ρ = 0) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 n Bias σ^2(ρ = 0) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='004 n Bias ρ^(ρ = 0) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='10 n RMSE η^1(ρ = 0) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='10 n RMSE η^2(ρ = 0) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4 n RMSE σ^1(ρ = 0) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4 n RMSE σ^2(ρ = 0) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='20 n RMSE ρ^(ρ = 0) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='96 n CP η^1(ρ = 0) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='96 n CP η^2(ρ = 0) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) η^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='96 n CP σ^1(ρ = 0) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^1(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='96 n CP σ^2(ρ = 0) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) σ^2(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) 0 100 300 500 700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='94 n CP ρ^(ρ = 0) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='25) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='75) ρ^(ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='95) Figure 1: Monte Carlo simulation results for the bivariate unit-log-normal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 6 Application to soccer data In this section, two real soccer data sets, corresponding to times elapsed until scored goals of UEFA Champions League and pass completions of 2022 FIFA World Cup, are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The UEFA Champions League data set was extracted from Meintanis (2007), whereas the 2022 FIFA World Cup data set is new and is analyzed for the first time here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 UEFA Champions League We consider a bivariate data set on the group stage of the UEFA Champions League for the seasons 2004/05 and 2005/06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Only matches with at least one goal scored directly from a kick by any team, and with at least one goal scored by the home team, are considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' see Meintanis (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The first variable (푊1) is the time (in minutes) elapsed until a first kick goal is scored by any team, and the second one 푊2 is the time (in minutes) elapsed until a first goal of any type is scored by the home team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The times are divided by 90 minutes (full game time) to obtain data on the unit square (0, 1) × (0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' see Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 3 provides descriptive statistics for the variables 푊1 and 푊2, including the minimum, median, mean, maximum, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS), and coefficient of kurtosis (CK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We observe in the variable 푊1, the mean and median to be, respectively, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='454 15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='456, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', the mean is almost equal to the median, which indicates symmetry in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The CV is 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='274%, which means a moderate level of dispersion around the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Furthermore, the CS value also confirms the symmetry nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The variable 푊1 has mean equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='365 and median equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='311, which indicates the small positively skewed feature in the distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, the CV value is 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='475%, showing a moderate level of dispersion around the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The CS confirms the small skewed nature and the CK value indicates the small kurtosis feature in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 3: Summary statistics for the UEFA Champions League data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Variables 푛 Minimum Median Mean Maximum SD CV CS CK 푊1 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='224 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='930 푊2 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='254 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='839 The ML estimates and the standard errors (in parentheses) for the bivariate unit-log-symmetric model parameters are presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The extra parameters, associated with log-Student-푡, log-hyperbolic and log-slash models, were estimated using the profile log-likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' see Saulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 4 also presents the log-likelihood value, and the values of the Akaike (AIC) and Bayesian (BIC) information criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We observe that the log-hyperbolic model provides better fit than other models based on the values of log-likelihood, AIC and BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Note, however, that the values of log-likelihood, AIC and BIC of all bivariate unit-log-symmetric models are quite close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 4: ML estimates (with standard errors in parentheses), log-likelihood, AIC and BIC values for the indicated bivariate unit-log-symmetric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Distribuiton �휂1 �휂2 �휎1 �휎2 �휌 �휈 Log-likelihood AIC BIC Log-normal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5288* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3414* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8865* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1355* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4956* – 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='693 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='386 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='441 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0771) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0637) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1031) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1320) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1240) Log-Student-푡 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5541* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3783* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7431* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9734* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4723* 7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='487 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='974 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='029 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0751) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0672) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1033) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1308) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1463) Log-hyperbolic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5458* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3816* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8456* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0950* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4893* 2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='470 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='940 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='996 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0752) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0677) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1162) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1462) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1428) Log-Laplace 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5680* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5679* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9928* 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3231* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5281* – 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='009 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='019 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='073 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0020) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0021) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1692) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2164) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1639) Log-slash 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5629* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3715* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6203* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8302* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4472* 5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='560 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='120 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='174 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0749) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0666) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0847) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1096) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1472) ∗ significant at 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Figure 2 shows the QQ plots of the Mahalanobis distance for the bivariate unit-log-symmetric models considered in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The QQ plot is a plot of the empirical quantiles of the Mahalanobis distance against the theoretical quantiles of the respective reference distribution (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Therefore, points falling along a straight line indicate a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' From Figure 2, we see clearly that, with the exception of the log-Student-푡 case, the Mahalanobis distances in the considered models conform relatively well with their reference distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We also see that, in all the cases, there is a point away from the reference line, which may be an outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='(e) Log-slash ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='Figure 2: QQ plot of the Mahalanobis distances for the indicated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 2022 FIFA World Cup In this subsection, data on the 2022 FIFA World Cup are used to illustrate the proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The data are available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The first variable (푊1) is the medium pass completion proportion, that is, successful passes between 14 and 18 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The second variable (푊2) is the long pass completion proportion, namely, passes longer than 37 meters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' see Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 5 provides descriptive statistics for the variables 푊1 and 푊2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We observe in the variable 푊1, the mean and median to be, respectively, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='454 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='456, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', the mean is almost equal to the median, which indicates symmetry in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The CV is 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='274%, which means a moderate level of dispersion around the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Furthermore, the CS value also confirms the symmetry nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The variable 푊2 has mean equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='365 and median equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='311, which indicates the small positively skewed feature in the distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Moreover, the CV value is 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='475%, showing a moderate level of dispersion around the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The CS confirms the small skewed nature and the CK value indicates the small kurtosis feature in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 17 Table 5: Summary statistics for the 2022 FIFA World Cup data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Variables 푛 Minimum Median Mean Maximum SD CV CS CK 푊1 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='038 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='194 푊2 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='075 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='425 Table 6 presents the estimation results for the bivariate unit-log-symmetric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The results of this table reveal that the log-normal model provides better adjustment than the other models based on the values of log-likelihood, AIC and BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Figure 3 shows the QQ plots of the Mahalanobis distances (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) for the bivariate unit- log-symmetric models considered in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We see clearly that the log-normal model provides better fit than the other bivariate unit-log-symmetric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Table 6: ML estimates (with standard errors in parentheses), log-likelihood, AIC and BIC values for the indicated bivariate unit-log-symmetric models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Distribuiton �휂1 �휂2 �휎1 �휎2 �휌 �휈 Log-likelihood AIC BIC Log-normal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='9872* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7953* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1364* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2089* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7343* – 20.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0482) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0295) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0154) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0237) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0844) ∗ significant at 5% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 7 Conclusions In the present paper we have proposed a family of bivariate distributions over the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By adequately defining the density generator we can transform any distribution over the real line into a bivariate distribution over the region (0, 1) × (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Such a model has several potential applications, since the simultaneous modeling of quantities like proportions, rates or indices frequently arises in applied sciences like economy, medicine engineering and social sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We have addressed diverse theoretical properties like stochastic representation, quantiles, conditional distributions, independence and moments and practical questions like estimation, a Monte Carlo Simulation Study and applications to soccer data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The present research can be extended in several possible directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By changing the density generator, numerous special forms of the BULS distribution can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Furthermore, generalizations to higher dimensions can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 18 0 2 4 6 8 0 2 4 6 8 theoretical quantile empirical quantile (a) Log-normal 0 2 4 6 8 0 2 4 6 8 10 theoretical quantile empirical quantile (b) Log-Student-푡 0 2 4 6 8 10 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='0 theoretical quantile empirical quantile (c) Log-hyperbolic 0 2 4 6 8 10 12 14 0 1 2 3 4 5 6 theoretical quantile empirical quantile (d) Log-Laplace 0 5 10 15 20 0 2 4 6 8 10 12 theoretical quantile empirical quantile (e) Log-slash Figure 3: QQ plot of the Mahalanobis distances for the indicated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Acknowledgements Roberto Vilaand Helton Saulo gratefully acknowledgefinancial support from CNPq, CAPES and FAP-DF, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Disclosure statement There are no conflicts of interest to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' References Abdous, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=', Fougères and A.' metadata={'source': 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+page_content=' Journal of Statistical Planning and Inference, 136:209–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 20 Appendix A Some additional results For the convenience of the reader, in this section, some complementary results related to the Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 are presented in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We say that a random variable 푋 follows a univariate generalised hyperbolic (GH) distri- bution, denoted by 푋 ∼ GH(휆, 훼, 훿), if its PDF is given by 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 휆, 훼, 훿) = ( √ 훼2 /훿)휆 √ 2휋 퐾휆(훿 √ 훼2 ) 퐾휆−1/2(훼 √ 훿2 + 푥2 ) ( √ 훿2 + 푥2 /훼)1/2−휆 , −∞ < 푥 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Here, 퐾푟 is the modified Bessel function of the third kind with index 푟, 휆 ∈ R, 훼 ∈ R and 훿 > 0 is a scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The following result has appeared in a multivariated version in Deng and Yao (2018, Proposition 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 (Hyperbolic generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If 푔푐(푥) = exp(−휈 √ 1 + 푥 ) then the conditional distribution of 푍2 given 푍1 = 푥 is GH(1, 휈, √ 1 + 푥2 ) and its unconditional distributions both are GH(3/2, 휈, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4), the joint PDF of 푍1 and 푍2 is (see Table 1) 푓푍1,푍2(푥, 푦) = 휈2 exp(휈) 2휋(휈 + 1) exp � −휈 � 1 + 푥2 + 푦2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) The marginal PDF of 푍1 is given by 푓푍1(푥) = ∫ ∞ −∞ 푓푍1,푍2(푥, 푦) d푦 = 휈2 exp(휈) 2휋(휈 + 1) ∫ ∞ −∞ exp � −휈 � 1 + 푥2 + 푦2 � d푦 = 2휈2 exp(휈) 2휋(휈 + 1) ∫ ∞ 0 exp � −휈 � 1 + 푥2 + 푦2 � d푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By using the Formula 6 of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='46-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='48 of reference Gradshteyn and Ryzhik (2000, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 364): ∫ ∞ 0 exp(−푎 √ 푏2 + 푥2 )d푥 = 푏퐾1(푎푏), the above integral is = 2휈2 exp(휈) 2휋(휈 + 1) � 1 + 푥2 퐾1(휈 � 1 + 푥2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2) Since 퐾3/2(휈) = 2√휋 exp(−휈)(휈 + 1)(휈/2)3/2/휈3, the above espression is written as = 휈3/2 √ 2휋 퐾3/2(휈) 퐾1(휈 √ 1 + 푥2 ) ( √ 1 + 푥2 /휈 )−1 = 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 3/2, 휈, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' This proves that 푍1 ∼ GH(3/2, 휈, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Analogously, we prove that 푍2 ∼ GH(3/2, 휈, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 21 On the other hand, employing (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2), and by using the the well-known identity 퐾1/2(푧) = � 휋/(2푧) exp(−푧), the conditional PDF of 푍2 given 푍1 = 푥 is 푓푍2 | 푍1(푦 | 푥) = exp(−휈 � 1 + 푥2 + 푦2 ) 2 √ 1 + 푥2 퐾1(휈 √ 1 + 푥2 ) = 휈/ √ 1 + 푥2 √ 2휋 퐾1(휈 √ 1 + 푥2 ) 퐾1/2(휈 � 1 + 푥2 + 푦2 ) ( � 1 + 푥2 + 푦2 /휈)−1/2 = 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1, 휈, � 1 + 푥2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We thus complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ The following result has also appeared in a multivariated version in the reference Kotz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (2001, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 253).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 (Laplace generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If 푔푐(푥) = 퐾0( √ 2푥 ) then the conditional distribution of 푍2 given 푍1 = 푥 is GH(1/2, √ 2 , |푥|) and its unconditional distributions both are Laplace(0, 1/ √ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) and by using the definitions of 퐾0 and 푍푔푐 in Table 1, 푓푍1,푍2(푥, 푦) = 1 휋 퐾0 �� 2(푥2 + 푦2) � = 1 2휋 ∫ ∞ 0 1 푡 exp � −푡 − 푥2 + 푦2 2푡 � d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3) An algebraic calculation shows that the marginal of 푍1 is 푓푍1(푥) = ∫ ∞ −∞ 푓푍1,푍2(푥, 푦) d푦 = 1 √ 2 exp(− √ 2 |푥|) = 푓L(푥), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) where 푓L(푥) = exp(− √ 2 |푥|)/ √ 2 is the Laplace PDF with scale parameter 1/ √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' That is, 푍1 ∼ Laplace(0, 1/ √ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Analogously we verify that 푍2 ∼ Laplace(0, 1/ √ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' On the other hand, by using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) and the well-known identity 퐾1/2(푧) = � 휋/(2푧) exp(−푧), the conditional PDF of 푍2 given 푍1 = 푥 can be written as 푓푍2 | 푍1(푦 | 푥) = 1 휋 퐾0 �� 2(푥2 + 푦2) � 1 √ 2 exp(− √ 2 |푥|) = ( √ 2 /|푥|)1/2 √ 2휋 퐾1/2( √ 2 |푥|) 퐾0 �√ 2 � 푥2 + 푦2 � = 푓GH(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 1/2, √ 2 , |푥|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Following the Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1 we get that 푍2 | 푍1 = 푥 ∼ GH(1/2, √ 2 , |푥|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' We say that a random variable 푋 follows a univariate extended slash (ESL) distribution, denoted by 푋 ∼ ESL(푎, 푞), if its PDF is given by 푓ESL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푎, 푞) = ∫ 1 0 푡푞휙(푡푎)휙(푡푥) d푡 ∫ 1 0 푢푞−1휙(푢푎) d푢 , −∞ < 푥 < ∞, 22 where 휙 denotes the PDF of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Choosing 푎 = 0 the classical slash (SL) PDF is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' The resulting density takes the form 푓SL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞) = 푞 ∫ 1 0 푡푞휙(푡푥) d푡, −∞ < 푥 < ∞ = 푞 2 푞 2 −1 √휋 |푥|−(푞+1) 훾 �푞 + 1 2 , 푥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' In this case we denote 푋 ∼ SL(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='3 (Slash generator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Let Z = (푍1, 푍2)⊤ be a random vector as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' If 푔푐(푥) = 푥−(푞+2)/2훾((푞 + 2)/2, 푥/2) then the conditional distribution of 푍2 given 푍1 = 푥 is ESL(푥, 푞 + 1) and its unconditional distributions both are SL(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='4) and by definition of 푍푔푐 in Table 1, the joint PDF of 푍1 and 푍2 is given by 푓푍1,푍2(푥, 푦) = 푞 2 푞 2 −1 휋 (푥2 + 푦2)− 푞+2 2 훾 �푞 + 2 2 , 푥2 + 푦2 2 � = 푞 ∫ 1 0 푡푞+1휙(푡푥)휙(푡푦) d푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) Then the marginal PDF of 푍1 is 푓푍1(푥) = ∫ ∞ −∞ 푓푍1,푍2(푥, 푦) d푦 = 푞 ∫ 1 0 푡푞+1휙(푡푥) �∫ ∞ −∞ 휙(푡푦) d푦 � d푡 = 푞 ∫ 1 0 푡푞휙(푡푥) d푡 = 푓SL(푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6) This proves that (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2), 푍1 ∼ SL(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Analogously it is proved that 푍2 ∼ SL(푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' On the other hand, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='5) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='6), the conditional PDF of 푍2 given 푍1 = 푥, is 푓푍2 | 푍1(푦 | 푥) = ∫ 1 0 푡푞+1휙(푡푥)휙(푡푦) d푡 ∫ 1 0 푢푞휙(푢푥) d푢 = 푓ESL(푦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' 푥, 푞 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' Following the Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='2 we have that 푍2 | 푍1 = 푥 ∼ ESL(푥, 푞 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' □ Appendix B Data sets 23 Table 7: UEFA Champions League and 2022 FIFA World Cup data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content=' UEFA FIFA W1 W2 W1 W2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='541 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E5T4oBgHgl3EQfaA9_/content/2301.05585v1.pdf'} +page_content='200 0.' 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a/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/2301.03843v1.pdf.txt b/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/2301.03843v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c5a98377b1a946c829897794c2041431e8db297 --- /dev/null +++ b/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/2301.03843v1.pdf.txt @@ -0,0 +1,410 @@ +A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models. +Rei Aso1 , Tatsuya Chuman1 and Hitoshi kiya1 +1Tokyo Metropolitan University +6-6 Asahigaoka, Hino-shi, Tokyo 191-0065, Japan +Abstract +In this paper, a privacy preserving image classification method +is proposed under the use of ConvMixer models. To protect the +visual information of test images, a test image is divided into +blocks, and then every block is encrypted by using a random +orthogonal matrix. Moreover, a ConvMixer model trained +with plain images is transformed by the random orthogonal +matrix used for encrypting test images, on the basis of the +embedding structure of ConvMixer. The proposed method +allows us not only to use the same classification accuracy as +that of ConvMixer models without considering privacy pro- +tection but to also enhance robustness against various attacks +compared to conventional privacy-preserving learning. +1. Introduction +Deep learning has been deployed in many applications in- +cluding security-critical ones. Generally, data contains sensi- +tive information such as personal informational, so privacy- +preserving methods for deep learning have become an urgent +problem [1]. To achieve privacy-preserving learning, vari- +ous methods have been proposed. One of them is Federated +Learning (FL) [2], which is a type of distributed learning. FL +allows us to train a model over multiple participants without +directly sharing their raw data. However, FL have not consid- +ered the protection of test data in cloud environments so far. +In this paper, we propose a novel method for protecting visual +information on test images. +To protect visual information on plain images in untrusted +cloud environments, many learnable encryption methods have +been studied so far [3]-[13]. Learnable encryption has to sat- +isfy three requirements in general: (a) having a high accuracy +that is almost the same as that of plain models, (b) being +robust enough against various attacks, and (c) easily updat- +ing a secret key. However, most of existing methods [3]-[11] +degrade the accuracy of models due to the use of encrypted +images, and moreover, need to retrain models to update the +key. In contrast, the similarity between block-wise encryption +and the architecture of isotropic networks has been pointed +out to enable us to perfectly stratify the two requirements that +the existing methods cannot [12][13]. Information on em- +beddings in isotropic networks such as the vision transformer +[14] and ConvMixer [15] is encrypted by random matrixes +generated with secret keys for privacy-preserving learning. +However, in the conventional methods [12][13], simple per- +mutation matrixes are used for image and model encryption, +so encrypted images are not robust enough against various +attacks. Accordingly, we propose the use of a novel random +matrix, which is called a random orthogonal one generated by +using the Gram-Schmidt orthonormalization. The proposed +method allows us to enhance the visual protection of images, +while maintaining the same as that of plain models and the +easy update of a secret key. +2. ConvMixer +Before discussing the proposed method, we summarize +ConvMixer and its properties briefly. ConvMixer is mainly +used for image classification tasks and is known for its high +classification performance[15]. The structure of ConvMixer +is inspired by the Vision Transformer (ViT)[14]. ViT consists +of two Embedding processes (Patch Embedding and Posi- +tion Embedding) and a Transformer structure. On the other +hand, ConvMixer consists of a Patch Embedding and a CNN +structure. Figure 2 shows the structure of ConvMixer, which +consists of two main structures: Patch Embedding and Con- +vMixer Layer. In this paper, we focus on Patch Embedding. +In Patch Embedding, an input image 𝑥 ∈ R𝐻×𝑊 ×𝐶 of height +𝐻, width 𝑊, and number of channels 𝐶 is divided into patches +of size 𝑝 × 𝑝. Each patch is then transformed into a vector +𝑥𝑖 +𝑝 ∈ R𝑝2𝐶, multiplied by the learnable filter 𝐸 and linearly +transform it into a vector of 𝑑-dimensions by taking the prod- +uct of 𝑥𝑖 +𝑝 ∈ R𝑝2𝐶 as +𝑧 = [𝑥1 +𝑝𝐸, ..., 𝑥𝑖 +𝑝𝐸, ..., 𝑥𝑁 +𝑝 𝐸] +(1) +𝑧 ∈ R𝑁 ×𝑑, 𝐸 ∈ R( 𝑝2𝑐)×𝑑 +In previous studies[12][13], it is known that it is possible +to protect the privacy of test images by transforming the filter +𝐸 with a secret key. In this paper, we propose a method to +achieve stronger privacy preserving of test images by using +random orthogonal matrices. +3. Proposed Method +arXiv:2301.03843v1 [cs.CV] 10 Jan 2023 + +Figure 1: Architecture of ConvMixer +3.1 Overview +Figure 2: Framework of proposed method +Figure 2 illustrates the framework of the proposed method. +The proposed method aims to protect visual information on +test images. To achieve this aim, we encrypt test images and a +transform model by using an random orthogonal matrix. The +framework is summarized as below. +• A third party (trusted) generates random num- +bers with a secret key (seed), and prepares a +random orthogonal matrix 𝐴 from the random +numbers and an inverse random orthogonal +matrix 𝐴−1. +• The third party trains a ConvMixer model 𝜑 +with plane images. The trained model 𝜑 is +transformed into an encrypted model 𝜑𝑇 by +using 𝐴−1. +• The third party provides the random orthog- +onal matrix 𝐴 to a client (trusted) and model +𝜑𝑇 to a provider (untrusted). +• The client transforms a test image 𝑥 into an +encrypted image ˆ𝑥 by using 𝐴. After that, the +client sends ˆ𝑥 to the provider. +• The provider inputs ˆ𝑥 into model 𝜑𝑇 , and +sends back a prediction result to the client. +Even if the provider is not trusted, the client does not give +visual information of test images and matrix 𝐴 used for im- +age encryption to the provider. Thus, the client can receive +prediction results while maintaining the privacy preserving of +test images. +3.2 Test Image Encryption +A test image 𝑥 ∈ R𝐻×𝑊 ×𝐶 is transformed into an encrypted +image ˆ𝑥 ∈ R𝐻×𝑊 ×𝐶 as below. +1. Divide 𝑥 into 𝑁 blocks with a size of 𝑝 × 𝑝 such that 𝐵 = +{𝐵1, ..., 𝐵𝑁 }, where 𝑝 × 𝑝 is the same size as the patch +size used in a ConvMixer model, and 𝑁 is (𝐻 × 𝑊)/𝑝2. +2. Flatten each block 𝐵𝑖 ∈ R𝑝×𝑝×𝐶 into a vector 𝑥𝑖 +𝑝 ∈ R𝑝2𝐶 +as +𝑥𝑖 +𝑝 = [𝑥𝑖 +𝑝(1), ..., 𝑥𝑖 +𝑝(𝑝2𝐶)]. +(2) +3. Generate a encrypted vector ˆ𝑥𝑖 +𝑝 ∈ R𝑝2𝐶 by multiplying +vector 𝑥𝑖 +𝑝 by matrix 𝐴 ∈ R( 𝑝2𝐶)×( 𝑝2𝐶) as +ˆ𝑥𝑖 +𝑝 = 𝑥𝑖 +𝑝 𝐴. +(3) +4. Rebuild vector ˆ𝑥𝑖 +𝑝 into block ˆ𝐵𝑖 in the reverse order of +step 2. +5. Concatenate ˆ𝐵 = { ˆ𝐵1, ..., ˆ𝐵𝑁 } into an encrypted test +image ˆ𝑥. +3.3 Model Encryption +To avoid the performance degradation caused by encryption +of test images, 𝐸 in Eq.(1) is transformed by using 𝐴−1 as +𝐸 ′ = 𝐴−1𝐸. +(4) +When replacing 𝐸 and 𝑥𝑖 +𝑝 with 𝐸 ′ and ˆ𝑥𝑖 +𝑝 , respectively, vector +z in Eq.(1) is reduced to as +𝑧′ = [ ˆ𝑥1 +𝑝𝐸 ′, ..., ˆ𝑥𝑖 +𝑝𝐸 ′, ..., ˆ𝑥𝑁 +𝑝 𝐸 ′]. +(5) +Thus, by substituting Eqs. (3) and (4) to Eq.(5), we obtain: +𝑧′ = [𝑥1 +𝑝 𝐴𝐴−1𝐸, ..., 𝑥𝑖 +𝑝 𝐴𝐴−1𝐸, ..., 𝑥𝑁 +𝑝 𝐴𝐴−1𝐸] += [𝑥1 +𝑝𝐸, ..., 𝑥𝑖 +𝑝𝐸, ..., 𝑥𝑁 +𝑝 𝐸] = 𝑧. +(6) +From Eq.(5), encrypted model 𝜑𝑇 allows us to have the same +performance as that of the model trained with plane images, +under the use of encrypted images. + +ConvMixer Layer +Residual connection +Global Average Pooling +PatchEmbedding +Fully-Connected +BatchNorm +BatchNorm +BatchNorm +Class +Depthwise +Convolution +Pointwise +Convolution +★ +GELU +GELU +GELU +★ +HxWxC +(H /p)x(W /p)xdThird Party(Trusted) +Generation of Random Orthogonal Matrices + random orthogonal matrix + inverse matrix of A +A-1 +A-1 +Trained +Encrypted +ConvMixer +ConvMixer +Train +Transform +Training Images +Provide +T +Provide A +Client +Provider(Untrusted) +Encrypted Images +A +Encrypted +ConvMixer +PT +Classification +Encryption +Encrypted Images +Result +Test Images +父 +x3.4 Generation of Random Orthogonal Matrices +A random orthogonal matrix 𝐴 can be generated by using +the Gram–Schmidt orthonormalization. +The procedure for +generating 𝐴 with a size of 𝑛 × 𝑛 is given as follows. +1. Generate an real matrix 𝑅 with a size of 𝑛 × 𝑛 by using a +random number generator with a seed. +2. Calculate 𝑑𝑒𝑡(𝑅), and proceed to 3 if 𝑑𝑒𝑡(𝑅) ≠ 0. Oth- +erwise, return to 1. +3. Compute a random orthogonal matrix 𝐴 from 𝑅 by using +the Gram-Schmidt orthogonalization. +In this framework, any regular matrix can be used as A for +image encryption. Several conventional methods for privacy- +preserving image classification use permutation matrices of +pixel values, in which many elements have zero values in +matrices as +𝐴 = +������ +0 +1 +0 +0 +0 +1 +1 +0 +0 +������ +. +(7) +In contrast, the proposed random orthogonal matrices include +no the zero values as elements. The use of such matrices +allows us not only to more strongly protect visual information +on plain images but to also enhance robustness against various +attacks, while maintaining the same performance as that of +models trained with plain images. In addition, 𝐴−1 can easily +be calculated as the transposed matrix of A. +4. Experiment Results +To verify the effectiveness of the proposed method, we ran +a number of experiments on the CIFAR-10 dataset. +4.1 Setup +We used the CIFAR-10 dataset, which consists of 60,000 +color images (dimension of 32 × 32 × 3) with 10 classes (6000 +images for each class) where 50,000 images are for training +and 10,000 for testing. ConvMixer was trained and tested on +the CIFAR-10 dataset. In the model setting, we set the patch +size to 4, the number of channels after patch embedding to +256, the kernel size of depth-wise convolution to 7 and the +number of ConvMixer layers to 8. models were trained for +200 epochs with the Adam optimizer, where the learning rate +was 0.001. We also used a random orthogonal matrix with a +size of 48 × 48 for the encryption of test images and models. +4.2 Visual Protection Performance +Figure 3 shows an example of images encrypted with a con- +ventional encryption method[12][13], in which pixel shuffling +and negative-positive transformation are carried out for image +encryption, and an example of images encrypted with the pro- +posed method, where the images had 𝐻×𝑊×𝐶 = 512×512×3 +as an image size, and the block sizes used for encryption were +𝑝 = 8 and 𝑝 = 16. When using an orthogonal matrix for +encryption, transformed pixel values are real values, so (b) in +Fig. 3 were displayed after normalizing the pixel values to the +range of [0.1]. From the figures, the selection of a larger the +block size gave smaller visual information. The use of random +orthogonal matrices was also demonstrated to have a stronger +visual protection performance than that of the conventional +method. In addition to visual protection, encrypted images +have to be robust enough against various attacks, which aim to +restore visual information from encrypted images. We already +confirmed that images encrypted with the proposed method +are more robust against attacks including jigsaw puzzle solver +attacks [16]. In particular, unlike ViT, ConvMixer models do +have position embedding, so the position of patches cannot be +changed. Therefore, privacy-preserving ConvMixer needs a +stronger encryption method than ViT. +(a) p = 8 +(b) p = 8 +(c) p = 16 +(d) p = 16 +Plane image +(a) +(b) +Figure 3: Example of encrypted images with (a) conventional +method[12][13] and (b) proposed method +4.3 Classification Performance +We evaluated the classification performance of the pro- +posed method as shown in Table 1, where plain and encrypted +indicate plane test images and an encrypted test images, +respectively, and plain model and encrypted model are +models trained with plain images, and models trained with +encrypted images. Table 1 shows the classification results for +each combination. From the table, the proposed method (the +combined use of Encrypted models and encrypted images) +had the same classification accuracy as that of the baseline +without privacy protection (plain models and plain images). +Accordingly, the proposed method can not only protect +the visual information of test image, but also classify the +encrypted image without any degradation of classification +accuracy. + +3.Table 1: Classification accuracy (%) +model\test image +plane +encypted +plane model +90.38 +13.2 +encypted model +10.27 +90.38 +5. Conclusion +In this paper, we proposed a novel method for protecting vi- +sual informational on test images under the use of ConvMixer +models. +The proposed method allows us to use a random +orthogonal matrix for image encryption, and it was demon- +strated not only to enhance the visual protection of images but +to also maintain the same accuracy as that of models trained +with plain images. +Ackowledgment +This study was partially supported by JSPS KAKENHI (Grant +Number JP21H01327). +References +[1] H. Kiya, MM. AprilPyone, Y. Kinoshita, S. Imaizumi, and +S. Shiota, “An Overview of Compressible and Learnable +Image Transformation with Secret Key and its Applica- +tions," APSIPA Transactions on Signal and Information +Processing, vol.11, no.1, e11, 2022. +[2] H.B. McMahan, E. Moore, D. Ramage, S. Hampson, +and B.A.y. Arcas, “Communication-Efficient Learning of +Deep Networks from Decentralized Data," Proceedings +of the 20th International Conference on Artificial Intelli- +gence and Statistics, PMLR, vol.54, pp.1273-1282, 2017. +[3] I. Nakamura, Y. Tonomura, H. Kiya, “Unitary Transform- +Based Template Protection and Its Application to l2-norm +Minimization Problems," IEICE Transactions on Infor- +mation and Systems, vol.99, pp.60-68, 2016 +[4] M. Tanaka, "Learnable Image Encryption," 2018 IEEE In- +ternational Conference on Consumer Electronics-Taiwan +(ICCE-TW), pp. 1-2, 2018 +[5] W. Sirichotedumrong and H. kiya, “Grayscale-based +block scrambling image encryption for social networking +services," IEEE international conference on multimedia +and expo (ICME), pp.1-6, 2018 +[6] W. Sirichotedumrong and H. kiya, “A gan-based image +transformation scheme for privacy-preserving deep neu- +ral networks," European Signal Processing Conference +(EUSIPCO), pp.745-749, 2020 +[7] H. Ito, Y. Kinoshita, MM. 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Kolter, “Patches are all you need?,” +arXiv:2201.09792, 2022. +[16] T.Chuman and H.Kiya, “Security evaluation of block- +based image encryption for vision transformer against jig- +saw puzzle solver attack," IEEE 4th Global Conference on +Life Sciences and Technologies (LifeTech), pp.448-451, +2022 + diff --git a/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/load_file.txt b/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf5846f62cc6e6d3e23df58006d794008f6c4266 --- /dev/null +++ b/UdE2T4oBgHgl3EQfXQfL/content/tmp_files/load_file.txt @@ -0,0 +1,286 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf,len=285 +page_content='A Privacy Preserving Method with a Random Orthogonal Matrix for ConvMixer Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Rei Aso1 , Tatsuya Chuman1 and Hitoshi kiya1 1Tokyo Metropolitan University 6-6 Asahigaoka, Hino-shi, Tokyo 191-0065, Japan Abstract In this paper, a privacy preserving image classification method is proposed under the use of ConvMixer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' To protect the visual information of test images, a test image is divided into blocks, and then every block is encrypted by using a random orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Moreover, a ConvMixer model trained with plain images is transformed by the random orthogonal matrix used for encrypting test images, on the basis of the embedding structure of ConvMixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The proposed method allows us not only to use the same classification accuracy as that of ConvMixer models without considering privacy pro- tection but to also enhance robustness against various attacks compared to conventional privacy-preserving learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Introduction Deep learning has been deployed in many applications in- cluding security-critical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Generally, data contains sensi- tive information such as personal informational, so privacy- preserving methods for deep learning have become an urgent problem [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' To achieve privacy-preserving learning, vari- ous methods have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' One of them is Federated Learning (FL) [2], which is a type of distributed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' FL allows us to train a model over multiple participants without directly sharing their raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' However, FL have not consid- ered the protection of test data in cloud environments so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In this paper, we propose a novel method for protecting visual information on test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' To protect visual information on plain images in untrusted cloud environments, many learnable encryption methods have been studied so far [3]-[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Learnable encryption has to sat- isfy three requirements in general: (a) having a high accuracy that is almost the same as that of plain models, (b) being robust enough against various attacks, and (c) easily updat- ing a secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' However, most of existing methods [3]-[11] degrade the accuracy of models due to the use of encrypted images, and moreover, need to retrain models to update the key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In contrast, the similarity between block-wise encryption and the architecture of isotropic networks has been pointed out to enable us to perfectly stratify the two requirements that the existing methods cannot [12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Information on em- beddings in isotropic networks such as the vision transformer [14] and ConvMixer [15] is encrypted by random matrixes generated with secret keys for privacy-preserving learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' However, in the conventional methods [12][13], simple per- mutation matrixes are used for image and model encryption, so encrypted images are not robust enough against various attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Accordingly, we propose the use of a novel random matrix, which is called a random orthogonal one generated by using the Gram-Schmidt orthonormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The proposed method allows us to enhance the visual protection of images, while maintaining the same as that of plain models and the easy update of a secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' ConvMixer Before discussing the proposed method, we summarize ConvMixer and its properties briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' ConvMixer is mainly used for image classification tasks and is known for its high classification performance[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The structure of ConvMixer is inspired by the Vision Transformer (ViT)[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' ViT consists of two Embedding processes (Patch Embedding and Posi- tion Embedding) and a Transformer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' On the other hand, ConvMixer consists of a Patch Embedding and a CNN structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Figure 2 shows the structure of ConvMixer, which consists of two main structures: Patch Embedding and Con- vMixer Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In this paper, we focus on Patch Embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In Patch Embedding, an input image 𝑥 ∈ R𝐻×𝑊 ×𝐶 of height 𝐻, width 𝑊, and number of channels 𝐶 is divided into patches of size 𝑝 × 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Each patch is then transformed into a vector 𝑥𝑖 𝑝 ∈ R𝑝2𝐶, multiplied by the learnable filter 𝐸 and linearly transform it into a vector of 𝑑-dimensions by taking the prod- uct of 𝑥𝑖 𝑝 ∈ R𝑝2𝐶 as 𝑧 = [𝑥1 𝑝𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑖 𝑝𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑁 𝑝 𝐸] (1) 𝑧 ∈ R𝑁 ×𝑑, 𝐸 ∈ R( 𝑝2𝑐)×𝑑 In previous studies[12][13], it is known that it is possible to protect the privacy of test images by transforming the filter 𝐸 with a secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In this paper, we propose a method to achieve stronger privacy preserving of test images by using random orthogonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Proposed Method arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='03843v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='CV] 10 Jan 2023 Figure 1: Architecture of ConvMixer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='1 Overview Figure 2: Framework of proposed method Figure 2 illustrates the framework of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The proposed method aims to protect visual information on test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' To achieve this aim, we encrypt test images and a transform model by using an random orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The framework is summarized as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' A third party (trusted) generates random num- bers with a secret key (seed), and prepares a random orthogonal matrix 𝐴 from the random numbers and an inverse random orthogonal matrix 𝐴−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The third party trains a ConvMixer model 𝜑 with plane images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The trained model 𝜑 is transformed into an encrypted model 𝜑𝑇 by using 𝐴−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The third party provides the random orthog- onal matrix 𝐴 to a client (trusted) and model 𝜑𝑇 to a provider (untrusted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The client transforms a test image 𝑥 into an encrypted image ˆ𝑥 by using 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' After that, the client sends ˆ𝑥 to the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The provider inputs ˆ𝑥 into model 𝜑𝑇 , and sends back a prediction result to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Even if the provider is not trusted, the client does not give visual information of test images and matrix 𝐴 used for im- age encryption to the provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Thus, the client can receive prediction results while maintaining the privacy preserving of test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='2 Test Image Encryption A test image 𝑥 ∈ R𝐻×𝑊 ×𝐶 is transformed into an encrypted image ˆ𝑥 ∈ R𝐻×𝑊 ×𝐶 as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Divide 𝑥 into 𝑁 blocks with a size of 𝑝 × 𝑝 such that 𝐵 = {𝐵1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝐵𝑁 }, where 𝑝 × 𝑝 is the same size as the patch size used in a ConvMixer model, and 𝑁 is (𝐻 × 𝑊)/𝑝2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Flatten each block 𝐵𝑖 ∈ R𝑝×𝑝×𝐶 into a vector 𝑥𝑖 𝑝 ∈ R𝑝2𝐶 as 𝑥𝑖 𝑝 = [𝑥𝑖 𝑝(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑖 𝑝(𝑝2𝐶)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Generate a encrypted vector ˆ𝑥𝑖 𝑝 ∈ R𝑝2𝐶 by multiplying vector 𝑥𝑖 𝑝 by matrix 𝐴 ∈ R( 𝑝2𝐶)×( 𝑝2𝐶) as ˆ𝑥𝑖 𝑝 = 𝑥𝑖 𝑝 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Rebuild vector ˆ𝑥𝑖 𝑝 into block ˆ𝐵𝑖 in the reverse order of step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Concatenate ˆ𝐵 = { ˆ𝐵1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', ˆ𝐵𝑁 } into an encrypted test image ˆ𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='3 Model Encryption To avoid the performance degradation caused by encryption of test images, 𝐸 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (1) is transformed by using 𝐴−1 as 𝐸 ′ = 𝐴−1𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (4) When replacing 𝐸 and 𝑥𝑖 𝑝 with 𝐸 ′ and ˆ𝑥𝑖 𝑝 , respectively, vector z in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (1) is reduced to as 𝑧′ = [ ˆ𝑥1 𝑝𝐸 ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', ˆ𝑥𝑖 𝑝𝐸 ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', ˆ𝑥𝑁 𝑝 𝐸 ′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (5) Thus, by substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (3) and (4) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (5), we obtain: 𝑧′ = [𝑥1 𝑝 𝐴𝐴−1𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑖 𝑝 𝐴𝐴−1𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑁 𝑝 𝐴𝐴−1𝐸] = [𝑥1 𝑝𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑖 𝑝𝐸, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=', 𝑥𝑁 𝑝 𝐸] = 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (6) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (5), encrypted model 𝜑𝑇 allows us to have the same performance as that of the model trained with plane images, under the use of encrypted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='ConvMixer Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Residual connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Global Average Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='PatchEmbedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Fully-Connected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Depthwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Pointwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='GELU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='GELU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='GELU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='HxWxC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='(H /p)x(W /p)xdThird Party(Trusted) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Generation of Random Orthogonal Matrices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='random orthogonal matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='inverse matrix of A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='A-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='A-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Encrypted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='ConvMixer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='ConvMixer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Transform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Training Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Provide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Provide A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Client ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Provider(Untrusted) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Encrypted Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Encrypted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='ConvMixer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='PT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Encryption ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Encrypted Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Result ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Test Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='父 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='4 Generation of Random Orthogonal Matrices A random orthogonal matrix 𝐴 can be generated by using the Gram–Schmidt orthonormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The procedure for generating 𝐴 with a size of 𝑛 × 𝑛 is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Generate an real matrix 𝑅 with a size of 𝑛 × 𝑛 by using a random number generator with a seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Calculate 𝑑𝑒𝑡(𝑅), and proceed to 3 if 𝑑𝑒𝑡(𝑅) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Oth- erwise, return to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Compute a random orthogonal matrix 𝐴 from 𝑅 by using the Gram-Schmidt orthogonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In this framework, any regular matrix can be used as A for image encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Several conventional methods for privacy- preserving image classification use permutation matrices of pixel values, in which many elements have zero values in matrices as 𝐴 = ������ 0 1 0 0 0 1 1 0 0 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (7) In contrast, the proposed random orthogonal matrices include no the zero values as elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The use of such matrices allows us not only to more strongly protect visual information on plain images but to also enhance robustness against various attacks, while maintaining the same performance as that of models trained with plain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In addition, 𝐴−1 can easily be calculated as the transposed matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Experiment Results To verify the effectiveness of the proposed method, we ran a number of experiments on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='1 Setup We used the CIFAR-10 dataset, which consists of 60,000 color images (dimension of 32 × 32 × 3) with 10 classes (6000 images for each class) where 50,000 images are for training and 10,000 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' ConvMixer was trained and tested on the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In the model setting, we set the patch size to 4, the number of channels after patch embedding to 256, the kernel size of depth-wise convolution to 7 and the number of ConvMixer layers to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' models were trained for 200 epochs with the Adam optimizer, where the learning rate was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' We also used a random orthogonal matrix with a size of 48 × 48 for the encryption of test images and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='2 Visual Protection Performance Figure 3 shows an example of images encrypted with a con- ventional encryption method[12][13], in which pixel shuffling and negative-positive transformation are carried out for image encryption, and an example of images encrypted with the pro- posed method, where the images had 𝐻×𝑊×𝐶 = 512×512×3 as an image size, and the block sizes used for encryption were 𝑝 = 8 and 𝑝 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' When using an orthogonal matrix for encryption, transformed pixel values are real values, so (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3 were displayed after normalizing the pixel values to the range of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' From the figures, the selection of a larger the block size gave smaller visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The use of random orthogonal matrices was also demonstrated to have a stronger visual protection performance than that of the conventional method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In addition to visual protection, encrypted images have to be robust enough against various attacks, which aim to restore visual information from encrypted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' We already confirmed that images encrypted with the proposed method are more robust against attacks including jigsaw puzzle solver attacks [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' In particular, unlike ViT, ConvMixer models do have position embedding, so the position of patches cannot be changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Therefore, privacy-preserving ConvMixer needs a stronger encryption method than ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' (a) p = 8 (b) p = 8 (c) p = 16 (d) p = 16 Plane image (a) (b) Figure 3: Example of encrypted images with (a) conventional method[12][13] and (b) proposed method 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='3 Classification Performance We evaluated the classification performance of the pro- posed method as shown in Table 1, where plain and encrypted indicate plane test images and an encrypted test images, respectively, and plain model and encrypted model are models trained with plain images, and models trained with encrypted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Table 1 shows the classification results for each combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' From the table, the proposed method (the combined use of Encrypted models and encrypted images) had the same classification accuracy as that of the baseline without privacy protection (plain models and plain images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Accordingly, the proposed method can not only protect the visual information of test image, but also classify the encrypted image without any degradation of classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='Table 1: Classification accuracy (%) model\\test image plane encypted plane model 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='38 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='2 encypted model 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='27 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content='38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Conclusion In this paper, we proposed a novel method for protecting vi- sual informational on test images under the use of ConvMixer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' The proposed method allows us to use a random orthogonal matrix for image encryption, and it was demon- strated not only to enhance the visual protection of images but to also maintain the same accuracy as that of models trained with plain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Ackowledgment This study was partially supported by JSPS KAKENHI (Grant Number JP21H01327).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UdE2T4oBgHgl3EQfXQfL/content/2301.03843v1.pdf'} +page_content=' Kiya, MM.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:000984cbc55aae74b78a19de37398f718e07ab925231b91e91fb00656a5d5e15 +size 459873 diff --git a/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/2301.02025v1.pdf.txt b/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/2301.02025v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..436b1588855629f4540e7220eb2f9b044edc03e8 --- /dev/null +++ b/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/2301.02025v1.pdf.txt @@ -0,0 +1,921 @@ +1 +Unconventional ferroelectricity in half-filling states of antiparallel +stacking of twisted WSe2 +Liheng An1,2#, Zishu Zhou1,2#, Xuemeng Feng1,2#, Meizhen Huang1,2, Xiangbin Cai1,2, Yong +Chen1,2, Pei Zhao3, Xi Dai1, Jingdi Zhang1, Wang Yao3, Junwei Liu1,*, Ning Wang1,2,* +1Department of Physics and Center for Quantum Materials, the Hong Kong University of Science and Technology, Hong +Kong, China; +2William Mong Institute of Nano Science and Technology, the Hong Kong University of Science and Technology, Hong +Kong, China; +3Department of Physics, University of Hong Kong, Hong Kong, China +#Contributed equally to this work. +*Corresponding authors (emails: phwang@ust.hk (Ning Wang); liuj@ust.hk (Junwei Liu)) +Abstract: We report on emergence of an abnormal electronic polarization in twisted double bilayer WSe2 in antiparallel +interface stacking geometry, where local centrosymmetry of atomic registries at the twist interface does not favor the +spontaneous electronic polarizations as recently observed in the parallel interface stacking geometry. The unconventional +ferroelectric behaviors probed by electronic transport measurement occur at half filling insulating states at 1.5 K and +gradually disappear at about 40 K. Single band Hubbard model based on the triangular moiré lattice and the interlayer charge +transfer controlled by insulating phase transition are proposed to interpret the formation of electronic polarization states near +half filling in twisted WSe2 devices. Our work highlights the prominent role of many-body electronic interaction in fostering +novel quantum states in moiré-structured systems. +Keywords: two-dimensional semiconductor, twist moiré, ferroelectricity, electron interaction, electronic transport + +2 +Stacking layered materials with small twist angles or with small lattice mismatches produce moiré +superlattices that realize modulation of potentials at much larger spatial scale resulting in remarkable +electronic properties [1, 2], for instance, correlated insulators [3], superconductivity states [4] and +ferromagnetism [5, 6], etc. More recently, a new type of ferroelectricity has been created and identified +through the stacking of layered van der Waals heterostructures of hexagonal boron nitride (BN) and +semiconducting transition metal dichalcogenides (TMDCs) [7-13], of which the bulk structures forbids +ferroelectricity. Ferroelectricity in twisted TMDCs is created by the structurally relaxed parallel stacking +geometry and the moiré interface formed by the parallel stacking is also called ‘R-stacking’. Two types of +locally R-stacking domains coexist in one moiré unit cell, featuring opposite spontaneous polarizations of the +same magnitude. The macroscopic ferroelectric polarization results from the dynamic bending of domain +walls that microscopically favors polarization in one direction over the other [14]. +According to the definition in previous publications [14-16], two identical 1H-WSe2 layers stacked to each +other at zero twisting angle will result in the parallel stacking interface. The antiparallel stacking interface is +formed by rotating one of these two identical 1H-WSe2 layers at 60o and stacking the two layers together. +Opposite to parallel stacking geometry, in antiparallel stacking WSe2, the electronic polarization locked to +the local atomic registries is expected to be vanishingly small when local centrosymmetric geometry is fully +respected [17]. Therefore, to date, neither theoretical nor experimental study has reported on the emergence +of ferroelectricity +in antiparallel +stacked +configuration. +As we +previously +identified +that +strong +electron-electron correlation effects occurred in twisted WSe2 [15, 16, 18], it is anticipated that the +correlation-driven insulating states in the moiré bands of antiparallel stacking WSe2 may modulate interlayer +charge transfer and even induce electronic polarization. It has been well acknowledged that the polarization +in crystalline structures can be divided into ionic polarization and electronic state polarization [19]. In +contrast to conventional displacive (ions or molecules) ferroelectricity, electronic polarization (the difference +in polarization between two different states in the same structure) is rare in materials and normally occurs in +a strongly correlated electronic system [20, 21]. +In this study, we choose double bilayer WSe2 to construct antiparallel stacking moiré superlattices and +field-effect devices, of which the moiré flat bands with  pocket holes contribute to the electrical transport. +Stacking two identical bilayer WSe2 with a twist angle near 0o results in the so-called antiparallel interface +(Figure 1). In each moiré unit cell, there are three important high-symmetry stacking sites as labeled by AB, +BW/W and BSe/Se separated by boundary regions (BR). AB sites are the energetically favorable regions (near +2H registry), whereas other regions correspond to higher energy states. The high symmetry sites periodically +modulate the electronic states in real space and therefore produce moiré bands [22]. +Atomically thin WSe2 is mechanically exfoliated from high-quality bulk crystals of WSe2. Field-effect +devices are fabricated based on twisted double bilayer WSe2 by using the tear-and-stack method [15, 18, 23]. +To achieve a large size of uniform moiré lattices, we select a relatively large twist angle of about 4o. +Atomically thin BN layers are used to form an encapsulated device structure. The electrical connection to the +twisted WSe2 channel is realized by Pt electrodes (with a matched work function to WSe2) which offer a +good efficiency for current injection. The device performance has been effectively improved by this kind of +electrode design. The measured channel resistance is about 5 k at a modest carrier density of 3×1012 cm-2 +and the field-effect carrier mobility approaches 2000 cm2V-1s-1. The device channel size is limited to 1×10 +µm to achieve a good uniformity of the twist angle. The strong interlayer coupling between WSe2 bilayers +results in the rise of the Γ valley band top (about 80 meV higher than that of the K valley) [18]. Details of +device fabrication and measurement principles are presented in Supplementary Information. +Our density functional calculation, in the absence of many-body Coulomb interaction, confirms the +dependence of spontaneous electrical polarization on stacking registry, as found in the parallel stacking +interface [14], but denies its existence in the antiparallel stacking interface of twisted double bilayer WSe2 of + +3 +arbitrary lattice match (Figure S2). Atomic-resolution electron microscopic imaging reveals several +noteworthy features in our samples (Figure 2(a) and 2(c)). We first verify the twist angle and moiré +periodicity by electron diffraction (Figure 2(b)). The general moiré structural features well match the model +shown in Figure 1(a). However, we find that the AB regions (marked by hexagons) are expandable and the +location of exact AB stacking (marked by the hexagonal dots and white arrows) is switchable. More analyses +can be found in Figure S3. These structural instability features deviate from the theoretical model, and +therefore hint on the necessity to incorporate new mechanisms for insights into the origin of the moiré band +modulation near half filling under different electrical fields as discussed in more detail later. +The electronic states of the antiparallel stacking bilayer WSe2 channel in the field-effect device shown in +Figure 3 are tuned by the top (VTG) and bottom (VBG) electrical gates. VTG and VBG together can linearly tune +the carrier density (n = (CBGVBG + CTGVTG)/e) and displacement field (D = (CBGVBG - CTGVTG)/2ε0) of this +p-type semiconductor electronic system, where C and V represent device capacitances and gate voltages, e is +the elementary charge and ε0 is the vacuum permittivity. A negative VTG (-12 V to -14 V) is first applied in +order to achieve good electrical contacts to the twisted WSe2 and injection of holes in the device channel. +Forward scanning of VBG (from -60 V to +60 V) is to release holes from the channel and increase D. At VBG += +60 V, the hole density in the channel is very low. The typical electrical transport characteristics of the +p-type antiparallel twisted WSe2 devices measured by a four-probe configuration at 1.5 K are shown in +Figure 3, in which a quick decrease of the resistance (Rxx) by decreasing VBG from +60 V indicates that the +Fermi level touches the edge of the topmost moiré band. By further decreasing VBG, two metallic and two +insulating states are detected. The metallic and insulating states are verified by measuring their Rxx at +different temperatures as demonstrated in Figure 4(a). We further verify that the two insulating states are +from the correlation-induced splitting of the topmost moiré band [2, 23, 24], corresponding to the half and +full filling states respectively. The full filling gap is resolved at a carrier density of n = 9.8×1012 cm-2 with +two holes per unit cell to fully occupy the first moiré band [25]. The resistance peak emerging near n = +5~6×1012 cm-2 represents the important half filling insulating states driven by strong correlation effects. By +forward-and-backward scanning VBG for several times, an obvious resistance hysteresis is repeatedly +observed around half filling states (Figure 3). At half filling, when VTG is set at -13.8 V, the forward +scanning (-60 V to +60 V) leads to a high resistance state of about 12.5 k, while the backward scanning +yields a low resistance state of about 10 k  . Such a resistance difference can be considered as the +appearance of internal electric fields [13] from charge polarization (perpendicular to the moiré lattice plane) +states in the system. We have carried out each measurement for five times to ensure the reproducibility of the +hysteresis loops. In addition, measurements of the I-V curves at the half filling states also confirm that the +forward and backward scanning of gate voltages result in different channel resistances (Figure S9). To rule +out the possibility that the BN dielectric layers or metal lead interfaces in the devices could potentially +generate a similar hysteretic resistivity, we fabricated and measured reference devices for comparison. We +also performed different gate scanning rates during hysteretic resistivity measurements and did not observe +any impurity/charge trapping effect. All these additional experiments evidence that BN or charge trapping +effects do not play a role in the observed hysteretic resistivity near half filling states in our samples (See +details in the Supporting Information). +The appearance of the electrical polarization characteristics in the antiparallel stacking interface structures +of WSe2 is abnormal. Different from conventional [19] and newly discovered layered van der Waals +ferroelectricity [7-13, 26-29], the electrical polarization in our samples is highly relevant to correlation +effects which are temperature dependent. As shown in Figure 4(a), the half filling insulating states gradually +disappear when temperature is higher than about 40 K. The largest hysteresis we observed at 1.5 K also +gradually disappears when temperature approaches to about 40 K (Figure 4(c)). All these phenomena are +very different from that of the ferroelectricity observed in the parallel stacking geometry of TMDCs where +the spontaneous electronic polarization survives at room temperature, not relevant to filling states at all, + +4 +further differentiating the origins of the ferroelectricity effects between the two twist moiré systems. +Here, we try to elucidate the abnormal electrical polarization characteristics based on the following facts. +The transport data are from the contribution of the Γ valley edge of WSe2 valance bands [18] and the +ultra-flat moiré bands (survived up to 4o twist angles [30]) are spatially associated with the AB sites in the +antiparallel stacking interface of WSe2 [16]. AB sites form a triangular lattice which can be considered as a +mimic triangular lattice of the Hubbard model (Figure 5(a)). The topmost moiré band is separated from the +rest and its charge distribution is tightly localized at the moiré potential minima of AB sites. Double +occupancy of AB sites is suppressed due to the strong on-site Coulomb repulsion potential U, resulting in +moiré band splitting to lower (LHB) and upper (UHB) Hubbard bands [31]. +Ideally, the carrier densities at full/half filling of the Hubbard bands are purely determined by the moiré +periodicity and obtained by n0 = 4/(√3λ2), where λ is the moiré superlattice constant. We first focus on the +Rxx peak positions of full filling states as a reference to test the functionality of the two gates. By setting VTG +and scanning VBG (Figure 4(b)), we confirm that the injected density n (hole carriers in p-type WSe2) at full +filling (the gap position) as characterized by Rxx peaks are all correctly tuned by the two gates. This is +reflected by the change of the full filling peak positions as outlined by the inclined straight dashed line in +Figure 4(b). By increasing VTG, the full filling peak positions shift to right side, meaning that along this +dashed line, the carrier density is unchanged. Because both full and half filling densities are fixed by the +moiré lattice geometry [32], the Rxx peaks at half filling should also follow a straight line parallel to the full +filling peak line. However, the peak positions at half filling obviously shift to left side (towards to higher +hole concentration regions). This suggests that when increasing the D field, more holes are needed to be +injected into the system in order to completely fill the LHB as displayed more clearly in Figure 5(c). In +addition, we noticed that the Rxx at half filling decreases by increasing D, which could be interpreted as that +the Hubbard gap U becomes narrow accordingly. These interesting experimental data suggest following two +possible physical pictures: (1) D enhances new electronic states involved in the half filling states in the moiré +system; or (2) increasing D could narrow the Hubbard band gap U. +We now provide a theoretical picture to intuitively interpret the correlation-effect-induced electronic +polarization based on the interlayer charge transfer controlled by electric fields and insulating phase +transition. For a large D field applied to the double bilayer twisted device, it is possible that the entire +low-energy moiré bands become layer polarized, as reported in the unconventional electronic polarization in +bilayer graphene heterostructures [9]. Similarly, electrons occupying a moiré band at low energy could locate +on a specific WSe2 layer in the device. It is also reasonable to assume that LHB/UHB locates at the interface +between the two inner WSe2 layers (Figure 3). In our device design, the electrodes directly connect to the +bottom WSe2 layer and holes are firstly filled preferably in the bottom layer. We then propose that the +observed D-dependent shift of the half filling peak is due to the involvement of new electronic states. Figure +5(b) schematically shows the additional band (B) coexisted with the Hubbard bands which contributes to the +new electronic states. This band can be considered as the contribution from the bottom WSe2 layer directly +connected to the electrodes for hole injection. The inner two layers which host LHB/UHB states are actually +not directly connected by the electrodes. +At a fixed D field, the backward scanning of VBG (starting from +60 V) means filling holes into the +electron-occupied LHB first (Figure 5(b)). In our p-type WSe2 moiré system, the charge carriers which +contribute to the transport measurement are holes in the valance band. Before adding holes into the valance +band, all states are occupied by electrons. We use solid color (black and yellow) to illustrate the electron +occupied states in the LHB and UHB in Figure 5(b). By turning the Fermi level, holes are firstly injected into +the LHB and then into the UHB. Because of the involvement of the B-band, the apparent Hubbard gap +position may shift to left side. During backward VBG scanning, hole carriers transfer from the B-band to LHB. +This interlayer charge transfer is driven by the potential associated with the gates and D field [33]. The +bottom WSe2 layer offers the B-band which is p-type under the same gating condition (larger than the + +5 +threshold gate voltage). Figure 5(b) shows the band alignment and the carrier transfer direction between the +bands. The hole filling process allows current to flow easily from B-band (p-type) to the Hubbard bands +(p-type) across the spatial layer interface (similar to a p-p isotype junction [34]). Near half filling, however, +the states at the bottom edge of LHB are electron-like (n-type). Holes transfer across a p-n anisotype junction +(the forward biasing of a p-n junction) is less resistive. Therefore, filling LHB/UHB driven by backward VBG +scanning is less resistive. For the forward VBG scanning, however, holes are released from LHB/UHB to the +B-band. Near half filling, the p-n anisotype characteristic at the interface severely restricts holes reversely +flowing through the anisotype junction (analogous to the reverse biasing of a p-n junction). The consequence +is that more holes are retained in the moiré layer when the forward scanning is near half filling, while the +holes in the B-layer are always released to the electrodes normally during VBG scanning. Therefore, the +charge polarization between the moiré layer and the B-layer is established. The extra holes in the moiré layer +countervail only a part of D field strength (Figure S11(b)), resulting in a higher Rxx compared to that in the +opposite direction of backward scanning VBG. This is because Rxx is inversely proportional to the strength of +the D field as revealed experimentally in Figure 5(c). This mechanism supports the transport data of the Rxx +hysteresis in Figure 3. We now conclude that the two electrical polarization states are the result of more +versus less occupation of the Hubbard bands driven by the interlayer carrier transfer modulated by the +insulating phase transition during D field scanning near half filling states. This abnormal electronic +polarization effect has been repeatedly observed in the devices with twist angles ranging from 3.8-4.2o (see +more data in the Supporting Information). +Electronic ferroelectricity caused by electron correlation has been proposed and studied previously based +on Hubbard models in different bulk material systems [20, 21], in which parameters of Coulomb repulsion +potential U and hopping energy t are critical. In our moiré system, the expandable AB sites driven by +correlation effects or D fields [26] could modulate U and t. This is because the local AB sites (near 2H +registry) in our samples is expandable (also deviating from an ideal local 2H registry) involving the atomic +rearrangement of the boundaries to surrounding regions. Therefore, the expanded local 2H registry together +with the boundaries surrounding these regions do not hold the centrosymmetric geometry according to our +electron microscopy results (see more details in the Supporting Information). The expansion of AB sites +(increasing the energy favorable area) driven by the D field could provide the driving force for decreasing U, +since increasing AB site area can effectively lower U to host two electrons with opposite spins in one AB +site. At present, it is still not clear whether the deviation of the half filling Rxx peak under different D fields is +solely due to the interplay between the Hubbard bands and B-band or partially due to the D field modulation +(e.g., D-field-induced electronic state instability) of the electronic states in the Hubbard bands. To this end, +further experimental and theoretical investigations are needed. Our discovery of the abnormal ferroelectricity +behaviors suggests a new platform for further exploring the flat band properties tunable by an electric field, +particularly the correlation-induced insulating states at the half filling states adjacent to the superconductivity +states in this two-dimensional moiré system. +Acknowledgements +We are grateful to the technical support from Dr. Yuan Cai from the Materials Characterization and +Preparation Facility at the Hong Kong University of Science and technology. +Funding +This work was supported by grants from the National Key R&D Program of China (2020YFA 0309600), the Hong Kong +Research Grants Council (Project Nos. AoE/P-701/20, C6025-19G, 16305919 ECS26302118, 16303720, 16305019, +16306220 and N_HKUST626/18), National Natural Science Foundation of China (NSFC20SC07) and the William Mong +Institute of Nano Science and Technology. + +6 +Author contributions +N.W. initiated and supervised the project. L.A. designed the research, device structure, measurement and data analysis. L.A., +Z.Z., X.F. fabricated device and collected data. L.A. and N.W. wrote the manuscript. M.H., J.Z., X.D. and J.L. contributed to +data analysis. P.Z. and W.Y. contributed to calculation. X.C. and Y.C. contributed to sample preparation. +Conflict of interest +The authors declare that they have no conflict of interest. +Data availability +The original data are available from corresponding authors upon reasonable request. Details of device fabrication, +measurement principles and proposed mechanisms are presented in Supplementary Information. +References +1 +Kennes DM, Claassen M, Xian L, et al. 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Mater. 2020; 6: +1900818 (1-30). +31 +Magorrian SJ, Enaldiev VV, Zólyomi V, et al. Multifaceted moiré superlattice physics in twisted WSe2 bilayers, arXiv +2021; 2106.06058. +32 +Zhang Y, Yuan NF and Fu L, Moiré quantum chemistry: Charge transfer in transition metal dichalcogenide superlattices, +Phys. Rev. B 2020; 102: 201115-6. +33 +Li T, Jiang S, Li L, et al., Continuous Mott transition in semiconductor moiré superlattices, Nature 2021; 597: 350-354. +34 +Liu Y, Zhang Meng S, et al. Electric Field Tunable Ultrafast Interlayer Charge Transfer in Graphene/WS2 Heterostructure, +Nano Lett. 2021; 21: 10, 44034409. +35 +Lourtioz JM, Abstreiter G, Meyerson B, Group IV Heterostructures, Physics and Devices (Si, Ge, C, Sn) 1998; 61: +Elsevier Science & Technology, Oxford, United Kingdom. + +8 +Figure 1 (a) Moiré superlattice structure constructed by antiparallel stacking of twisted double bilayer WSe2. The circles denote the high +symmetry stacking sites. The hexagons (dashed lines) indicate the near 2H registry regions. BR indicates the boundary regions. (b) The +atomic configurations of the high symmetry stacking sites of AB, BW/W and BSe/Se and their side views of these sites. Blue and red arrows +denote the atom positions shifts from top and bottom layers of WSe2. + +Near2Hregistry +(a) +(q) +Atom position +regions +40 +shift by distortion +BR +AB: +AB +BR +BR +Top-view +AB(2H) +BW/W +BSe/Se +A +AB +B +A +B +Side-view9 +Figure 2 (a) Transmission electron microscopy (TEM) image of the antiparallel twisted double bilayer WSe2. (b) Electron diffraction pattern +taken from the area in (a). The twist angle is determined to be about 3.8o. (c) An enlarge image of the region marked by the white dashed +rectangle in (a). The positions of exact 2H stacking are marked by yellow hexagons (dashed lines). The white thick arrows indicate the +directions of the 2H registry shifts. The scale bar is 3 nm. + +(a) +(b) +3.80 +(100)A +(100]B +(000) +BR +BR +BR +AB +(c) +2Hregistry10 +Figure 3 The structure of the field-effect device built based on antiparallel stacking of twisted double bilayer WSe2 and the typical resistance +hysteresis measured in the device. The device is double gated by the top and bottom gates in order to tune the carrier concentration and the +displacement field (D). By fixing the top gate (VTG) at -13.8 V, scanning the bottom gate (VBG) results in the change of the filling states. +Forward-backward scans (indicated by the dashed/solid lines) of VBG show a clear resistance hysteresis. + +Top gate +Twist +WSe2 +BN +Charge +VTG +transfer +VBG +Bottom +SiO2+BN +layer +Bottom gate +Half filling +Full filling +Metallic +Metallic11 +Figure 4 (a) Phase diagram plotted based on transport measurement. The half filling states gradually disappear when temperature is higher +than 40 K. (b) Displacement field effects (by setting VTG at different voltages and scanning VBG) on the full filling and half filling states. The +full and half filling states behave differently by changing the displacement fields. The black/red thick dashed lines indicate the ideal full/half +filling peak positions. The black thin dashed line indicate the peak positions of half filling states measured. (c) Temperature dependence of +the resistance hysteresis. The hysteresis loops gradually disappear at about 40 K. + +Half filling +Full +filling12 +Figure 5 (a) A mimic triangular lattice of the Hubbard model for the AB sites and the variation of U and t due to AB site expansion. (b) +Carrier transfer between the B-band and LHB/UHB. Forward scanning of VBG results in the releasing of holes and the backward scanning of +VBG results in the filling of holes in the bands. Charge transferring between the B-band and LHB/UHB shows different behaviors near the +gap region of the Hubbard bands. (c) Displacement field effects on the full filling and half filling gap positions. The full filling states are +independent of the D field as indicated by the thin solid line. However, the half filling is D-dependent which deviates from the ideal positions +as indicated by the right side thin solid line. + +(a) +U +Decrease +AB +AB +AB +ti +Decrease +(b) +BackwardVBG +Electron like (n) +(filling holes) +UHB +U +LHB +Full filling +1 +e +B-band +Forward VBG +Charge +(releasing holes) +transfer +(c) +Vtc = -12.8V to -14V +1/2 filling +50 +0 = 3.8°± 0.1° +40 +Increase D +(k2) +30 +R +Full filling +20 +10 +0 +-10 +-8 +6 +-4 +n (1012/cm²)13 +Supplementary Information +Unconventional ferroelectricity in half filling states of +antiparallel stacking of twisted WSe2 +Liheng An1,2#, Zishu Zhou1,2#, Xuemeng Feng1,2#, Meizhen Huang1,2, +Xiangbin Cai1,2, Yong Chen1,2, Pei Zhao3, Xi Dai1, Jingdi Zhang1, Wang Yao3, +Junwei Liu1, *, Ning Wang1,2,* +(# These authors contribute equally to this work) +*Corresponding authors (emails: phwang@ust.hk (Ning Wang); liuj@ust.hk (Junwei Liu)) +1Department of Physics and Center for Quantum Materials, The Hong Kong University of +Science and Technology, Hong Kong, China. +2 William Mong Institute of Nano Science and Technology, The Hong Kong University of +Science and Technology, Hong Kong, P.R. China. +3Department of Physics, University of Hong Kong, Hong Kong, China. +A. Device fabrication +Atomically thin WSe2 is mechanically exfoliated from bulk crystals (from 2D +Semiconductors). The bilayer WSe2 is identified by optical microscopy and photoluminescence +techniques. The bottom hexagonal boron nitride (hBN) is around 25 nm thick. The bottom hBN +is pre-patterned and selectively etched down to the SiO2 by plasma treatment in CHF3. Then, +Cr/Pt 10 nm/20 nm is deposited to form the bottom electrodes. We fabricated the twisted double +bilayer WSe2 devices by using the tear and stack method as previous work introduced [S1-3]. +Then polypropylene carbonate is used to tear one part of the bilayer WSe2. The bottom flake is +rotated by a small angle and the top flake is used to pick up the bottom flake. The whole stacked +structure is placed on the pre-patterned bottom electrodes to form the contacts to the twist +structure (FIG. S1). Finally, another flake of hBN (30 nm–50 nm) is transferred onto the top +surface of twist WSe2, and a thin layer of Cr/Au (10/70 nm) is deposited on the top surface of +hBN to form the top gate. Pt layers directly deposited onto the top surface of the bottom hBN +generally result in gas trapped around the bottom electrodes on the sample interfaces. The +trapped gas, after performing a thermal annealing, accumulates at the electrode edges, leading to +detach of the metal and the sample interface. The large real space gaps between the metal and +the sample is the main reason for generating a huge contact resistance and thus degrade the +device quality. Here, by deep etching of the bottom hBN and depositing Pt electrodes, we +improved the evenness of the bottom Pt electrodes, hBN bottom surfaces and the current +injection efficiency from the metal electrodes into the atomically thin WSe2. For electron +microscopy investigation, the twisted structure was dropped onto a holey carbon grid, cleaned +by different chemical solutions, and then dried in vacuum. + +14 +FIG. S1. (a) Device structure of antiparallel stacking of bilayer WSe2. (b) Twist WSe2 structure +placed on the metal electrodes before covering the hBN and metal gate. (c) Cross-sectional view +of the twist WSe2 device design. +B. Improvement of device performance +To ensure the quality of the devices, two layers of hBN are used to fabricate the fully +encapsulated structure (FIG. S1(c)). The misalignment of the top and bottom hBN is carefully +considered to avoid the formation of any small angle twisted hBN/WSe2 interfaces. In fact, the + +TG-10 +5 +10μm +TiIAu +BN +WSe2 +WSe2 +Pt +Pt +Pt +Pt +BN +SiO2 +n-Si15 +transport characteristics measured in the devices are dominated by the moiré interface properties +especially the strong half filling (υ=1, one hole per moiré unit cell) insulating states [S4], thus +the hBN interface effects are ignorable. We use the bottom-electrode design to make the +contacts to the channel of the bottom twist layer (See FIG. S1). We use the top and bottom gates +to control the out-of-plane DC electric field applied to the moiré channel and change the carrier +concentration. The device performance has been effectively improved by this kind of device +design. The measured channel resistance is about 5 k at a modest carrier density of 3×1012 +cm-2 and the field-effect carrier mobility approaches 2000 cm-2V-1s-1. The device channel size is +limited to 1×10 µm to achieve a good uniformity of the twist angle. Such small-sized samples +integrated with bottom and top metal electric gates therefore limit the detection of the +ferroelectric property at cryogenic temperatures by conventional techniques. Our transport data +are from the Γ valley. The strong interlayer coupling between WSe2 bilayers results in the rise +of the Γ valley band top (about 80meV higher than that of the K valley) [S3]. +C. Computation results +Based on the same theoretical model presented in Ref. +[S8], we examined the difference vacuum levels +vac on +the two sides of the 2L+2L WSe2 with lattice-matched +antiparallel stacking. We find that +vac is 0 (cf. FIG. +S2), indicating the electrical polarization is forbidden. +Notably, the electrical polarization at the interface of +2L+2L WSe2 has invisible dependence on each stacking +registry ���t. In this calculation, interaction effects have not +been considered. +FIG. S2. The difference in the vacuum levels on the two +sides of the 2L+2L WSe2, ������ indicates the stacking registry +of double bilayer WSe2. +We have carried out measurements in different samples +with antiparallel 2L+2L stacking geometry (FIG. S2) at +twist angles ranging from 3.8-4.2 degrees. These samples did not show hysteretic resistivity at +room temperature since the half filling states did not form at room temperature. While at +cryogenic temperatures, half filling sates emerge in these samples accompanied with the +hysteretic resistivity effects. Theoretically, this ideal antiparallel 2L+2L stacking geometry with +a local 2H registry does not have out-of-plane polarization. Therefore, in principle, the gate +voltage should not cause expansion of the local 2H registry. This also implies that “ideal” half +filling states should be displacement field independent, and there should not be electrical +polarization/ferroelectricity effect in this system. However, interaction effects may induce +symmetry breaking and generate electronic polarization. +D. Atomic structure of 2L+2L twisted WSe2 +We identified by high-resolution electron microscopy that the local AB sites (near 2H +registry) in our samples is non-uniformly expandable (also deviating from an ideal local 2H +registry) involving the atomic rearrangement of the boundaries to surrounding regions. +Therefore, the expanded local 2H registry together with the boundaries surrounding these +regions do not hold the centrosymmetric geometry according to our microscopy results. On the + +4.70 +4.68- +4.66- +E, +E2 +AE +=E-E, +4.64 +vac +4.62- +4.60 +4.58 +4.56- +0 +102030 +50 +z (A)16 +other hand, it is reasonable to assume that symmetry breaking (or atom position deviation) +effects could generate charge transfer between layers and induce electronic polarization. As +demonstrated by electron microscopy, the expandable 2H sites already break the local 2H +centrosymmetric geometry. +FIG. S3. (a) Schematic illustration of the expandable and switchable 2H registry induced by +atomic position shifting. (b) An enlarge TEM image showing that the exact 2H registry positions +as marked by yellow hexagons. The white thick arrows indicate the expansion or shift of the 2H +registry is non-uniform. The scale bars are 3 nm. +By analyzing the atom positions in transmission electron microscopy (TEM) images, we +identify the details of the 2H registry sites (the AB regions marked by hexagons). An ideal 2H +registry site is displayed as one “hexagonal-flower-like” pattern in the atomic image with +strong and sharp contrast (centered at a small yellow hexagon), in which the strong bright dots +indicate highly out-of-plane aligned atoms. The larger the area covered by the sharp +hexagonal-flowers, the more expended area occupied by the 2H registry. Obviously, many 2H +sites show two or three strong and sharp contrast of the hexagonal-flowers emerging together +due to the expansion of the 2H registry area. FIG. S3(a) illustrates the mechanism of the +expandable or switchable 2H registry which is useful for understanding the modulation of the +Hubbard band gap variation. The expansion of AB sites (increasing the energy favorable area) +driven by D field provides the driving force for decreasing U, since increasing AB site area can + +(a) +Atomic +shifting +BR +(b) +BW/W! +BR +-BR +AB +2Hegistry17 +effectively lower U to host two electrons with opposite spins in one AB site. In this scenario, +shrinking the sizes of AB sites should cost extra energy. +E. Transport measurement and carrier density estimation +For transport measurement, an AC excitation is set within the range from 0.2 mV to 5 mV +at a frequency of 4.579 Hz. The current signal is probed by a lock-in amplifier. The voltage +signals are detected through a low noise preamplifier SR550. The carrier density (if there is no +internal polarization field in the sample) can be estimated by ��� = (������������������ + ������������������)/���, and +the perpendicular electric field is expressed by ��� = (������������������ − ������������������)/2���0 = ������ − ������ . +(CTG/CBG, top/bottom gate capacitance; VTG/VBG, top/bottom gate voltages). The top gate electric +field is applied through the top few-layer hBN (defined by ������ = ������������������/2���0), and the back +gate electric field is applied through the 300nm-SiO2 layer on the Si substrate (defined by ������ = +������������������/2���0). +The dual-gate structure allows us to tune the charge density and the out-of-plane electric +displacement field [S5]. Different from the multilayer graphene sensing scheme for detecting the +interfacial ferroelectricity [S5, S6], the out-of-plane electrical polarization and/or potential +difference between opposite polarization of domains in twisted TMDCs can be translated into +the gate-controlled doping effect which is directly reflected by the electron transport hysteresis +of the twisted TMDCs field-effect transistor devices [S7]. In our work, we measure electrical +resistance at different carrier concentration under different DC gate scanning. The out-of-plane +electronic polarization effects in our devices is presented by resistance hysteresis loops. For +pristine few layer WSe2 samples, no resistance hysteresis loop has been observed during +forward-backward gate scanning (see FIG. S4). The moiré periodicity can be described by λ = +���/( +2sin��� +2 +), where a is the lattice constant of WSe2. The full filling carrier density is related to +the moiré wavelength with ���0 = 2/ +3 +2 λ2. Then, we can further calibrate the twisted angle based +on the estimation of the carrier densities. +F. Interface and charge impurity effects +We have carried out different experiments to rule out the possibility that hBN interfaces or +charge trapping effects could potentially generate a similar hysteretic resistivity as what we +reported. Charge trapping effects are normally resulted from interface impurities. The interfaces +in our devices are formed between hBN and WSe2. We provide here more experimental data and +analyses. +1. The hBN moiré interface effects: The hBN and WSe2 layers have no specific twist angles in +our devices. Therefore, there is no comparable moiré lattice formed at the hBN/WSe2 +interfaces. In fact, the lattice parameters of WSe2 (a = b = 0.3297 nm) and hBN (a = b = +0.2502 nm) are very different. There is no comparable moiré superlattice formed in our +devices. +2. hBN/WSe2/hBN interfaces: We fabricated hBN-sandwiched WSe2 field-effect devices using +the same interface structure and measured the gate dependent transport properties. There is +no hysteretic resistivity characteristics in these kind of samples during scanning the gate +voltage (FIG. S4). Therefore, we confirmed that hBN/WSe2 interfaces and the metal lead +interfaces in our devices do not generate any hysteretic resistivity. + +18 +FIG. S4. Transport property of the reference hBN-sandwiched p-type WSe2 field-effect device +built based on the same interface structure used for twisted WSe2. There is no hysteretic +resistivity characteristic as measured by scanning the gate voltage at 1.6 K. +3. Charge trapping effects can normally be distinguished by changing the gate scanning rates. +This method has been widely used in identifying charge trapping effects in graphene devices +[S9]. We performed different gate scanning rates in our twist WSe2 devices and did not +observe any obvious difference as shown in FIG. S5. This is a strong evidence that charge +trapping effects did not play a role in the observed hysteretic resistivity at half filling states. +4. +Charge trapping can normally exist at room temperature such as in graphene [S9], in +particular near the Dirac point. In this case, the charge trapping behavior becomes obvious. +Near the resistance peak of a graphene device, charge traps cause shifts of the peak position +(left-or-right). The amplitude of the peak has almost no change. This is because of the charge +nature of the impurity at the device interfaces. However, in our device, the main feature of +the hysteresis and resistance peaks is the amplitude change (high-or-low), a very different +behavior. + +0.20 + Backward +-Forward +0.15 +(sw) +0.10- +b +0.05 +0.00 +-11 +-10 +6- +-8 +-7 +-6 +-5 +-4 +Vg(V)19 +FIG. S5. Transport measurements at different gate scanning rates in a 4.2o twist WSe2 device. +The half filling peaks and the hysteretic resistivity data do not show any obvious difference at +different scanning rates. +The hysteretic effect in our sample behaves differently compared to the normal charge +trapping effects since it decays quickly when sample temperature is increased to about 40 K. +However, the charge trapping effects caused by impurities normally can still exist at room +temperature. In fact, charge trap effects usually occur globally in a device (not just at certain +carrier concentration) in poor conducting materials, while our devices show high field-effect +mobility with a high current injection efficiency between the electrodes and the twist bilayer +WSe2. The reproducibility of our device performance suggests that the hysteretic effects in our +samples are not due to charge trapping. +G. More transport results +Although the target of the twist angles is set to 4o, the fabricated devices often have a +deviation from this angle. The twist angles of the antiparallel stacking moiré super-lattices we +obtained range from 3.8 to 4.2o. Here are more experimental data we obtained from different +devices. In FIG. S6(a), we show the half filling peaks measured at different gate voltage from a +device with a 3.9o twist angle. FIG. S6(b) shows the corresponding resistance hysteresis. FIG. +S7 shows the transport data taken from a device with a 4.2o twist angle. We find that for the +samples with relatively large twist angles, their half filling peaks look sharp and the resistance +hysteresis characteristics are more localized around the half filling density position. To calculate +the displacement field D, we need to first estimate the top and bottom capacitances. By changing +VBG and VTG, we determine the D filed by: +One of the results is shown in FIG. S8. + +20 +FIG. S6. (a) The half filling peaks measured at different displacement fields from a device with a +3.9o twist angle. (b) The resistance hysteresis near the half filling states. +FIG. S7. Transport data taken from a 4.2o twist angle device. (a) Temperature effects on the half +filling states. (b) The resistance hysteresis near the half filling states. + +iiiifilling factor +(a) +(b) +-1.0 +-0.8 + (k2) +20 +Half filling +Backward +160 +80 +...... Forward +90 +15. +Half flling +20 +(kΩ) +60 +← 10. +40 +5. +20 +-6.0 -5.5 -5.0 -4.5 -4.0 +-60 +-30 +0 +30 +n (x1012/cm2) +VBG (V)21 +FIG. S8. Demonstration of both the strength of D field and the carrier density n dependent Rxx. +FIG. S9. I-V characteristics at the half filling states. (a) Transport measurements of I-V +characteristics performed at the half filling states under the fixed top and back gate voltages. +(b-c) I-V measurements of forward/backward scanning at ferroelectric state under different gate +voltages. (d) Different resistances observed from forward/backward scanning. +We performed more transport measurements of I-V characteristics in the half filling states at +fixed top gate (-9.5 V to -10.2 V) and back gate (-35 V to -41 V) voltages. We observed that for +Vds larger than 0.01 mV, the I-V shows linear relationship. We also carried out I-V + +n +D +n +S22 +measurements on forward/backward scanning at the half filling ferroelectric state and observed +different resistances. +In this study, we are not able to demonstrate a complete map of the resistant hysteresis +loops as a function of D field because the top gate voltage has a very limited variation range. +The main reason is that our devices are built with top and bottom gates. However, because there +are totally four layers WSe2, and the metal electrodes are designed to contact with the bottom +layer of WSe2, we need to apply a relative large top gate voltage in order to achieve a good +ohmic contact to the semiconductor twist WSe2. The ohmic contact behavior can only be kept in +a narrow range of the top gate voltage (-10 V to -15 V). Top gate voltages larger than -15 V may +cause breakdown of the dielectric layer. A top gate voltage smaller than -10 V resulting in poor +electrical contact. Therefore, the scan including both top and bottom gate is mainly limited by +the top gate. Scanning top gate in a large range will change the electrical contact characteristics +and the experimental data are not reliable. Although we cannot provide a complete map of the +resistant hysteresis loops as a function of D field, we add experimental data to partially show the +resistant hysteresis changes at limited range of the D field. FIG. S10 shows the changes of +resistance around half filling as a function of D field. +FIG. S10. Experimental data showing the changes of resistance around half filling as a function +of D field. +H. Interlayer charge transfer mechanism +FIG. S11(a) schematically shows the two bands associated with the bottom WSe2 layer +(directly connected to the electrodes) and the moiré super-lattice layer without considering +interaction effects. Both bands are from the  valley of WSe2 with p-type characteristics. They +form an isotype junction [S12, 13] for charge transfer between the two layers during +forward/backward scanning of VBG. +Due to the strong interaction occurring in the moiré super-lattice layer, LHB and UHB +formed (FIG. S11(b)). For backward scanning of VBG (starting from +60 V), holes start to +transfer from B-band to LHB first. In this case, both bands are p-type. Charge transfer across the +p-p isotype junction is less resistive. Noticed that when holes occupy more than half of the states +in LHB, the bottom band edge of LHB is electron-like (n-type semiconductor). Such a band +alignment between LHB and B-band generates a n-p junction (anisotype junction). Holes +transfer from B-band (p-type) to the n-type moiré layer is less resistive (similar to the forward +biasing of a n-p junction). However, the opposite direction of holes transferring across the n-p + +n = 5.8*1012/cm2 +Backward +--- Forward +V=1 +200 +(kΩ) +150 +100 +0.36 +0.37 +0.38 +0.39 +0.40 +0.41 +0.42 +0.43 +D (V/nm)23 +junction is restricted. Charge transfer between B-band to UHB is less resistive since their +junction is always isotype. +FIG. S11. (a) Two bands associated with the bottom WSe2 layer and the moiré super-lattice +without interaction. (b) Strong interaction induces LHB and UHB of the Hubbard bands. +Charge transfers between the Hubbard bands and B-band near the half filling states is +modulated by the formation of the Hubbard band gap. +For forward scanning of VBG, holes transfer back to the B-layer. In this case, charge +transfer across all isotype junctions is less resistive. However, near half filling states, the charge +transfer is severely restricted (analogous to the reverse biasing of a n-p junction). The +consequence is that there are more holes retained in the moiré layer (in the same time the holes +in the B-layer are released to the electrodes normally) when a forward scanning approaches the +half filling region. Therefore, the charge polarization between the moiré layer and the B-layer is +established. The extra holes in the moiré layer countervail only a part of the strength of the D +field (FIG. S11(b)), resulting in a higher Rxx compared to that in the opposite direction of +backward scanning VBG. This is because Rxx is inversely proportional to the D field as revealed +experimentally in Fig. 5(c) in the main text. +We noticed that there is also resistance hysteresis in the metallic phase. First, bilayer WSe2 +is semiconductor. The so-called insulating states at half filling are generated by the correlation +effects in the twist moiré system. The nearby metallic states are also called strongly correlated +metallic phases, such as the superconductor states in twisted WSe2 bilayer we discovered. +Second, since the electron polarization states are relevant to the correlated states, it is reasonable +to believe that the resistive hysteresis occurring at states away from the half filling states is also +related to correlation effect. However, the resistive hysteresis occurring at the state far away +from the half filling states is indeed complicated. According to our model, the alignment of the +Hubbard band and the additional B-band in the “metallic states” is isotype junction. If there is +a slight barrier between the moiré interface layers and the B-band layer which might be due to +the formation of the moiré bands, the electron polarization could happen, resulting in the +resistance hysteresis. This situation is similar to that of the anisotype junction. + +No interaction +Stronginteraction +Hole (p-type) +Isotype junction +Half filling +Anisotype +P-type +junction (n-p) +P-type +P-type +Isotype junction +Hole (p-type) +Moire +Bottom +laver +Wse2 +laver +Charge polarization +from extra holes in +the moire layer +Isotvpe junction ++24 +References +[S1] +X. Cai, A. Liheng, X. Feng, S. Wang, Z. Zhou, Y. Chen, Y. Cai, C. Cheng, X.-Q. Pan, +and N. Wang, Nanoscale +(2021). +[S2] +L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. +Kennes, Y. Bai, and B. Kim, Nature materials 19, 861 (2020). +[S3] +L. An, X. Cai, D. Pei, M. Huang, Z. Wu, Z. Zhou, J. Lin, Z. Ying, Z. Ye, and X. Feng, +Nanoscale horizons 5, 1309 (2020). [15] +[S4] +Y. Liu, J. Guo, E. Zhu, L. Liao, S.-J. Lee, M. Ding, I. Shakir, V. Gambin, Y. +Huang, and X. Duan, Nature 557, 696 (2018). +[S5] +Z. Zheng, Q. Ma, Z. Bi, S. de la Barrera, M.-H. Liu, N. Mao, Y. Zhang, N. Kiper, +K. Watanabe, and T. Taniguchi, J. Kong, W. A. Tisdale, R. Ashoori, N. G., L. Fu, S. Y. +Xu, P. Jarillo-Herrero, Nature 588, 71 (2020). +[S6] +X. Wang, K. Yasuda, Y. Zhang, S. Liu, K. Watanabe, T. Taniguchi, J. Hone, L. Fu, +and P. Jarillo-Herrero, arXiv preprint arXiv:2108.07659 +(2021). +[S7] +A. Weston, E. G. Castanon, V. Enaldiev, F. Ferreira, S. Bhattacharjee, S. Xu, H. +Corte-Leon, Z. Wu, N. Clark, and A. Summerfield, T. Hashimoto, Y. Gao, W. Wang, M. +Hamer, H. Read, L. Fumagalli, A. V. Kretinin, S. J. Haigh, O. Kazakova, A. K. Geim, V. +I. Fal'ko, R. Gorbachev arXiv preprint arXiv:2108.06489 (2021). +[S8] +Pei Zhao, Chengxin Xiao and Wang Yao, npj 2D Materials and Applications 5, 38 +(2021). +[S9] +Haomin Wang, Yihong Wu, Chunxiao Cong, Jingzhi Shang, and Ting Yu, ACS Nano +4, 12, 7221(2010). +[S10] +T. Li, S. Jiang, L. Li, Y. Zhang, K. Kang, J. Zhu, K. Watanabe, T. Taniguchi, D. +Chowdhury, L. Fu, J. Shan and K. F. Mak, Nature 597, 350 (2021). +[S11] +Kenji Yasuda, Xirui Wang, Kenji Watanabe, Takashi Taniguchi, Pablo Jarillo-Herrero, +Science 372, 1458 (2021). +[S12] +J.-M. Lourtioz , By (author) +G. Abstreiter , By (author) +B. Meyerson， “Group IV +Heterostructures, Physics and Devices (Si, Ge, C, Sn)”: Volume 61，Elsevier Science & +Technology， Oxford, United Kingdom， 1998. +[S13] +Opdorp, van, C. J. M. “Si-Ge isotype heterojunctions”. Technische Hogeschool +Eindhoven. 1969. + diff --git a/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/load_file.txt b/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..030a27046a94d837bb9831e9bf18bf7e627040fe --- /dev/null +++ b/V9A0T4oBgHgl3EQfE_8w/content/tmp_files/load_file.txt @@ -0,0 +1,753 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf,len=752 +page_content='1 Unconventional ferroelectricity in half-filling states of antiparallel stacking of twisted WSe2 Liheng An1,2#, Zishu Zhou1,2#, Xuemeng Feng1,2#, Meizhen Huang1,2, Xiangbin Cai1,2, Yong Chen1,2, Pei Zhao3, Xi Dai1, Jingdi Zhang1, Wang Yao3, Junwei Liu1,*, Ning Wang1,2,* 1Department of Physics and Center for Quantum Materials, the Hong Kong University of Science and Technology, Hong Kong, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 2William Mong Institute of Nano Science and Technology, the Hong Kong University of Science and Technology, Hong Kong, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 3Department of Physics, University of Hong Kong, Hong Kong, China #Contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Corresponding authors (emails: phwang@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='hk (Ning Wang);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' liuj@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='hk (Junwei Liu)) Abstract: We report on emergence of an abnormal electronic polarization in twisted double bilayer WSe2 in antiparallel interface stacking geometry, where local centrosymmetry of atomic registries at the twist interface does not favor the spontaneous electronic polarizations as recently observed in the parallel interface stacking geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The unconventional ferroelectric behaviors probed by electronic transport measurement occur at half filling insulating states at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 K and gradually disappear at about 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Single band Hubbard model based on the triangular moiré lattice and the interlayer charge transfer controlled by insulating phase transition are proposed to interpret the formation of electronic polarization states near half filling in twisted WSe2 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Our work highlights the prominent role of many-body electronic interaction in fostering novel quantum states in moiré-structured systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Keywords: two-dimensional semiconductor, twist moiré, ferroelectricity, electron interaction, electronic transport 2 Stacking layered materials with small twist angles or with small lattice mismatches produce moiré superlattices that realize modulation of potentials at much larger spatial scale resulting in remarkable electronic properties [1, 2], for instance, correlated insulators [3], superconductivity states [4] and ferromagnetism [5, 6], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' More recently, a new type of ferroelectricity has been created and identified through the stacking of layered van der Waals heterostructures of hexagonal boron nitride (BN) and semiconducting transition metal dichalcogenides (TMDCs) [7-13], of which the bulk structures forbids ferroelectricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Ferroelectricity in twisted TMDCs is created by the structurally relaxed parallel stacking geometry and the moiré interface formed by the parallel stacking is also called ‘R-stacking’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Two types of locally R-stacking domains coexist in one moiré unit cell, featuring opposite spontaneous polarizations of the same magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The macroscopic ferroelectric polarization results from the dynamic bending of domain walls that microscopically favors polarization in one direction over the other [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' According to the definition in previous publications [14-16], two identical 1H-WSe2 layers stacked to each other at zero twisting angle will result in the parallel stacking interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The antiparallel stacking interface is formed by rotating one of these two identical 1H-WSe2 layers at 60o and stacking the two layers together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Opposite to parallel stacking geometry, in antiparallel stacking WSe2, the electronic polarization locked to the local atomic registries is expected to be vanishingly small when local centrosymmetric geometry is fully respected [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, to date, neither theoretical nor experimental study has reported on the emergence of ferroelectricity in antiparallel stacked configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' As we previously identified that strong electron-electron correlation effects occurred in twisted WSe2 [15, 16, 18], it is anticipated that the correlation-driven insulating states in the moiré bands of antiparallel stacking WSe2 may modulate interlayer charge transfer and even induce electronic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' It has been well acknowledged that the polarization in crystalline structures can be divided into ionic polarization and electronic state polarization [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In contrast to conventional displacive (ions or molecules) ferroelectricity, electronic polarization (the difference in polarization between two different states in the same structure) is rare in materials and normally occurs in a strongly correlated electronic system [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this study, we choose double bilayer WSe2 to construct antiparallel stacking moiré superlattices and field-effect devices, of which the moiré flat bands with \uf047 pocket holes contribute to the electrical transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Stacking two identical bilayer WSe2 with a twist angle near 0o results in the so-called antiparallel interface (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In each moiré unit cell, there are three important high-symmetry stacking sites as labeled by AB, BW/W and BSe/Se separated by boundary regions (BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' AB sites are the energetically favorable regions (near 2H registry), whereas other regions correspond to higher energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The high symmetry sites periodically modulate the electronic states in real space and therefore produce moiré bands [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Atomically thin WSe2 is mechanically exfoliated from high-quality bulk crystals of WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Field-effect devices are fabricated based on twisted double bilayer WSe2 by using the tear-and-stack method [15, 18, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' To achieve a large size of uniform moiré lattices, we select a relatively large twist angle of about 4o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Atomically thin BN layers are used to form an encapsulated device structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The electrical connection to the twisted WSe2 channel is realized by Pt electrodes (with a matched work function to WSe2) which offer a good efficiency for current injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The device performance has been effectively improved by this kind of electrode design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The measured channel resistance is about 5 k\uf057 at a modest carrier density of 3×1012 cm-2 and the field-effect carrier mobility approaches 2000 cm2V-1s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The device channel size is limited to 1×10 µm to achieve a good uniformity of the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The strong interlayer coupling between WSe2 bilayers results in the rise of the Γ valley band top (about 80 meV higher than that of the K valley) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Details of device fabrication and measurement principles are presented in Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Our density functional calculation, in the absence of many-body Coulomb interaction, confirms the dependence of spontaneous electrical polarization on stacking registry, as found in the parallel stacking interface [14], but denies its existence in the antiparallel stacking interface of twisted double bilayer WSe2 of 3 arbitrary lattice match (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Atomic-resolution electron microscopic imaging reveals several noteworthy features in our samples (Figure 2(a) and 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We first verify the twist angle and moiré periodicity by electron diffraction (Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The general moiré structural features well match the model shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, we find that the AB regions (marked by hexagons) are expandable and the location of exact AB stacking (marked by the hexagonal dots and white arrows) is switchable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' More analyses can be found in Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' These structural instability features deviate from the theoretical model, and therefore hint on the necessity to incorporate new mechanisms for insights into the origin of the moiré band modulation near half filling under different electrical fields as discussed in more detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The electronic states of the antiparallel stacking bilayer WSe2 channel in the field-effect device shown in Figure 3 are tuned by the top (VTG) and bottom (VBG) electrical gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' VTG and VBG together can linearly tune the carrier density (n = (CBGVBG + CTGVTG)/e) and displacement field (D = (CBGVBG - CTGVTG)/2ε0) of this p-type semiconductor electronic system, where C and V represent device capacitances and gate voltages, e is the elementary charge and ε0 is the vacuum permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' A negative VTG (-12 V to -14 V) is first applied in order to achieve good electrical contacts to the twisted WSe2 and injection of holes in the device channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Forward scanning of VBG (from -60 V to +60 V) is to release holes from the channel and increase D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' At VBG = +60 V, the hole density in the channel is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The typical electrical transport characteristics of the p-type antiparallel twisted WSe2 devices measured by a four-probe configuration at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 K are shown in Figure 3, in which a quick decrease of the resistance (Rxx) by decreasing VBG from +60 V indicates that the Fermi level touches the edge of the topmost moiré band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By further decreasing VBG, two metallic and two insulating states are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The metallic and insulating states are verified by measuring their Rxx at different temperatures as demonstrated in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We further verify that the two insulating states are from the correlation-induced splitting of the topmost moiré band [2, 23, 24], corresponding to the half and full filling states respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The full filling gap is resolved at a carrier density of n = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8×1012 cm-2 with two holes per unit cell to fully occupy the first moiré band [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The resistance peak emerging near n = 5~6×1012 cm-2 represents the important half filling insulating states driven by strong correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By forward-and-backward scanning VBG for several times, an obvious resistance hysteresis is repeatedly observed around half filling states (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' At half filling, when VTG is set at -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8 V, the forward scanning (-60 V to +60 V) leads to a high resistance state of about 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 k\uf057, while the backward scanning yields a low resistance state of about 10 k \uf057 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Such a resistance difference can be considered as the appearance of internal electric fields [13] from charge polarization (perpendicular to the moiré lattice plane) states in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We have carried out each measurement for five times to ensure the reproducibility of the hysteresis loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In addition, measurements of the I-V curves at the half filling states also confirm that the forward and backward scanning of gate voltages result in different channel resistances (Figure S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' To rule out the possibility that the BN dielectric layers or metal lead interfaces in the devices could potentially generate a similar hysteretic resistivity, we fabricated and measured reference devices for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We also performed different gate scanning rates during hysteretic resistivity measurements and did not observe any impurity/charge trapping effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' All these additional experiments evidence that BN or charge trapping effects do not play a role in the observed hysteretic resistivity near half filling states in our samples (See details in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The appearance of the electrical polarization characteristics in the antiparallel stacking interface structures of WSe2 is abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Different from conventional [19] and newly discovered layered van der Waals ferroelectricity [7-13, 26-29], the electrical polarization in our samples is highly relevant to correlation effects which are temperature dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' As shown in Figure 4(a), the half filling insulating states gradually disappear when temperature is higher than about 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The largest hysteresis we observed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 K also gradually disappears when temperature approaches to about 40 K (Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' All these phenomena are very different from that of the ferroelectricity observed in the parallel stacking geometry of TMDCs where the spontaneous electronic polarization survives at room temperature, not relevant to filling states at all, 4 further differentiating the origins of the ferroelectricity effects between the two twist moiré systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Here, we try to elucidate the abnormal electrical polarization characteristics based on the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The transport data are from the contribution of the Γ valley edge of WSe2 valance bands [18] and the ultra-flat moiré bands (survived up to 4o twist angles [30]) are spatially associated with the AB sites in the antiparallel stacking interface of WSe2 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' AB sites form a triangular lattice which can be considered as a mimic triangular lattice of the Hubbard model (Figure 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The topmost moiré band is separated from the rest and its charge distribution is tightly localized at the moiré potential minima of AB sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Double occupancy of AB sites is suppressed due to the strong on-site Coulomb repulsion potential U, resulting in moiré band splitting to lower (LHB) and upper (UHB) Hubbard bands [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Ideally, the carrier densities at full/half filling of the Hubbard bands are purely determined by the moiré periodicity and obtained by n0 = 4/(√3λ2), where λ is the moiré superlattice constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We first focus on the Rxx peak positions of full filling states as a reference to test the functionality of the two gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By setting VTG and scanning VBG (Figure 4(b)), we confirm that the injected density n (hole carriers in p-type WSe2) at full filling (the gap position) as characterized by Rxx peaks are all correctly tuned by the two gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is reflected by the change of the full filling peak positions as outlined by the inclined straight dashed line in Figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By increasing VTG, the full filling peak positions shift to right side, meaning that along this dashed line, the carrier density is unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Because both full and half filling densities are fixed by the moiré lattice geometry [32], the Rxx peaks at half filling should also follow a straight line parallel to the full filling peak line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, the peak positions at half filling obviously shift to left side (towards to higher hole concentration regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This suggests that when increasing the D field, more holes are needed to be injected into the system in order to completely fill the LHB as displayed more clearly in Figure 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In addition, we noticed that the Rxx at half filling decreases by increasing D, which could be interpreted as that the Hubbard gap U becomes narrow accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' These interesting experimental data suggest following two possible physical pictures: (1) D enhances new electronic states involved in the half filling states in the moiré system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' or (2) increasing D could narrow the Hubbard band gap U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We now provide a theoretical picture to intuitively interpret the correlation-effect-induced electronic polarization based on the interlayer charge transfer controlled by electric fields and insulating phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For a large D field applied to the double bilayer twisted device, it is possible that the entire low-energy moiré bands become layer polarized, as reported in the unconventional electronic polarization in bilayer graphene heterostructures [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Similarly, electrons occupying a moiré band at low energy could locate on a specific WSe2 layer in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' It is also reasonable to assume that LHB/UHB locates at the interface between the two inner WSe2 layers (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In our device design, the electrodes directly connect to the bottom WSe2 layer and holes are firstly filled preferably in the bottom layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We then propose that the observed D-dependent shift of the half filling peak is due to the involvement of new electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Figure 5(b) schematically shows the additional band (B) coexisted with the Hubbard bands which contributes to the new electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This band can be considered as the contribution from the bottom WSe2 layer directly connected to the electrodes for hole injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The inner two layers which host LHB/UHB states are actually not directly connected by the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' At a fixed D field, the backward scanning of VBG (starting from +60 V) means filling holes into the electron-occupied LHB first (Figure 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In our p-type WSe2 moiré system, the charge carriers which contribute to the transport measurement are holes in the valance band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Before adding holes into the valance band, all states are occupied by electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We use solid color (black and yellow) to illustrate the electron occupied states in the LHB and UHB in Figure 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By turning the Fermi level, holes are firstly injected into the LHB and then into the UHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Because of the involvement of the B-band, the apparent Hubbard gap position may shift to left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' During backward VBG scanning, hole carriers transfer from the B-band to LHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This interlayer charge transfer is driven by the potential associated with the gates and D field [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The bottom WSe2 layer offers the B-band which is p-type under the same gating condition (larger than the 5 threshold gate voltage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Figure 5(b) shows the band alignment and the carrier transfer direction between the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The hole filling process allows current to flow easily from B-band (p-type) to the Hubbard bands (p-type) across the spatial layer interface (similar to a p-p isotype junction [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Near half filling, however, the states at the bottom edge of LHB are electron-like (n-type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Holes transfer across a p-n anisotype junction (the forward biasing of a p-n junction) is less resistive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, filling LHB/UHB driven by backward VBG scanning is less resistive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For the forward VBG scanning, however, holes are released from LHB/UHB to the B-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Near half filling, the p-n anisotype characteristic at the interface severely restricts holes reversely flowing through the anisotype junction (analogous to the reverse biasing of a p-n junction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The consequence is that more holes are retained in the moiré layer when the forward scanning is near half filling, while the holes in the B-layer are always released to the electrodes normally during VBG scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, the charge polarization between the moiré layer and the B-layer is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The extra holes in the moiré layer countervail only a part of D field strength (Figure S11(b)), resulting in a higher Rxx compared to that in the opposite direction of backward scanning VBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is because Rxx is inversely proportional to the strength of the D field as revealed experimentally in Figure 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This mechanism supports the transport data of the Rxx hysteresis in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We now conclude that the two electrical polarization states are the result of more versus less occupation of the Hubbard bands driven by the interlayer carrier transfer modulated by the insulating phase transition during D field scanning near half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This abnormal electronic polarization effect has been repeatedly observed in the devices with twist angles ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2o (see more data in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Electronic ferroelectricity caused by electron correlation has been proposed and studied previously based on Hubbard models in different bulk material systems [20, 21], in which parameters of Coulomb repulsion potential U and hopping energy t are critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In our moiré system, the expandable AB sites driven by correlation effects or D fields [26] could modulate U and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is because the local AB sites (near 2H registry) in our samples is expandable (also deviating from an ideal local 2H registry) involving the atomic rearrangement of the boundaries to surrounding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, the expanded local 2H registry together with the boundaries surrounding these regions do not hold the centrosymmetric geometry according to our electron microscopy results (see more details in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The expansion of AB sites (increasing the energy favorable area) driven by the D field could provide the driving force for decreasing U, since increasing AB site area can effectively lower U to host two electrons with opposite spins in one AB site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' At present, it is still not clear whether the deviation of the half filling Rxx peak under different D fields is solely due to the interplay between the Hubbard bands and B-band or partially due to the D field modulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=', D-field-induced electronic state instability) of the electronic states in the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' To this end, further experimental and theoretical investigations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Our discovery of the abnormal ferroelectricity behaviors suggests a new platform for further exploring the flat band properties tunable by an electric field, particularly the correlation-induced insulating states at the half filling states adjacent to the superconductivity states in this two-dimensional moiré system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Acknowledgements We are grateful to the technical support from Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Yuan Cai from the Materials Characterization and Preparation Facility at the Hong Kong University of Science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Funding This work was supported by grants from the National Key R&D Program of China (2020YFA 0309600), the Hong Kong Research Grants Council (Project Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' AoE/P-701/20, C6025-19G, 16305919 ECS26302118, 16303720, 16305019, 16306220 and N_HKUST626/18), National Natural Science Foundation of China (NSFC20SC07) and the William Mong Institute of Nano Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 6 Author contributions N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' initiated and supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' designed the research, device structure, measurement and data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=', Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' fabricated device and collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' contributed to data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' contributed to calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' contributed to sample preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Conflict of interest The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Data availability The original data are available from corresponding authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Details of device fabrication, measurement principles and proposed mechanisms are presented in Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 35 Lourtioz JM, Abstreiter G, Meyerson B, Group IV Heterostructures, Physics and Devices (Si, Ge, C, Sn) 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 61: Elsevier Science & Technology, Oxford, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 8 Figure 1 (a) Moiré superlattice structure constructed by antiparallel stacking of twisted double bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The circles denote the high symmetry stacking sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The hexagons (dashed lines) indicate the near 2H registry regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' BR indicates the boundary regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) The atomic configurations of the high symmetry stacking sites of AB, BW/W and BSe/Se and their side views of these sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Blue and red arrows denote the atom positions shifts from top and bottom layers of WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Near2Hregistry (a) (q) Atom position regions 40 shift by distortion BR AB: AB BR BR Top-view AB(2H) BW/W BSe/Se A AB B A B Side-view9 Figure 2 (a) Transmission electron microscopy (TEM) image of the antiparallel twisted double bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) Electron diffraction pattern taken from the area in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The twist angle is determined to be about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (c) An enlarge image of the region marked by the white dashed rectangle in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The positions of exact 2H stacking are marked by yellow hexagons (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The white thick arrows indicate the directions of the 2H registry shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The scale bar is 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) (b) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='80 (100)A (100]B (000) BR BR BR AB (c) 2Hregistry10 Figure 3 The structure of the field-effect device built based on antiparallel stacking of twisted double bilayer WSe2 and the typical resistance hysteresis measured in the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The device is double gated by the top and bottom gates in order to tune the carrier concentration and the displacement field (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By fixing the top gate (VTG) at -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8 V, scanning the bottom gate (VBG) results in the change of the filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Forward-backward scans (indicated by the dashed/solid lines) of VBG show a clear resistance hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Top gate Twist WSe2 BN Charge VTG transfer VBG Bottom SiO2+BN layer Bottom gate Half filling Full filling Metallic Metallic11 Figure 4 (a) Phase diagram plotted based on transport measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The half filling states gradually disappear when temperature is higher than 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) Displacement field effects (by setting VTG at different voltages and scanning VBG) on the full filling and half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The full and half filling states behave differently by changing the displacement fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The black/red thick dashed lines indicate the ideal full/half filling peak positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The black thin dashed line indicate the peak positions of half filling states measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (c) Temperature dependence of the resistance hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The hysteresis loops gradually disappear at about 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Half filling Full filling12 Figure 5 (a) A mimic triangular lattice of the Hubbard model for the AB sites and the variation of U and t due to AB site expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) Carrier transfer between the B-band and LHB/UHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Forward scanning of VBG results in the releasing of holes and the backward scanning of VBG results in the filling of holes in the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge transferring between the B-band and LHB/UHB shows different behaviors near the gap region of the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (c) Displacement field effects on the full filling and half filling gap positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The full filling states are independent of the D field as indicated by the thin solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, the half filling is D-dependent which deviates from the ideal positions as indicated by the right side thin solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) U Decrease AB AB AB ti Decrease (b) BackwardVBG Electron like (n) (filling holes) UHB U LHB Full filling 1 e B-band Forward VBG Charge (releasing holes) transfer (c) Vtc = -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8V to -14V 1/2 filling 50 0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8°± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='1° 40 Increase D (k2) 30 R Full filling 20 10 0 10 8 6 4 n (1012/cm²)13 Supplementary Information Unconventional ferroelectricity in half filling states of antiparallel stacking of twisted WSe2 Liheng An1,2#, Zishu Zhou1,2#, Xuemeng Feng1,2#, Meizhen Huang1,2, Xiangbin Cai1,2, Yong Chen1,2, Pei Zhao3, Xi Dai1, Jingdi Zhang1, Wang Yao3, Junwei Liu1, *, Ning Wang1,2,* (# These authors contribute equally to this work) Corresponding authors (emails: phwang@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='hk (Ning Wang);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' liuj@ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='hk (Junwei Liu)) 1Department of Physics and Center for Quantum Materials, The Hong Kong University of Science and Technology, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 2 William Mong Institute of Nano Science and Technology, The Hong Kong University of Science and Technology, Hong Kong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 3Department of Physics, University of Hong Kong, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Device fabrication Atomically thin WSe2 is mechanically exfoliated from bulk crystals (from 2D Semiconductors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The bilayer WSe2 is identified by optical microscopy and photoluminescence techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The bottom hexagonal boron nitride (hBN) is around 25 nm thick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The bottom hBN is pre-patterned and selectively etched down to the SiO2 by plasma treatment in CHF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Then, Cr/Pt 10 nm/20 nm is deposited to form the bottom electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We fabricated the twisted double bilayer WSe2 devices by using the tear and stack method as previous work introduced [S1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Then polypropylene carbonate is used to tear one part of the bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The bottom flake is rotated by a small angle and the top flake is used to pick up the bottom flake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The whole stacked structure is placed on the pre-patterned bottom electrodes to form the contacts to the twist structure (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Finally, another flake of hBN (30 nm–50 nm) is transferred onto the top surface of twist WSe2, and a thin layer of Cr/Au (10/70 nm) is deposited on the top surface of hBN to form the top gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Pt layers directly deposited onto the top surface of the bottom hBN generally result in gas trapped around the bottom electrodes on the sample interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The trapped gas, after performing a thermal annealing, accumulates at the electrode edges, leading to detach of the metal and the sample interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The large real space gaps between the metal and the sample is the main reason for generating a huge contact resistance and thus degrade the device quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Here, by deep etching of the bottom hBN and depositing Pt electrodes, we improved the evenness of the bottom Pt electrodes, hBN bottom surfaces and the current injection efficiency from the metal electrodes into the atomically thin WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For electron microscopy investigation, the twisted structure was dropped onto a holey carbon grid, cleaned by different chemical solutions, and then dried in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) Device structure of antiparallel stacking of bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) Twist WSe2 structure placed on the metal electrodes before covering the hBN and metal gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (c) Cross-sectional view of the twist WSe2 device design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Improvement of device performance To ensure the quality of the devices, two layers of hBN are used to fabricate the fully encapsulated structure (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The misalignment of the top and bottom hBN is carefully considered to avoid the formation of any small angle twisted hBN/WSe2 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In fact, the TG-10 5 10μm TiIAu BN WSe2 WSe2 Pt Pt Pt Pt BN SiO2 n-Si15 transport characteristics measured in the devices are dominated by the moiré interface properties especially the strong half filling (υ=1, one hole per moiré unit cell) insulating states [S4], thus the hBN interface effects are ignorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We use the bottom-electrode design to make the contacts to the channel of the bottom twist layer (See FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We use the top and bottom gates to control the out-of-plane DC electric field applied to the moiré channel and change the carrier concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The device performance has been effectively improved by this kind of device design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The measured channel resistance is about 5 k\uf057 at a modest carrier density of 3×1012 cm-2 and the field-effect carrier mobility approaches 2000 cm-2V-1s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The device channel size is limited to 1×10 µm to achieve a good uniformity of the twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Such small-sized samples integrated with bottom and top metal electric gates therefore limit the detection of the ferroelectric property at cryogenic temperatures by conventional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Our transport data are from the Γ valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The strong interlayer coupling between WSe2 bilayers results in the rise of the Γ valley band top (about 80meV higher than that of the K valley) [S3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Computation results Based on the same theoretical model presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' [S8], we examined the difference vacuum levels vac on the two sides of the 2L+2L WSe2 with lattice-matched antiparallel stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We find that vac is 0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S2), indicating the electrical polarization is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Notably, the electrical polarization at the interface of 2L+2L WSe2 has invisible dependence on each stacking registry ���t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this calculation, interaction effects have not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The difference in the vacuum levels on the two sides of the 2L+2L WSe2, ������ indicates the stacking registry of double bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We have carried out measurements in different samples with antiparallel 2L+2L stacking geometry (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S2) at twist angles ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' These samples did not show hysteretic resistivity at room temperature since the half filling states did not form at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' While at cryogenic temperatures, half filling sates emerge in these samples accompanied with the hysteretic resistivity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Theoretically, this ideal antiparallel 2L+2L stacking geometry with a local 2H registry does not have out-of-plane polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, in principle, the gate voltage should not cause expansion of the local 2H registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This also implies that “ideal” half filling states should be displacement field independent, and there should not be electrical polarization/ferroelectricity effect in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, interaction effects may induce symmetry breaking and generate electronic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Atomic structure of 2L+2L twisted WSe2 We identified by high-resolution electron microscopy that the local AB sites (near 2H registry) in our samples is non-uniformly expandable (also deviating from an ideal local 2H registry) involving the atomic rearrangement of the boundaries to surrounding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, the expanded local 2H registry together with the boundaries surrounding these regions do not hold the centrosymmetric geometry according to our microscopy results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' On the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='68- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='66- E, E2 AE =E-E, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='64 vac 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='62- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='56- 0 102030 50 z (A)16 other hand, it is reasonable to assume that symmetry breaking (or atom position deviation) effects could generate charge transfer between layers and induce electronic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' As demonstrated by electron microscopy, the expandable 2H sites already break the local 2H centrosymmetric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) Schematic illustration of the expandable and switchable 2H registry induced by atomic position shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) An enlarge TEM image showing that the exact 2H registry positions as marked by yellow hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The white thick arrows indicate the expansion or shift of the 2H registry is non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The scale bars are 3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By analyzing the atom positions in transmission electron microscopy (TEM) images, we identify the details of the 2H registry sites (the AB regions marked by hexagons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' An ideal 2H registry site is displayed as one “hexagonal-flower-like” pattern in the atomic image with strong and sharp contrast (centered at a small yellow hexagon), in which the strong bright dots indicate highly out-of-plane aligned atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The larger the area covered by the sharp hexagonal-flowers, the more expended area occupied by the 2H registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Obviously, many 2H sites show two or three strong and sharp contrast of the hexagonal-flowers emerging together due to the expansion of the 2H registry area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S3(a) illustrates the mechanism of the expandable or switchable 2H registry which is useful for understanding the modulation of the Hubbard band gap variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The expansion of AB sites (increasing the energy favorable area) driven by D field provides the driving force for decreasing U, since increasing AB site area can (a) Atomic shifting BR (b) BW/W!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' BR BR AB 2Hegistry17 effectively lower U to host two electrons with opposite spins in one AB site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this scenario, shrinking the sizes of AB sites should cost extra energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Transport measurement and carrier density estimation For transport measurement, an AC excitation is set within the range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2 mV to 5 mV at a frequency of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='579 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The current signal is probed by a lock-in amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The voltage signals are detected through a low noise preamplifier SR550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The carrier density (if there is no internal polarization field in the sample) can be estimated by ��� = (������������������ + ������������������)/���, and the perpendicular electric field is expressed by ��� = (������������������ − ������������������)/2���0 = ������ − ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (CTG/CBG, top/bottom gate capacitance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' VTG/VBG, top/bottom gate voltages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The top gate electric field is applied through the top few-layer hBN (defined by ������ = ������������������/2���0), and the back gate electric field is applied through the 300nm-SiO2 layer on the Si substrate (defined by ������ = ������������������/2���0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The dual-gate structure allows us to tune the charge density and the out-of-plane electric displacement field [S5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Different from the multilayer graphene sensing scheme for detecting the interfacial ferroelectricity [S5, S6], the out-of-plane electrical polarization and/or potential difference between opposite polarization of domains in twisted TMDCs can be translated into the gate-controlled doping effect which is directly reflected by the electron transport hysteresis of the twisted TMDCs field-effect transistor devices [S7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In our work, we measure electrical resistance at different carrier concentration under different DC gate scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The out-of-plane electronic polarization effects in our devices is presented by resistance hysteresis loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For pristine few layer WSe2 samples, no resistance hysteresis loop has been observed during forward-backward gate scanning (see FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The moiré periodicity can be described by λ = ���/( 2sin��� 2 ), where a is the lattice constant of WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The full filling carrier density is related to the moiré wavelength with ���0 = 2/ 3 2 λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Then, we can further calibrate the twisted angle based on the estimation of the carrier densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Interface and charge impurity effects We have carried out different experiments to rule out the possibility that hBN interfaces or charge trapping effects could potentially generate a similar hysteretic resistivity as what we reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge trapping effects are normally resulted from interface impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The interfaces in our devices are formed between hBN and WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We provide here more experimental data and analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The hBN moiré interface effects: The hBN and WSe2 layers have no specific twist angles in our devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, there is no comparable moiré lattice formed at the hBN/WSe2 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In fact, the lattice parameters of WSe2 (a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='3297 nm) and hBN (a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2502 nm) are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' There is no comparable moiré superlattice formed in our devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' hBN/WSe2/hBN interfaces: We fabricated hBN-sandwiched WSe2 field-effect devices using the same interface structure and measured the gate dependent transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' There is no hysteretic resistivity characteristics in these kind of samples during scanning the gate voltage (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, we confirmed that hBN/WSe2 interfaces and the metal lead interfaces in our devices do not generate any hysteretic resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Transport property of the reference hBN-sandwiched p-type WSe2 field-effect device built based on the same interface structure used for twisted WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' There is no hysteretic resistivity characteristic as measured by scanning the gate voltage at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge trapping effects can normally be distinguished by changing the gate scanning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This method has been widely used in identifying charge trapping effects in graphene devices [S9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We performed different gate scanning rates in our twist WSe2 devices and did not observe any obvious difference as shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is a strong evidence that charge trapping effects did not play a role in the observed hysteretic resistivity at half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge trapping can normally exist at room temperature such as in graphene [S9], in particular near the Dirac point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this case, the charge trapping behavior becomes obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Near the resistance peak of a graphene device, charge traps cause shifts of the peak position (left-or-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The amplitude of the peak has almost no change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is because of the charge nature of the impurity at the device interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, in our device, the main feature of the hysteresis and resistance peaks is the amplitude change (high-or-low), a very different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='20 Backward Forward 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='15 (sw) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='10- b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='00 11 10 6- 8 7 6 5 4 Vg(V)19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Transport measurements at different gate scanning rates in a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2o twist WSe2 device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The half filling peaks and the hysteretic resistivity data do not show any obvious difference at different scanning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The hysteretic effect in our sample behaves differently compared to the normal charge trapping effects since it decays quickly when sample temperature is increased to about 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, the charge trapping effects caused by impurities normally can still exist at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In fact, charge trap effects usually occur globally in a device (not just at certain carrier concentration) in poor conducting materials, while our devices show high field-effect mobility with a high current injection efficiency between the electrodes and the twist bilayer WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The reproducibility of our device performance suggests that the hysteretic effects in our samples are not due to charge trapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' More transport results Although the target of the twist angles is set to 4o, the fabricated devices often have a deviation from this angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The twist angles of the antiparallel stacking moiré super-lattices we obtained range from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Here are more experimental data we obtained from different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S6(a), we show the half filling peaks measured at different gate voltage from a device with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='9o twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S6(b) shows the corresponding resistance hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S7 shows the transport data taken from a device with a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2o twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We find that for the samples with relatively large twist angles, their half filling peaks look sharp and the resistance hysteresis characteristics are more localized around the half filling density position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' To calculate the displacement field D, we need to first estimate the top and bottom capacitances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' By changing VBG and VTG, we determine the D filed by: One of the results is shown in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) The half filling peaks measured at different displacement fields from a device with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='9o twist angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) The resistance hysteresis near the half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Transport data taken from a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2o twist angle device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) Temperature effects on the half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) The resistance hysteresis near the half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' iiiifilling factor (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8 (k2) 20 Half filling Backward 160 80 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='. Forward 90 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Half flling 20 (kΩ) 60 ← 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='0 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='0 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='0 60 30 0 30 n (x1012/cm2) VBG (V)21 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Demonstration of both the strength of D field and the carrier density n dependent Rxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' I-V characteristics at the half filling states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) Transport measurements of I-V characteristics performed at the half filling states under the fixed top and back gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b-c) I-V measurements of forward/backward scanning at ferroelectric state under different gate voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (d) Different resistances observed from forward/backward scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We performed more transport measurements of I-V characteristics in the half filling states at fixed top gate (-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='5 V to -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='2 V) and back gate (-35 V to -41 V) voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We observed that for Vds larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='01 mV, the I-V shows linear relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We also carried out I-V n D n S22 measurements on forward/backward scanning at the half filling ferroelectric state and observed different resistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this study, we are not able to demonstrate a complete map of the resistant hysteresis loops as a function of D field because the top gate voltage has a very limited variation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The main reason is that our devices are built with top and bottom gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, because there are totally four layers WSe2, and the metal electrodes are designed to contact with the bottom layer of WSe2, we need to apply a relative large top gate voltage in order to achieve a good ohmic contact to the semiconductor twist WSe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The ohmic contact behavior can only be kept in a narrow range of the top gate voltage (-10 V to -15 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Top gate voltages larger than -15 V may cause breakdown of the dielectric layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' A top gate voltage smaller than -10 V resulting in poor electrical contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, the scan including both top and bottom gate is mainly limited by the top gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Scanning top gate in a large range will change the electrical contact characteristics and the experimental data are not reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Although we cannot provide a complete map of the resistant hysteresis loops as a function of D field, we add experimental data to partially show the resistant hysteresis changes at limited range of the D field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S10 shows the changes of resistance around half filling as a function of D field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Experimental data showing the changes of resistance around half filling as a function of D field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Interlayer charge transfer mechanism FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S11(a) schematically shows the two bands associated with the bottom WSe2 layer (directly connected to the electrodes) and the moiré super-lattice layer without considering interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Both bands are from the \uf047 valley of WSe2 with p-type characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' They form an isotype junction [S12, 13] for charge transfer between the two layers during forward/backward scanning of VBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Due to the strong interaction occurring in the moiré super-lattice layer, LHB and UHB formed (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S11(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For backward scanning of VBG (starting from +60 V), holes start to transfer from B-band to LHB first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this case, both bands are p-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge transfer across the p-p isotype junction is less resistive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Noticed that when holes occupy more than half of the states in LHB, the bottom band edge of LHB is electron-like (n-type semiconductor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Such a band alignment between LHB and B-band generates a n-p junction (anisotype junction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Holes transfer from B-band (p-type) to the n-type moiré layer is less resistive (similar to the forward biasing of a n-p junction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, the opposite direction of holes transferring across the n-p n = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='8*1012/cm2 Backward --- Forward V=1 200 (kΩ) 150 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='43 D (V/nm)23 junction is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge transfer between B-band to UHB is less resistive since their junction is always isotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (a) Two bands associated with the bottom WSe2 layer and the moiré super-lattice without interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' (b) Strong interaction induces LHB and UHB of the Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Charge transfers between the Hubbard bands and B-band near the half filling states is modulated by the formation of the Hubbard band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' For forward scanning of VBG, holes transfer back to the B-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' In this case, charge transfer across all isotype junctions is less resistive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, near half filling states, the charge transfer is severely restricted (analogous to the reverse biasing of a n-p junction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The consequence is that there are more holes retained in the moiré layer (in the same time the holes in the B-layer are released to the electrodes normally) when a forward scanning approaches the half filling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Therefore, the charge polarization between the moiré layer and the B-layer is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The extra holes in the moiré layer countervail only a part of the strength of the D field (FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' S11(b)), resulting in a higher Rxx compared to that in the opposite direction of backward scanning VBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This is because Rxx is inversely proportional to the D field as revealed experimentally in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' 5(c) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' We noticed that there is also resistance hysteresis in the metallic phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' First, bilayer WSe2 is semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The so-called insulating states at half filling are generated by the correlation effects in the twist moiré system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' The nearby metallic states are also called strongly correlated metallic phases, such as the superconductor states in twisted WSe2 bilayer we discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Second, since the electron polarization states are relevant to the correlated states, it is reasonable to believe that the resistive hysteresis occurring at states away from the half filling states is also related to correlation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' However, the resistive hysteresis occurring at the state far away from the half filling states is indeed complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' According to our model, the alignment of the Hubbard band and the additional B-band in the “metallic states” is isotype junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' If there is a slight barrier between the moiré interface layers and the B-band layer which might be due to the formation of the moiré bands, the electron polarization could happen, resulting in the resistance hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' This situation is similar to that of the anisotype junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' No interaction Stronginteraction Hole (p-type) Isotype junction Half filling Anisotype P-type junction (n-p) P-type P-type Isotype junction Hole (p-type) Moire Bottom laver Wse2 laver Charge polarization from extra holes in the moire layer Isotvpe junction +24 References [S1] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Cai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Liheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Feng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Cai, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Duan, Nature 557, 696 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' [S5] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Zheng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' Bi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content=' de la Barrera, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} +page_content='-H.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9A0T4oBgHgl3EQfE_8w/content/2301.02025v1.pdf'} diff --git a/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/2301.11858v1.pdf.txt b/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/2301.11858v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..875148ba9cb241fb9a4469606da1e4e0c8a92cbd --- /dev/null +++ b/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/2301.11858v1.pdf.txt @@ -0,0 +1,918 @@ +MNRAS 000, 000–000 (0000) +Preprint Monday 30th January, 2023 +Compiled using MNRAS LATEX style file v3.0 +3D MHD models of the centrifugal magnetosphere from a massive +star with an oblique dipole field +Asif ud-Doula1⋆, Stanley P. Owocki2, Christopher Russell2, Marc Gagn´e3 and +Simon Daley-Yates4 +1 Penn State Scranton, Dunmore, PA 18512, USA. +2 Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA +3Department of Earth and Space Sciences, West Chester University, PA, USA +4 School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, Fife, Scotland KY16 YSS, UK +01:30Monday 30th January, 2023 +ABSTRACT +We present results from new self-consistent 3D MHD simulations of the magnetospheres from +massive stars with a dipole magnetic axis that has a non-zero obliquity angle (β) to the star’s +rotation axis. As an initial direct application, we compare the global structure of co-rotating +disks for nearly aligned (β = 5o) versus half-oblique (β = 45o) models, both with moderately +rapid rotation (∼ 0.5 critical). We find that accumulation surfaces broadly resemble the forms +predicted by the analytic Rigidly Rotating Magnetosphere (RRM) model, but the mass buildup +to near the critical level for centrifugal breakout against magnetic confinement distorts the +field from the imposed initial dipole. This leads to an associated warping of the accumulation +surface toward the rotational equator, with the highest density concentrated in wings centered +on the intersection between the magnetic and rotational equators. These MHD models can be +used to synthesize rotational modulation of photometric absorption and Hα emission for a +direct comparison with observations. +Key words: (magnetohydrodynamics) MHD — Stars: winds, outflows — Stars: magnetic +fields — +1 +INTRODUCTION +Hot luminous, massive stars of spectral type O and B have dense, +high-speed, radiatively driven stellar winds (Castor et al. 1975). In +the subset (∼10%; Grunhut et al. (2017); Sikora et al. (2019)) of +massive stars with strong (> 100 G; Shultz et al. (2019)), glob- +ally ordered (often significantly dipolar; Kochukhov et al. (2019)) +magnetic fields, the trapping of this wind outflow by closed mag- +netic loops leads to the formation of a circumstellar magnetosphere +(Petit et al. 2013). Because of the large angular momentum loss +associated with their relatively strong, magnetised wind (ud-Doula +et al. 2009), magnetic O-type stars are typically slow rotators, with +trapped wind material falling back on a dynamical timescale, giv- +ing what’s known as a “dynamical magnetosphere” (DM). +However, in the case of magnetic B-type stars, such angular +momentum loss is greatly reduced due to their relatively weak stel- +lar winds, implying longer spin-down times. Thus, not surprisingly, +a significant fraction of B-type stars still retain a moderately rapid +rotation. For such cases, the associated Keplerian co-rotation radius +RK lies within the Alfv´en radius RA that characterises the maxi- +mum extent of the magnetosphere. The rotational support within +⋆ Email: asif@psu.edu +the magnetosphere leads to formation of a “centrifugal magneto- +sphere” (CM), wherein the much longer confinement time allows +material to build up to much higher density than in DM’s. +For the special case of a dipole field aligned with the rotation +axis, ud-Doula et al. (2008) carried out 2D MHD simulations of the +resulting axisymmetric CM with material concentrated along the +common rotational and magnetic equator. The central aim of the +current paper is to present results from new 3D MHD simulations +for the sample case of an oblique dipole field that has a tilt angle +β = 45o, and characterize its more complex, inherently 3D CM. +For models without rotation, initial 2D MHD simulations of +such wind-fed magnetospheres by ud-Doula & Owocki (2002) +showed that, for a star with radius R∗ and dipole field of surface +strength Beq at the magnetic equator, the competition of the field +with a stellar wind of mass loss rate +˙M and terminal speed V∞ +can be characterized in terms of a dimensionless, wind-magnetic- +confinement parameter η∗ ≡ B2 +eqR2 +∗/ ˙MV∞, with the Alfv´en ra- +dius then scaling as RA ≈ R∗η1/4 +∗ +. The follow-on study of aligned +rotation by ud-Doula et al. (2008) parameterized its effects by the +dimensionless ratio W ≡ Vrot/Vorb between the equatorial ro- +tation speed and the orbital speed near the equatorial surface. In +the inner region where the field maintains rigid-body rotation, this +© 0000 The Authors +arXiv:2301.11858v1 [astro-ph.SR] 27 Jan 2023 + +2 +A. ud-Doula et al. +gives a Kepler co-rotation radius RK = R∗W −2/3, at which grav- +itational and centrifugal forces are in balance. +The observational compilation by Petit et al. (2013, see in par- +ticular their figure 3) shows that many B-stars have RA > RK, with +confinement parameters ranging even to η∗ > 106. For example, +the prototypical CM star σ Ori E, has η∗ ≈ 106 and W ≈ 0.34, +implying RA/R∗ ≈ 31 ≫ RK/R∗ ≈ 2.1, and thus an extensive +CM. +The field stiffness and associated high Alfv´en speed of a star +with such large η∗ imply very small Courant time step in direct +MHD simulations, and so far this has limited MHD models to +η∗ ≲ 103. Alternatively, by considering the limit of an arbitrar- +ily strong field (effectively with RA → ∞), semi-analytic anal- +yses based on an idealization of purely rigid fields have led to a +rigidly rotating magnetosphere (RRM) formalism (Townsend et al. +2005), which derives how CM material accumulates on surfaces +set by minima of a combined centrifugal and gravitational poten- +tial, under the assumed condition of rigid-body rotation. Such RRM +models have shown great potential for explaining key observational +signatures of CM stars, e.g. rotational modulation of Balmer line +emission (Townsend et al. 2005; Oksala et al. 2012). +However, central weakness of this RRM approach is that it +provides no description for how the stellar-wind-fed accumulation +surfaces of the CM are ultimately emptied. Recent theoretical anal- +yses (Owocki et al. 2020, 2022), developed to explain empirical +scaling for H-alpha (Shultz et al. 2020) and radio emission (Leto +et al. 2021; Shultz et al. 2022) from CM stars, provide strong ev- +idence that this emptying occurs through frequent, low-level, cen- +trifugal breakout (CBO) events. These are triggered when the accu- +mulated mass exceeds a critical level for which the centrifugal force +overwhelms the confinement of the magnetic field tension force. +Applying the estimated CBO critical density distribution within the +RRM formalism for accumulation surfaces, Berry et al. (2022) re- +cently modeled the photometric light variation associated with ab- +sorption and scattering emission from magnetic clouds around the +prototypical CM star σ Ori E. +The CBO-limited density distribution derived by Owocki et al. +(2020) was actually based on analysis for the simplified special +case of an aligned dipole, calibrated against the 2D MHD simu- +lations by ud-Doula et al. (2008). Specifically, assuming W = 1/4 +or 1/2 and the strongest allowed confinement η∗ = 1000, the MHD +models show that, in the CM region extending above RK, the crit- +ical surface density accumulated along the common magnetic and +rotational equators declines in radius as r−6, consistent with the an- +alytic CBO analysis that this should follow the decline in magnetic +tension ∼ B2. +But for the many CM’s with a nonzero tilt angle β between +the field and rotation axes, it is not clear how the density on the ac- +cumulation surface should vary in azimuth away from the direction +set by intersection of the rotational and magnetic equators. +The paper here presents new 3D MHD simulations of the CM +formed by a tilted dipole with axis that makes an angle β = 45o +with the rotation axis. The standard parameterization of moderately +rapid rotation (W = 1/2) and strong confinement (η∗ = 103) pro- +vides, for the first time, a 3D MHD, oblique-dipole model with +an extended CM region1, ranging here from RK ≈ 1.6R∗ to +RA ≈ 5.6R∗. A particular emphasis is to characterize the resulting +1 By comparison, the recent 3D simulations by (Subramanian et al. 2022) +are limited to modest η∗ = 50 with RA ≈ 2.7R∗ ≳ RK and have too +small CM regions to enable direct comparison with RRM models. +3D, dynamical distribution of density, and compare that with ex- +pected accumulation surfaces from the semi-analytic RRM model. +A specific goal is to test and calibrate the RRM density parameteri- +zation used by Berry et al. (2022), which was inspired by the CBO +analysis and preliminary versions of the 3D MHD simulations pre- +sented here. +To lay the basis for results presented in Section 3, Section 2 +first reviews the general numerical MHD approach, numerical grid, +boundary conditions and parameter domain. Section 4 concludes +with a summary and outline for future work. +2 +NUMERICAL SETUP AND INITIAL CONDITIONS +Most of our previous numerical models were performed using +ZEUS-3D or ZEUS-MP codes, but here we use publicly available, +massively parallel MHD code PLUTO (version 4.4) (Mignone et al. +2007), because of its highly versatile, modular structure that is well +suited for modern Linux clusters. +The winds of massive stars are highly ionized and the compe- +tition between photoionization heating and radiative cooling keeps +the wind close to the stellar effective temperature (Pauldrach 1987; +Drew 1989). As such, we can approximate the wind to be isother- +mal and model it with standard magnetohydrodynamics (MHD) +equations in cgs units: +∂ρ +∂t + ∇ · (ρv) = 0 +(1) +∂v +∂t + (v · ∇) v + +1 +4πρB × (∇ × B) + 1 +ρ∇p = g + glines + Fco +(2) +∂B +∂t + ∇ × (B × v) = 0. +(3) +Here ρ, v, B, p, g and glines are, the density, velocity, magnetic +field, pressure, and accelerations due to respectively gravity and +line-scattering of radiation. The comoving frame acceleration Fco +is the sum of both the centrifugal and Coriolis forces, given respec- +tively by +Fcentrifugal = − [Ωfr × (Ωfr × R)] +(4) +and +Fcoriolis = −2 (Ωfr × v) , +(5) +where Ωfr is the angular frequency of the rotating frame with r the +radial distance vector. +As the wind is assumed to be isothermal at the stellar surface +temperature T, we close equations (1 - 3) using the ideal gas equa- +tion of state, +p = ρkBT +µ += ρc2 +iso, +(6) +where kB is the Boltzmann constant, µ = 0.6mp is the molecular +weight, and the last equality casts this in terms of the isothermal +sound speed ciso. +2.1 +Radial line-driving of wind outflow +The radial outflow described in the previous section arises from +the strong radial driving of the line-force, glines. As in ud-Doula & +Owocki (2002), we model this here in terms of the standard Castor, +MNRAS 000, 000–000 (0000) + +Oblique Rotators +3 +Abbott & Klein (1975, hereafter CAK) formalism, corrected for the +finite cone angle of the star, using a spherical expansion approxi- +mation for the local flow gradients (Pauldrach, Puls, Hummer & +Kudritzki 1985; Friend & Abbott 1986) and ignoring non-radial +line-force components that can arise in a non-spherical outflow. +Although such non-radial terms are typically only a few percent +of the radial force (Owocki & ud-Doula 2004), in non-magnetic +models of rotating winds, they act without much competition in the +lateral force balance, and so can have surprisingly strong effects +on the wind channeling and rotation (Owocki, Cranmer & Gayley +1996; Gayley & Owocki 2000). But in magnetic models with an +already strong component of non-radial force, such terms are not +very significant, and since their full inclusion substantially compli- +cates both the numerical computation and the analysis of simula- +tion results, we have elected to defer further consideration of such +non-radial line-force terms to future studies. +By limiting our study to moderately fast rotation, half or less +of the critical rate, we are also able to neglect the effects of stellar +oblateness and gravity darkening. +2.2 +Simulation +For our numerical scheme, we choose a method that is fully unsplit +and 2nd order accurate in space and time, using linear reconstruc- +tion, Runge-Kutta time stepping and the HLL Riemann solver. The +extended GLM divergence cleaning algorithm was used to ensure +the ∇ · B = 0 condition. For highly magnetized plasma, such as +the ones discussed here, it is advantageous to use a background +magnetic field (typically dipolar, as is the case here) and evolve its +deviation in time rather than the actual magnetic field. +2.3 +Numerical grid +For all our models, we use a stretched rectilinear spherical polar +grid extending from r = R∗ to r = 25R∗ in which the physical +volume is discretised with 250 cells in r, 64 cells in θ and 128 cells +in φ. This leads to a cell size in the r direction which stretches +from ∆r1 ≈ 0.008 R∗ to ∆r250 ≈ 0.542 R∗ with a constant +stretching factor of 1.023. Both the θ and φ directions have uniform +spacing. The stretching regime in the radial direction is required to +resolve the sonic point of the wind, which is very close to the stellar +surface. Typically, at least 5-10 grid points are required to resolve +the sonic point to ensure accurate base mass flow. +2.4 +Boundary conditions +Boundary conditions are challenging in MHD modelling and great +care must be taken to avoid any unphysical outcomes. For the +most part, we closely follow boundary conditions outlined in ud- +Doula & Owocki (2002) and Daley-Yates et al. (2019), with the +latter describing the first obliquely rotating massive-star winds, al- +though their focus was on radio emission from such objects. Sim- +ilar boundary conditions are also employed by Subramanian et al. +(2022) albeit for the geodesic mesh-based RIEMANN GEOMESH +code. +In brief, the outer radial boundary of all our simulations is set +to outflow, which allows material to freely leave the computational +domain. The inner radial boundary is set to ‘inflow’ such that the +star is continually feeding material to the wind and therefore re- +plenishing material in the simulation. +The velocity in the lower radial boundary is specified by lin- +early extrapolating back from the first 2 computational cells above +the boundary, allowing the flow into the computational active zone +to adjust to the conditions of the wind and permitting material to +also re-enter the stellar surface as magnetically confined material +follows field lines back to the stellar surface. We limit the maxi- +mum radial inflow/outflow speeds to the fixed sound speed. Speci- +fying the boundary in this manner also allows the mass loading of +the wind to self consistently adapt to the rotation of the star. Large +rotational velocities can impact the mass-loss of a star. This is due +to the effective gravity at the rotational equator being reduced rel- +ative to the poles, leading to material being lifted from the surface +more easily. +The boundary of the lower and upper azimuthal direction is +assumed to be periodic. The upper and lower boundary of the polar +direction was set to reflective so as not to act as a sink for material. +This final boundary condition is non-physical and a reflective polar +boundary can lead to spurious heating or jets along the polar axis. +There are several methods designed to overcome this numerical dif- +ficulty. One such method is known as π-boundary conditions in +which the fluid quantities are translated π around the axis and vec- +tor values transformed such that material effectively passes over the +pole. This method is implemented in the public codes ATHENA++ +(White et al. 2016) and MPI-AMRVAC (Xia et al. 2018). PLUTO +provides a similar functionality called ‘polaraxis’ which we utilize +in our simulations here. +Since we use an isothermal equation of state for the wind, we +neglect behaviour due to both shock heating and radiative cooling, +both of which can play a role in the wind dynamics (ud-Doula et al. +2008, 2013). Inclusion of a full energy balance with radiative cool- +ing is thus a goal for future studies. +2.5 +Stellar Parameters +We performed a number of simulations both in 3D and 2D to ensure +the results are consistent. Here, we focus specifically on 3D models +with inclination angle between the magnetic field and rotation axes +of 45o and 5o; the latter mimics a field-aligned model, but with +a small inclination to ensure azimuthal symmetry is numerically +broken. +Following previous studies (ud-Doula & Owocki 2002; ud- +Doula et al. 2008), we use the stellar parameters of ζ Pup, a pro- +totypical O-supergiant. In the absence of magnetic field, its mass +loss is assumed to be about 3.0 × 10−6M⊙/yr. For our model here +we assume a dipolar magnetic field with polar strength of 9300 G +corresponding to magnetic confinement of η∗ = 1000. Although +these differ from the parameters of a typical Bp star, the models +still mimic general trends in magnetospheric structure with the key +magnetic confinement parameter (ud-Doula & Owocki 2002). +Our assumed rotation is half the critical, corresponding to +about 250 km/s, with axis aligned with z-axis. Following the ap- +proach of Daley-Yates et al. (2019), the magnetic pole is rotated +about y-axis, and so lies in xz-plane, as indicated by the blue ar- +row in figures 1 - 3. +2.6 +Initial conditions +The initial conditions of the simulations are specified using the +density and velocity profile equations assuming a spherically sym- +metric wind with CAK mass loss rate (Owocki & ud-Doula 2004) +and ‘β’-law, i.e. v(r) = v∞(1 − R∗/r)β with β = 0.8 and +v∞ ∝ vescape. +MNRAS 000, 000–000 (0000) + +4 +A. ud-Doula et al. +0 ks +100 ks +200 ks +300 ks +400 ks +500 ks +1000 ks +1500 ks +1940 ks +Figure 1. Time evolution of the standard model (η∗ = 103, W = 1/2, +β = 45o), showing volume renderings of the density structure as viewed +from the rotational pole (red dot), with projected magnetic axis (blue arrow) +directed upward. After initial transits, the dragonfly wing-like co-rotating +structure settles into a quasi-steady state with occasional outbreaks due to +centrifugal forces. . +The magnetic field is initialised as an ideal dipole, centred at +the origin and rotated about the y-axis, in the xz-plane. We then +evolve the model by letting the wind and magnetic field compete +against each other. +3 +RESULTS +3.1 +Time Evolution of the Standard Model +To characterize our 3D models we use a combination of 2D slices +and 3D projection plots. For our standard model with dipole tilt +angle β = 45o, figure 1 shows for example the time evolution of +3D density as viewed from an inclination i = 0 over the rotational +pole (here marked by a red dot, indicating a rotational vector point- +ing directly toward the observer). Since we are primarily interested +in the 3D structure of the CM that forms above RK, we have for il- +lustrative clarity chosen to hollow out the density for radii r < RK, +ignoring the DM part of the magnetosphere which has been exten- +sively discussed in previous studies (e.g. see, ud-Doula et al. (2008, +2013)). +The blue arrow pointing upward marks the magnetic dipole +axis, with the projection of the magnetic equator thus along the +horizontal. Starting from the initial condition of a spherical outflow, +the combination of rotation and strong field confinement progres- +sively channels material toward greater concentration, forming two +opposing wings that straddle the common rotational and magnetic +equator. This wing structure is already clear in the 400 ks snap- +shot, after which there are relatively modest variations about this +basic shape, extending here to nearly 5 times longer, to the well- +relaxed, final simulation time of 1940 ks. By comparison, for a typ- +ical wind speed of Vw ≈ 1000 km/s, the dynamical wind crossing +time through a stellar radius R∗ is just tw = R∗/Vw ≈ 14 ks. +i=90 +o +i=0o +β = 5o +β = 45o +phase=0.25 +phase=0 +phase=0.5 +phase=0 +0.25 +0.5 +Figure 2. As in figure 1, volume renderings of density structure, now com- +paring representative evolved time step (t = 1000 ks) structures of two dif- +ferent models with identical magnetic field, rotation but different obliquity, +β : 45o (right) and 5o (left). Notice how higher obliquity model restricts +density in the azimuthal direction. This is a result of complex dynamics +between rotation and strong tilted magnetic field. +3.2 +Contrast with nearly aligned case: Disk vs. Wings +To highlight further the distinct wing structure formed for this +β = 45o oblique dipole, figure 2 compares a representative evolved +time (right column) with a similarly evolved model for the nearly +aligned case β = 5o (left column). The top row again shows the +view from inclination i = 0 over the rotational pole, with the dipole +axis upward and the dotted horizontal lines marking the intersection +between the magnetic and rotation equators. +The nearly aligned model on the left has its density in a nearly +azimuthally symmetric, equatorial disk. In contrast, the oblique +dipole on the right has density that is azimuthally concentrated in +wings that straddle the common magnetic/rotation equator. Note, +however, the modest prograde asymmetry, with a somewhat higher +density in the direction toward the stellar rotation (here counter- +clockwise). +The bottom 3 rows show equatorial views (i = 90o) at 3 +rotational phases, corresponding to the dipole axis inclination to- +ward (phase=0), perpendicular to (phase=0.25), and away from +(phase=0.5) the observer. These show that the wings are quite near +the rotational equator, but with a distinct warping out of the equato- +rial plane at larger radii. For phase=0.25, showing views along the +common equator, the foreground wing partially obscures the stellar +disk, but with a mark upward offset from this warping. At phase=0, +the prograde extension of high density also leads to some obscura- +tion; but a half-period later, at phase=0.75, there is now little mate- +rial in front of the star, since the view now through the trailing gap +between the two wings. +MNRAS 000, 000–000 (0000) + +{t=, 0.] +{t=, 100.] +{t=, 200.] +{t=, 300.] +{t=, 400.] +{t=, 500.]{t=, 1000.] +[t=, 1500.] +{t=, 1940.](t=, 995.] +{t=, 1000.](t=, 995.] +{t=, 1000.]Oblique Rotators +5 +Figure 3. For an observer with inclination i = 45o, the red curves compare +the magnetic field lines for the initially imposed dipole (top) versus the +dynamically evolved field at the final time step (bottom). These are both +superposed on the associated phase variation of volume-rendered density +for this final time step to highlight the distortion of the magnetic field due +to wind dynamics. +Figure 4. For the final time snapshot t = 1940 ks of the β = 45o tilted +dipole, 2D slices showing density, field lines, and mass flux vectors in the +x = 0 (left) and y = 0 (right) planes, where y is along the common equator +(dashed line), z is the rotation axis, and the field tilt is in the xz plane. The +blue vector in the right panel shows the dipole axis in this xz plane. The +vertical lines mark offsets from the rotation axis by one Kepler radius RK. +Clearly, such details will have important consequences for ro- +tationally modulated light curves from stars with tilted CM’s. +3.3 +Centrifugal distortion of dipole field by CM plasma +This warped-wing structure results from trapping of centrifugal ma- +terial by the strong magnetic field, which in turn distorts the field +from its dipole form. To illustrate this, figure 3 adds magnetic field +lines (red curves) to the density surfaces, as viewed now from ob- +server inclination i = 45o at the labeled rotational phases. The +upper panel shows the initial tilted dipole, while the lower panel +shows the MHD dynamical field at the final time. While the two +are quite similar, the differences show the result of the distortions of +the field from the CM plasma, in particular the centrifugal stretch- +ing from the trapped material in the dense wings. +For example, at phase=0, which gives a view looking down +the magnetic pole, the dipole field lines project into simple X-cross, +whereas the dynamical fields show a rotational twist between lines +into and out of the page. +At phase=0.5, note that closed dipole loops in the upper panel +show an outward stretching in the lower panel. Such stretching +of the magnetic field leads to eventual centrifugal breakout of the +trapped material. +Comparison of phase=0.25 and 0.75 show a simple left/right +swap for the dipole fields; but there is an asymmetric distortion in +the closed dynamical field lines, reflecting a notable asymmetry in +the density structure as well. +3.4 +2D slices in zy and zx planes +To complement this 3D rendition of the field, which can be difficult +to track visually, figures 4 and 5 show 2D slices of the density and +field line structure in the yz (x = 0) and xz (y = 0) planes. The +mass flux vectors (ρv) in figure 4 show how magnetically chan- +neled outflow compresses material into high density structures. The +vertical, z-axis is along the rotation vector (red arrows), while the +y-axis is along the common magneto-rotational equator, marked by +the horizontal dashed line in the left panel for x = 0. The closed +loops in this plane confine the dense plasma against centrifugal +forces, forming the center of the dense wing structure in 3D. By +contrast, in the right panel with y = 0, mass flux along equator- +ward boundary of the closed to open field channels material along +a diagonal to the loop tops. As shown in the outwardly distorted +form of closed loops at the lower right and upper left, this leads to +centrifugal stretching and eventually breakout. +In both panels, the density shows a nearly spherical form in +the DM within RK, but transitions to equatorial concentration just +beyond RK. In the x = 0 plane this equatorial material is trapped +by closed loops, while in the y = 0 plane it can flow along the field +toward the loop top, where it stretches the field toward breakout. +The series of final five snapshots in figure 5 illustrate the time +sequence of mass build up, field line stressing, and centrifugal +breakout. The upshot is that material above RK is confined in wings +near the y-axis, but escapes in perpendicular directions, leading to +the lower-density gaps between these wings. +3.5 +Implications for Rigidly Rotating Magnetosphere model +The plasma concentration into a warped-wing form, and the asso- +ciated centrifugal distortion of the confining magnetic field, both +have implications for the Rigidly Rotating Magnetosphere (RRM) +model introduced by Townsend & Owocki (2005). As with the +Rigid-Field Hydrodynamics (RFHD) formalism subsequently de- +veloped by Townsend et al. (2007), this approach assumes the mag- +netic field is so dominant that it acts like rigid pipes that channel +outflowing wind plasma to accumulation surfaces, set by minima +in the combined centrifugal and gravitational potential. This rigid +field notion seems well justified by the very high wind-magnetic +confinement parameters ( η∗ > 106) inferred for many B-stars with +moderate to rapid rotation. +However, recent analysis (Owocki et al. 2020) for how cen- +trifugal breakout likely sets the limit for CM mass build up implies +that, in practice, the confining fields are not in fact rigid, but rather +are distorted by centrifugal forces near the mass accumulation limit +set by breakout. +In this context, figure 6 compares equatorial views of the pro- +jected density distribution of two different versions of the RRM +model (top, bottom) with the results of the final time of the present +MNRAS 000, 000–000 (0000) + +i=β= 450 +phase=0 +0.25 +0.5 +0.754 +2 +z +0 +-2 +.4 +-2 +0 +2 +4 +y2 +Z +0 +y=0 +-2 +-4 +-4 +-2 +0 +2 +4 +x6 +A. ud-Doula et al. +Figure 5. Time evolution over the last five snapshots (t = 1900 − 1940 ks) of log density and field lines, showing both yz (top) and xz (bottom) slices, now +extending over spatial range of ±10R∗. In the yz cut note the progressive field line stretching and mass ejection associated with a centrifugal breakout event. +Breakouts also occur in the xz cut but they are more conspicuous in a different snapshot due to slight left-right asymmetry in the numerical model. Evidence +of such breakout events are quite apparent in figure 7 as well and occur on about 50 ks timescale . +MHD simulations (middle), for rotation phases 0, 0.125, and 0.025. +The RRM model in the top panel follows the original description +from Townsend & Owocki (2005), in which the local density along +the accumulation surface is just set proportional to local feeding +rate by the stellar wind along that field line. Since the area of the +flow tube scales inversely with the field strength, A ∼ 1/B, mass +flux conservation for a dipole field gives a surface density that de- +clines with radius as σ ∼ B ∼ 1/r3. For a tilted dipole, the inner +edge of the CM is closest to the star (roughly at RK) along the line +(y-axis) of common magneto-rotation equator, giving it a somewhat +higher density compared to other azimuths. However, as shown in +the top row of figure 6, the overall azimuthal variation in density +is modest, much less than from the distinct wing structure of the +MHD model in the middle row. +There are also differences in the 3D form of the RRM surface +vs. the MHD wings, with the former being distinctly offset from +the rotational equator, and latter warped about that equator. As a +result, there are quite notable differences in the phase variations, +for example in the timing and degree of occultation of the star by +the respective CM’s. +Moreover, much as found in the CBO analysis of the aligned +rotation case (Owocki et al. 2020), the radial decline in surface den- +sity in this MHD simulation of the oblique rotator is much steeper +than the assumed σ ∼ B ∼ 1/r3 scaling of original RRM analysis, +instead following closer the CBO scaling with magnetic tension, +σ ∼ B2 ∼ 1/r6. +To account for this, as well as the stronger azimuthal variation, +the bottom panel of figure 6 show an RRM with density following +a CBO-adjusted form given by eqn. (2) of Berry et al. (2022), re- +produced below as eqn. (7). Specifically, this uses their standard +value for radial power index p = 5, along with a scaling parame- +ter χ = 0.1, which sets the azimuthal decline in CM density away +from the common magneto-rotational equator2. While the geomet- +ric differences remain, there is now an improved correspondence in +the density distribution, and the associated occultations of the star. +3.6 +Distribution of Mass flux and Density +To give further insight into the overall structure and evolution of +the MHD simulations for the tilted dipole (β = 45o) case, figure +7 shows the time evolution and spatial variation of the latitudinally +integrated mass distribution in radius dM/dr, plotted versus ra- +dius r and azimuth φ for the time snapshots denoted. From initial +development over times 100−400 ks shown along the top row, the +structure settles in a quasi-steady form, with episodes of mass ejec- +tion distributed about the y-axis positions (φ = 90o and φ = 270o) +representing the intersection between the magnetic and rotational +equators. The horizontal dotted line at r = 2.8R∗ marks the bound- +ary between magnetically confined material, and the onset of CBO +events. +Figure 8 shows mass flux plotted as a function of the azimuth +and co-latitude at the labeled radii and times. The yellow shows re- +gions of infall that surrounded dark region of compressed outflow; +at radii at and below the confinement radius r = 2.8R∗ this com- +pressed outflow is near the rotational equator, but at larger radii, it +shifts closer to the magnetic equator. +For the same samples in radius and time, Figure 9 now shows +the density, again plotted as a function of the azimuth and co- +latitude. This again shows that material is compressed near the +rotational equator at radii at and below the confinement radius +r = 2.8R∗, but closer to the magnetic equator at larger radii. The +2 The χ = 0.1 used here is twice the value assumed by Berry et al. (2022), +giving a somewhat weaker drop in density away from the common equator. +The original RRM model effectively assumes χ ≫ 1. +MNRAS 000, 000–000 (0000) + +1900ks +1910ks +1920ks +1930ks +1940ksOblique Rotators +7 +Figure 6. For standard oblique dipole case (β = 45o and W = 0.5), +comparison of the phase variation of volume-rendered density for the final +time (t = 1940 ks) of the MHD simulation (center row) with predictions of +the analytic Rigidly Rotating Magnetosphere (RRM) model. The top row +depicts the density distribution resulting from wind feeding over a fixed +time, as assumed in the original RRM analysis of Townsend & Owocki +(2005). The bottom row shows a modified scaling to mimic predictions of +a centrifugal breakout (CBO) analysis, with higher concentration along the +common magnetic-rotational equator (y-axis) and a radial decline in surface +density, σ ∼ 1/r5, that is steeper than the σ ∼ 1/r3 for fixed-time wind- +feeding along a dipole field. The blue vectors depict the magnetic dipole +axis, and the observer inclination is i = 90o, and so perpendicular to the +red vectors representing the fixed stellar rotation axis. +Figure 7. For MHD simulation of tilted dipole β = 45o, time evolution +of the latitudinally averaged mass distribution in radius dM/dr, plotted +versus radius and azimuth. After about 500 ks, the model reaches a quasi- +steady state with episodic mass ejections distributed about the y-axis (φ = +90o and φ = 270o), where the magnetic and rotational equatorial planes +intersect. The horizontal dotted line at r = 2.8R∗ marks the boundary +between magnetically confined material, and the onset of CBO events. +bottom row compares the density for the CBO-RRM model, show- +ing that the density concentration near r = 2.8R∗ is, in contrast +to MHD result, intermediate between the magnetic and rotational +equators. +4 +DISCUSSION AND FUTURE WORK +4.1 +Result summary +The central aim of this paper is to use 3D MHD simulations to +characterize the centrifugal magnetospheres (CM) of strongly mag- +Figure 8. For tilted dipole β = 45o, mass flux plotted as a function of +the azimuth and co-latitude at various radii. The columns show slices at the +4 labeled radii, while rows show snapshots at the 4 labeled times. Yellow +denotes regions of infall. +Figure 9. For tilted dipole β = 45o, radial slices of density plotted as a +function of azimuth and co-latitude at the various radii labeled at the top, +with time progression of MHD model in four rows at labeled time snap- +shots. The bottom row shows for comparison the corresponding density +distribution of the CBO-RRM model. +netic, rapidly rotating hot-stars for which the assumed dipole field +has a significant tilt (β = 45o) to the star’s rotation axis. Start- +ing from an initial condition with a spherically symmetric, line- +driven stellar wind, the MHD trapping and corotation of the wind +outflow leads over many dynamical flow times to gradual build- +up of material into a complex 3D CM, characterized by distinct +wings of enhanced density, roughly centered on the line intersect- +ing the magnetic and rotational equatorial planes. The asymptotic, +quasi-steady-state includes repeated, small-scale centrifugal break- +out (CBO) events, roughly centered about the direction of common +equator, through which the ongoing wind feeding of the CM is bal- +anced by CBO mass ejections. +MNRAS 000, 000–000 (0000) + +r = rk=1.6R* +r=2.8R* +r=rA=5.6R* +r= 10.2R* +t=100 ks +p(t, r, 0, Φ) +p(t = 0,r) +2.0 +t=300 ks +1.5 +1.0 +0.5 +t=500 ks +0.0 +180 +t=1500 ks +90 +90 180 270 360 +d +CBO-RRMphase=0 +phase=0.125 +phase=0.25 +RRM +MHD +CBO +RRM100 ks +200 ks +300 ks +400 ks +dMIdr (t, r, Φ) +2 × 10-9 +500 ks +1000 ks +1500 ks +1940 ks +10 +1 × 10-9 +7 +r +2.8 +0 +1 +90 180 270 360r=rk =1.6R* +r=2.8R* +r=rA=5.6R* +=10.2R +t=100 ks +M(t, r, 0, Φ) +M(t = 0,r) +1.0 +t=300 ks +0.8 +0.6 +0.4 +0.2 +t=500 ks +0 +180 +t=1500 ks +90 +0 +90 180 270 3608 +A. ud-Doula et al. +The geometry of this dynamically fed CM follows roughly +the minimum potential surfaces derived by the hydrostatic, rigidly +rotating magnetosphere (RRM) model developed by Townsend & +Owocki (2005), with however some key differences. In particular, +the surface density follows a steeper σ ∼ 1/r5 radial decline, re- +flecting the similar drop in magnetic tension B2 ∼ 1/r6, in con- +trast with the σ ∼ B ∼ 1/r3 scaling assumed for the original RRM +model. Moreover, the density is more concentrated azimuthally, +into two wings centered on the common equatorial axis. Both ef- +fects can be roughly captured by the parameterization introduced +by Berry et al. (2022, their eqn. 2), in which the surface density at a +minimum potential location with radius r and magnetic co-latitude +θo is given by +σ(r, θo) = σK +�RK +r +�p +exp(− cos2 θo/χ) . +(7) +Here the surface density at the Kepler radius RK is given in terms +of the magnetic field and gravity there, +σK = 0.3 B2 +K +4πgK . +(8) +Specifically, the comparisons in figure 6 show that adopting p = 5 +and χ = 0.1 gives an overall density distribution (bottom row) +that agrees better with MHD results (middle row) than the standard +RRM result (top row). +But this figure also shows that the overall geometric form of +the dynamical CM in the middle panel has some moderate devi- +ations from the minimum-potential, hydrostatic accumulation sur- +face assumed in even the CBO-modified RRM model shown in the +lowermost panel. This reflects the fact that, in contrast to the per- +fectly rigid dipole field assumed in the RRM paradigm, the dynam- +ical CM naturally builds up to a limiting density that distorts this +initial dipole, culminating in episodic CBO events and associated +magnetic reconnection. +This field distortion leads to an associated dynamical contor- +tion of the CM. Instead of following the minimum total poten- +tial surface that generally lies between the magnetic and rotational +equators, the inner regions of the dynamical CM lie closer to the ro- +tational equator. However, in the outer regions this transitions to a +dense wind outflow that is concentrated toward the magnetic equa- +tor, and the associated wind current sheet that separates regions of +opposite magnetic polarity. +4.2 +Open questions and future work +Within these interesting new results and insights into the dynam- +ical form of CM’s, there remain several outstanding questions, +grounded in limitations and approximations of these 3D MHD +sims. +For example, the Courant limit on the time-step imposed by +Alfv´en propagation across grid cells has so far limited the simu- +lations to only moderately strong magnetic confinement parameter +η∗ ≲ 103, much smaller than the η∗ ≳ 106 estimated for known +CM stars like σ Ori E. In the associated stronger, stiffer magnetic +field, it is possible that the dynamical distortion effects identified +here would be less pronounced. On the hand, in the view that this +distortion stems from the inexorable build-up of CM density to- +ward breakout, instead of the direct competition between field and +wind outflow, then the CM contortion derived here may well be +applicable to observed CM stars. To distinguish between these dif- +ferent pictures, future work should carry out a parameter study in +η∗, including extension to strong confinement, e.g., η∗ ≲ 104. +Future work should also explore a broader range of field tilt +angles, including the extreme case of fully oblique dipoles, β → +90o, which RRM analyses show to have a distinct “cone-sheet” +form for the minimum-potential surfaces (Townsend & Owocki +2005). This sheet represents the asymptotic form of “leaves” that +form at large tilt angles, and it will be of interest to determine if +these localized minima show plasma accumulation in full MHD +simulations. +A further priority will be to derive observational diagnos- +tics. For example, the RRM model predicts quite distinctive dy- +namical spectra for the rotational modulation of Hα line emission +(Townsend et al. 2005), and it will be interesting how this may be +altered by the dynamical distortion effects found in these MHD +models. It will also be of interest to see if CBO-induced magnetic +reconnection events in the MHD models can reproduce the empir- +ical scaling of incoherent, circularly polarized radio emission in +massive stars (Leto et al. 2021; Shultz et al. 2022; Owocki et al. +2022) with potential implications for radio emission in Hot Jupiters +(e.g. Weber et al. 2017). +Synthesis of X-ray emission will require replacing the isother- +mal models here with a full energy equation. A key issue regards +the outliers found by Naz´e et al. (2014) in their correlation of +observed X-ray luminosity with predictions from the dynamical +magnetosphere (DM) model that applies for slow stellar rotation +(Owocki et al. 2016). These outliers generally have relatively rapid +rotation, and so are better modeled as having CM’s than DM’s. A +key question is whether the stronger observed X-rays might arise +from stronger shocks with a higher duty cycle in CM’s than DM’s, +or whether the CBO-induced magnetic reconnection might con- +tribute to the inferred enhanced X-rays. +Finally, in our focus here on the dynamical form of the CM in +these 3D MHD simulations, we have not yet examined the loss of +angular momentum associated with the magnetic stresses and mass +outflow in open field regions. For the field-aligned case (β = 0), +MHD models have provided a simple analytic scaling law for how +this angular momentum loss scales with magnetic field strength, +mass loss rate, and stellar rotation (ud-Doula et al. 2009). But a key, +open question, so far only tentatively explored for 3D MHD models +with modest magnetic confinement parameter (Subramanian et al. +2022), is how the non-zero tilt angle between the magnetic and +rotation axes might alter this spindown scaling law. This will thus +be a central focus of planned parameter studies of models with a +range of tilt angles β. +DATA AVAILABILITY STATEMENT +The data underlying this article will be shared on reasonable re- +quest to the corresponding author. +ACKNOWLEDGEMENTS +This work is supported in part by the National Aeronautics and +Space Administration under Grant No. 80NSSC22K0628 issued +through the Astrophysics Theory Program. AuD and MRG ac- +knowledge support by the National Aeronautics and Space Admin- +istration through Chandra Award Numbers TM-22001 and GO2- +23003X, issued by the Chandra X-ray Center, which is operated +by the Smithsonian Astrophysical Observatory for and on behalf +of the National Aeronautics Space Administration under contract +NAS8-03060. This work used the Bridges2 cluster at the Pittsburgh +MNRAS 000, 000–000 (0000) + +Oblique Rotators +9 +Supercomputer Center through allocation AST200002 from the Ex- +treme Science and Engineering Discovery Environment (XSEDE), +which was supported by National Science Foundation grant num- +ber 1548562. +REFERENCES +Berry I. D., Owocki S. P., Shultz M. E., ud-Doula A., 2022, MNRAS, 511, +4815 +Castor J. I., Abbott D. C., Klein R. I., 1975, ApJ, 195, 157 +Daley-Yates S., Stevens I. R., ud-Doula A., 2019, MNRAS, 489, 3251 +Drew J. E., 1989, ApJS, 71, 267 +Friend D. B., Abbott D. C., 1986, ApJ, 311, 701 +Gayley K. G., Owocki S. 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D., +2013, MNRAS, 428, 2723 +MNRAS 000, 000–000 (0000) + diff --git a/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/load_file.txt b/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d9a92a8a3c23de7772b3cbb5ccfca559327f3e5 --- /dev/null +++ b/VdFKT4oBgHgl3EQfmi5y/content/tmp_files/load_file.txt @@ -0,0 +1,598 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf,len=597 +page_content='MNRAS 000, 000–000 (0000) Preprint Monday 30th January, 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 3D MHD models of the centrifugal magnetosphere from a massive star with an oblique dipole field Asif ud-Doula1⋆, Stanley P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Owocki2, Christopher Russell2, Marc Gagn´e3 and Simon Daley-Yates4 1 Penn State Scranton, Dunmore, PA 18512, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2 Bartol Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' University of Delaware,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Newark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' DE 19716,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' USA 3Department of Earth and Space Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' West Chester University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' USA 4 School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' University of St Andrews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' North Haugh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' St Andrews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Fife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Scotland KY16 YSS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' UK 01:30Monday 30th January,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2023 ABSTRACT We present results from new self-consistent 3D MHD simulations of the magnetospheres from massive stars with a dipole magnetic axis that has a non-zero obliquity angle (β) to the star’s rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As an initial direct application, we compare the global structure of co-rotating disks for nearly aligned (β = 5o) versus half-oblique (β = 45o) models, both with moderately rapid rotation (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 critical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' We find that accumulation surfaces broadly resemble the forms predicted by the analytic Rigidly Rotating Magnetosphere (RRM) model, but the mass buildup to near the critical level for centrifugal breakout against magnetic confinement distorts the field from the imposed initial dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This leads to an associated warping of the accumulation surface toward the rotational equator, with the highest density concentrated in wings centered on the intersection between the magnetic and rotational equators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' These MHD models can be used to synthesize rotational modulation of photometric absorption and Hα emission for a direct comparison with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Key words: (magnetohydrodynamics) MHD — Stars: winds, outflows — Stars: magnetic fields — 1 INTRODUCTION Hot luminous, massive stars of spectral type O and B have dense, high-speed, radiatively driven stellar winds (Castor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the subset (∼10%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Grunhut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Sikora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2019)) of massive stars with strong (> 100 G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Shultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2019)), glob- ally ordered (often significantly dipolar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Kochukhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2019)) magnetic fields, the trapping of this wind outflow by closed mag- netic loops leads to the formation of a circumstellar magnetosphere (Petit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Because of the large angular momentum loss associated with their relatively strong, magnetised wind (ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2009), magnetic O-type stars are typically slow rotators, with trapped wind material falling back on a dynamical timescale, giv- ing what’s known as a “dynamical magnetosphere” (DM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' However, in the case of magnetic B-type stars, such angular momentum loss is greatly reduced due to their relatively weak stel- lar winds, implying longer spin-down times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Thus, not surprisingly, a significant fraction of B-type stars still retain a moderately rapid rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For such cases, the associated Keplerian co-rotation radius RK lies within the Alfv´en radius RA that characterises the maxi- mum extent of the magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The rotational support within ⋆ Email: asif@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='edu the magnetosphere leads to formation of a “centrifugal magneto- sphere” (CM), wherein the much longer confinement time allows material to build up to much higher density than in DM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For the special case of a dipole field aligned with the rotation axis, ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2008) carried out 2D MHD simulations of the resulting axisymmetric CM with material concentrated along the common rotational and magnetic equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The central aim of the current paper is to present results from new 3D MHD simulations for the sample case of an oblique dipole field that has a tilt angle β = 45o, and characterize its more complex, inherently 3D CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For models without rotation, initial 2D MHD simulations of such wind-fed magnetospheres by ud-Doula & Owocki (2002) showed that, for a star with radius R∗ and dipole field of surface strength Beq at the magnetic equator, the competition of the field with a stellar wind of mass loss rate ˙M and terminal speed V∞ can be characterized in terms of a dimensionless, wind-magnetic- confinement parameter η∗ ≡ B2 eqR2 ∗/ ˙MV∞, with the Alfv´en ra- dius then scaling as RA ≈ R∗η1/4 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The follow-on study of aligned rotation by ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2008) parameterized its effects by the dimensionless ratio W ≡ Vrot/Vorb between the equatorial ro- tation speed and the orbital speed near the equatorial surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the inner region where the field maintains rigid-body rotation, this © 0000 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='11858v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='SR] 27 Jan 2023 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' gives a Kepler co-rotation radius RK = R∗W −2/3, at which grav- itational and centrifugal forces are in balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The observational compilation by Petit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2013, see in par- ticular their figure 3) shows that many B-stars have RA > RK, with confinement parameters ranging even to η∗ > 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For example, the prototypical CM star σ Ori E, has η∗ ≈ 106 and W ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='34, implying RA/R∗ ≈ 31 ≫ RK/R∗ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1, and thus an extensive CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The field stiffness and associated high Alfv´en speed of a star with such large η∗ imply very small Courant time step in direct MHD simulations, and so far this has limited MHD models to η∗ ≲ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Alternatively, by considering the limit of an arbitrar- ily strong field (effectively with RA → ∞), semi-analytic anal- yses based on an idealization of purely rigid fields have led to a rigidly rotating magnetosphere (RRM) formalism (Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2005), which derives how CM material accumulates on surfaces set by minima of a combined centrifugal and gravitational poten- tial, under the assumed condition of rigid-body rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Such RRM models have shown great potential for explaining key observational signatures of CM stars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' rotational modulation of Balmer line emission (Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Oksala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' However, central weakness of this RRM approach is that it provides no description for how the stellar-wind-fed accumulation surfaces of the CM are ultimately emptied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Recent theoretical anal- yses (Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2020, 2022), developed to explain empirical scaling for H-alpha (Shultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2020) and radio emission (Leto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Shultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2022) from CM stars, provide strong ev- idence that this emptying occurs through frequent, low-level, cen- trifugal breakout (CBO) events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' These are triggered when the accu- mulated mass exceeds a critical level for which the centrifugal force overwhelms the confinement of the magnetic field tension force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Applying the estimated CBO critical density distribution within the RRM formalism for accumulation surfaces, Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022) re- cently modeled the photometric light variation associated with ab- sorption and scattering emission from magnetic clouds around the prototypical CM star σ Ori E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The CBO-limited density distribution derived by Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2020) was actually based on analysis for the simplified special case of an aligned dipole, calibrated against the 2D MHD simu- lations by ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Specifically, assuming W = 1/4 or 1/2 and the strongest allowed confinement η∗ = 1000, the MHD models show that, in the CM region extending above RK, the crit- ical surface density accumulated along the common magnetic and rotational equators declines in radius as r−6, consistent with the an- alytic CBO analysis that this should follow the decline in magnetic tension ∼ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' But for the many CM’s with a nonzero tilt angle β between the field and rotation axes, it is not clear how the density on the ac- cumulation surface should vary in azimuth away from the direction set by intersection of the rotational and magnetic equators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The paper here presents new 3D MHD simulations of the CM formed by a tilted dipole with axis that makes an angle β = 45o with the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The standard parameterization of moderately rapid rotation (W = 1/2) and strong confinement (η∗ = 103) pro- vides, for the first time, a 3D MHD, oblique-dipole model with an extended CM region1, ranging here from RK ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R∗ to RA ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' A particular emphasis is to characterize the resulting 1 By comparison, the recent 3D simulations by (Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2022) are limited to modest η∗ = 50 with RA ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='7R∗ ≳ RK and have too small CM regions to enable direct comparison with RRM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3D, dynamical distribution of density, and compare that with ex- pected accumulation surfaces from the semi-analytic RRM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' A specific goal is to test and calibrate the RRM density parameteri- zation used by Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022), which was inspired by the CBO analysis and preliminary versions of the 3D MHD simulations pre- sented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' To lay the basis for results presented in Section 3, Section 2 first reviews the general numerical MHD approach, numerical grid, boundary conditions and parameter domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Section 4 concludes with a summary and outline for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2 NUMERICAL SETUP AND INITIAL CONDITIONS Most of our previous numerical models were performed using ZEUS-3D or ZEUS-MP codes, but here we use publicly available, massively parallel MHD code PLUTO (version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='4) (Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2007), because of its highly versatile, modular structure that is well suited for modern Linux clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The winds of massive stars are highly ionized and the compe- tition between photoionization heating and radiative cooling keeps the wind close to the stellar effective temperature (Pauldrach 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Drew 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As such, we can approximate the wind to be isother- mal and model it with standard magnetohydrodynamics (MHD) equations in cgs units: ∂ρ ∂t + ∇ · (ρv) = 0 (1) ∂v ∂t + (v · ∇) v + 1 4πρB × (∇ × B) + 1 ρ∇p = g + glines + Fco (2) ∂B ∂t + ∇ × (B × v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (3) Here ρ, v, B, p, g and glines are, the density, velocity, magnetic field, pressure, and accelerations due to respectively gravity and line-scattering of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The comoving frame acceleration Fco is the sum of both the centrifugal and Coriolis forces, given respec- tively by Fcentrifugal = − [Ωfr × (Ωfr × R)] (4) and Fcoriolis = −2 (Ωfr × v) , (5) where Ωfr is the angular frequency of the rotating frame with r the radial distance vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As the wind is assumed to be isothermal at the stellar surface temperature T, we close equations (1 - 3) using the ideal gas equa- tion of state, p = ρkBT µ = ρc2 iso, (6) where kB is the Boltzmann constant, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6mp is the molecular weight, and the last equality casts this in terms of the isothermal sound speed ciso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1 Radial line-driving of wind outflow The radial outflow described in the previous section arises from the strong radial driving of the line-force, glines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As in ud-Doula & Owocki (2002), we model this here in terms of the standard Castor, MNRAS 000, 000–000 (0000) Oblique Rotators 3 Abbott & Klein (1975, hereafter CAK) formalism, corrected for the finite cone angle of the star, using a spherical expansion approxi- mation for the local flow gradients (Pauldrach, Puls, Hummer & Kudritzki 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Friend & Abbott 1986) and ignoring non-radial line-force components that can arise in a non-spherical outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Although such non-radial terms are typically only a few percent of the radial force (Owocki & ud-Doula 2004), in non-magnetic models of rotating winds, they act without much competition in the lateral force balance, and so can have surprisingly strong effects on the wind channeling and rotation (Owocki, Cranmer & Gayley 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Gayley & Owocki 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' But in magnetic models with an already strong component of non-radial force, such terms are not very significant, and since their full inclusion substantially compli- cates both the numerical computation and the analysis of simula- tion results, we have elected to defer further consideration of such non-radial line-force terms to future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' By limiting our study to moderately fast rotation, half or less of the critical rate, we are also able to neglect the effects of stellar oblateness and gravity darkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2 Simulation For our numerical scheme, we choose a method that is fully unsplit and 2nd order accurate in space and time, using linear reconstruc- tion, Runge-Kutta time stepping and the HLL Riemann solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The extended GLM divergence cleaning algorithm was used to ensure the ∇ · B = 0 condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For highly magnetized plasma, such as the ones discussed here, it is advantageous to use a background magnetic field (typically dipolar, as is the case here) and evolve its deviation in time rather than the actual magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='3 Numerical grid For all our models, we use a stretched rectilinear spherical polar grid extending from r = R∗ to r = 25R∗ in which the physical volume is discretised with 250 cells in r, 64 cells in θ and 128 cells in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This leads to a cell size in the r direction which stretches from ∆r1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='008 R∗ to ∆r250 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='542 R∗ with a constant stretching factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Both the θ and φ directions have uniform spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The stretching regime in the radial direction is required to resolve the sonic point of the wind, which is very close to the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Typically, at least 5-10 grid points are required to resolve the sonic point to ensure accurate base mass flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='4 Boundary conditions Boundary conditions are challenging in MHD modelling and great care must be taken to avoid any unphysical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For the most part, we closely follow boundary conditions outlined in ud- Doula & Owocki (2002) and Daley-Yates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2019), with the latter describing the first obliquely rotating massive-star winds, al- though their focus was on radio emission from such objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Sim- ilar boundary conditions are also employed by Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022) albeit for the geodesic mesh-based RIEMANN GEOMESH code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In brief, the outer radial boundary of all our simulations is set to outflow, which allows material to freely leave the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The inner radial boundary is set to ‘inflow’ such that the star is continually feeding material to the wind and therefore re- plenishing material in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The velocity in the lower radial boundary is specified by lin- early extrapolating back from the first 2 computational cells above the boundary, allowing the flow into the computational active zone to adjust to the conditions of the wind and permitting material to also re-enter the stellar surface as magnetically confined material follows field lines back to the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' We limit the maxi- mum radial inflow/outflow speeds to the fixed sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Speci- fying the boundary in this manner also allows the mass loading of the wind to self consistently adapt to the rotation of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Large rotational velocities can impact the mass-loss of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This is due to the effective gravity at the rotational equator being reduced rel- ative to the poles, leading to material being lifted from the surface more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The boundary of the lower and upper azimuthal direction is assumed to be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The upper and lower boundary of the polar direction was set to reflective so as not to act as a sink for material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This final boundary condition is non-physical and a reflective polar boundary can lead to spurious heating or jets along the polar axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' There are several methods designed to overcome this numerical dif- ficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' One such method is known as π-boundary conditions in which the fluid quantities are translated π around the axis and vec- tor values transformed such that material effectively passes over the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This method is implemented in the public codes ATHENA++ (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2016) and MPI-AMRVAC (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' PLUTO provides a similar functionality called ‘polaraxis’ which we utilize in our simulations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Since we use an isothermal equation of state for the wind, we neglect behaviour due to both shock heating and radiative cooling, both of which can play a role in the wind dynamics (ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2008, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Inclusion of a full energy balance with radiative cool- ing is thus a goal for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 Stellar Parameters We performed a number of simulations both in 3D and 2D to ensure the results are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Here, we focus specifically on 3D models with inclination angle between the magnetic field and rotation axes of 45o and 5o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' the latter mimics a field-aligned model, but with a small inclination to ensure azimuthal symmetry is numerically broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Following previous studies (ud-Doula & Owocki 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ud- Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2008), we use the stellar parameters of ζ Pup, a pro- totypical O-supergiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the absence of magnetic field, its mass loss is assumed to be about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 × 10−6M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For our model here we assume a dipolar magnetic field with polar strength of 9300 G corresponding to magnetic confinement of η∗ = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Although these differ from the parameters of a typical Bp star, the models still mimic general trends in magnetospheric structure with the key magnetic confinement parameter (ud-Doula & Owocki 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Our assumed rotation is half the critical, corresponding to about 250 km/s, with axis aligned with z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Following the ap- proach of Daley-Yates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2019), the magnetic pole is rotated about y-axis, and so lies in xz-plane, as indicated by the blue ar- row in figures 1 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6 Initial conditions The initial conditions of the simulations are specified using the density and velocity profile equations assuming a spherically sym- metric wind with CAK mass loss rate (Owocki & ud-Doula 2004) and ‘β’-law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' v(r) = v∞(1 − R∗/r)β with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8 and v∞ ∝ vescape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 0 ks 100 ks 200 ks 300 ks 400 ks 500 ks 1000 ks 1500 ks 1940 ks Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Time evolution of the standard model (η∗ = 103, W = 1/2, β = 45o), showing volume renderings of the density structure as viewed from the rotational pole (red dot), with projected magnetic axis (blue arrow) directed upward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' After initial transits, the dragonfly wing-like co-rotating structure settles into a quasi-steady state with occasional outbreaks due to centrifugal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The magnetic field is initialised as an ideal dipole, centred at the origin and rotated about the y-axis, in the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' We then evolve the model by letting the wind and magnetic field compete against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3 RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1 Time Evolution of the Standard Model To characterize our 3D models we use a combination of 2D slices and 3D projection plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For our standard model with dipole tilt angle β = 45o, figure 1 shows for example the time evolution of 3D density as viewed from an inclination i = 0 over the rotational pole (here marked by a red dot, indicating a rotational vector point- ing directly toward the observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Since we are primarily interested in the 3D structure of the CM that forms above RK, we have for il- lustrative clarity chosen to hollow out the density for radii r < RK, ignoring the DM part of the magnetosphere which has been exten- sively discussed in previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' see, ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2008, 2013)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The blue arrow pointing upward marks the magnetic dipole axis, with the projection of the magnetic equator thus along the horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Starting from the initial condition of a spherical outflow, the combination of rotation and strong field confinement progres- sively channels material toward greater concentration, forming two opposing wings that straddle the common rotational and magnetic equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This wing structure is already clear in the 400 ks snap- shot, after which there are relatively modest variations about this basic shape, extending here to nearly 5 times longer, to the well- relaxed, final simulation time of 1940 ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' By comparison, for a typ- ical wind speed of Vw ≈ 1000 km/s, the dynamical wind crossing time through a stellar radius R∗ is just tw = R∗/Vw ≈ 14 ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' i=90 o i=0o β = 5o β = 45o phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25 phase=0 phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 phase=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As in figure 1, volume renderings of density structure, now com- paring representative evolved time step (t = 1000 ks) structures of two dif- ferent models with identical magnetic field, rotation but different obliquity, β : 45o (right) and 5o (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Notice how higher obliquity model restricts density in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This is a result of complex dynamics between rotation and strong tilted magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2 Contrast with nearly aligned case: Disk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Wings To highlight further the distinct wing structure formed for this β = 45o oblique dipole, figure 2 compares a representative evolved time (right column) with a similarly evolved model for the nearly aligned case β = 5o (left column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The top row again shows the view from inclination i = 0 over the rotational pole, with the dipole axis upward and the dotted horizontal lines marking the intersection between the magnetic and rotation equators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The nearly aligned model on the left has its density in a nearly azimuthally symmetric, equatorial disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In contrast, the oblique dipole on the right has density that is azimuthally concentrated in wings that straddle the common magnetic/rotation equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Note, however, the modest prograde asymmetry, with a somewhat higher density in the direction toward the stellar rotation (here counter- clockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The bottom 3 rows show equatorial views (i = 90o) at 3 rotational phases, corresponding to the dipole axis inclination to- ward (phase=0), perpendicular to (phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25), and away from (phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5) the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' These show that the wings are quite near the rotational equator, but with a distinct warping out of the equato- rial plane at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25, showing views along the common equator, the foreground wing partially obscures the stellar disk, but with a mark upward offset from this warping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' At phase=0, the prograde extension of high density also leads to some obscura- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' but a half-period later, at phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='75, there is now little mate- rial in front of the star, since the view now through the trailing gap between the two wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) {t=, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ]{t=, 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] [t=, 1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 1940.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ](t=, 995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ](t=, 995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='] {t=, 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ]Oblique Rotators 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For an observer with inclination i = 45o, the red curves compare the magnetic field lines for the initially imposed dipole (top) versus the dynamically evolved field at the final time step (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' These are both superposed on the associated phase variation of volume-rendered density for this final time step to highlight the distortion of the magnetic field due to wind dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For the final time snapshot t = 1940 ks of the β = 45o tilted dipole, 2D slices showing density, field lines, and mass flux vectors in the x = 0 (left) and y = 0 (right) planes, where y is along the common equator (dashed line), z is the rotation axis, and the field tilt is in the xz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The blue vector in the right panel shows the dipole axis in this xz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The vertical lines mark offsets from the rotation axis by one Kepler radius RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Clearly, such details will have important consequences for ro- tationally modulated light curves from stars with tilted CM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='3 Centrifugal distortion of dipole field by CM plasma This warped-wing structure results from trapping of centrifugal ma- terial by the strong magnetic field, which in turn distorts the field from its dipole form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' To illustrate this, figure 3 adds magnetic field lines (red curves) to the density surfaces, as viewed now from ob- server inclination i = 45o at the labeled rotational phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The upper panel shows the initial tilted dipole, while the lower panel shows the MHD dynamical field at the final time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' While the two are quite similar, the differences show the result of the distortions of the field from the CM plasma, in particular the centrifugal stretch- ing from the trapped material in the dense wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For example, at phase=0, which gives a view looking down the magnetic pole, the dipole field lines project into simple X-cross, whereas the dynamical fields show a rotational twist between lines into and out of the page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' At phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5, note that closed dipole loops in the upper panel show an outward stretching in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Such stretching of the magnetic field leads to eventual centrifugal breakout of the trapped material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Comparison of phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='75 show a simple left/right swap for the dipole fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' but there is an asymmetric distortion in the closed dynamical field lines, reflecting a notable asymmetry in the density structure as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='4 2D slices in zy and zx planes To complement this 3D rendition of the field, which can be difficult to track visually, figures 4 and 5 show 2D slices of the density and field line structure in the yz (x = 0) and xz (y = 0) planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The mass flux vectors (ρv) in figure 4 show how magnetically chan- neled outflow compresses material into high density structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The vertical, z-axis is along the rotation vector (red arrows), while the y-axis is along the common magneto-rotational equator, marked by the horizontal dashed line in the left panel for x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The closed loops in this plane confine the dense plasma against centrifugal forces, forming the center of the dense wing structure in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' By contrast, in the right panel with y = 0, mass flux along equator- ward boundary of the closed to open field channels material along a diagonal to the loop tops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As shown in the outwardly distorted form of closed loops at the lower right and upper left, this leads to centrifugal stretching and eventually breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In both panels, the density shows a nearly spherical form in the DM within RK, but transitions to equatorial concentration just beyond RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the x = 0 plane this equatorial material is trapped by closed loops, while in the y = 0 plane it can flow along the field toward the loop top, where it stretches the field toward breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The series of final five snapshots in figure 5 illustrate the time sequence of mass build up, field line stressing, and centrifugal breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The upshot is that material above RK is confined in wings near the y-axis, but escapes in perpendicular directions, leading to the lower-density gaps between these wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 Implications for Rigidly Rotating Magnetosphere model The plasma concentration into a warped-wing form, and the asso- ciated centrifugal distortion of the confining magnetic field, both have implications for the Rigidly Rotating Magnetosphere (RRM) model introduced by Townsend & Owocki (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As with the Rigid-Field Hydrodynamics (RFHD) formalism subsequently de- veloped by Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2007), this approach assumes the mag- netic field is so dominant that it acts like rigid pipes that channel outflowing wind plasma to accumulation surfaces, set by minima in the combined centrifugal and gravitational potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This rigid field notion seems well justified by the very high wind-magnetic confinement parameters ( η∗ > 106) inferred for many B-stars with moderate to rapid rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' However, recent analysis (Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2020) for how cen- trifugal breakout likely sets the limit for CM mass build up implies that, in practice, the confining fields are not in fact rigid, but rather are distorted by centrifugal forces near the mass accumulation limit set by breakout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In this context, figure 6 compares equatorial views of the pro- jected density distribution of two different versions of the RRM model (top, bottom) with the results of the final time of the present MNRAS 000, 000–000 (0000) i=β= 450 phase=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='754 2 z 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='4 2 0 2 4 y2 Z 0 y=0 2 4 4 2 0 2 4 x6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Time evolution over the last five snapshots (t = 1900 − 1940 ks) of log density and field lines, showing both yz (top) and xz (bottom) slices, now extending over spatial range of ±10R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the yz cut note the progressive field line stretching and mass ejection associated with a centrifugal breakout event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Breakouts also occur in the xz cut but they are more conspicuous in a different snapshot due to slight left-right asymmetry in the numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Evidence of such breakout events are quite apparent in figure 7 as well and occur on about 50 ks timescale .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' MHD simulations (middle), for rotation phases 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='125, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The RRM model in the top panel follows the original description from Townsend & Owocki (2005), in which the local density along the accumulation surface is just set proportional to local feeding rate by the stellar wind along that field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Since the area of the flow tube scales inversely with the field strength, A ∼ 1/B, mass flux conservation for a dipole field gives a surface density that de- clines with radius as σ ∼ B ∼ 1/r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For a tilted dipole, the inner edge of the CM is closest to the star (roughly at RK) along the line (y-axis) of common magneto-rotation equator, giving it a somewhat higher density compared to other azimuths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' However, as shown in the top row of figure 6, the overall azimuthal variation in density is modest, much less than from the distinct wing structure of the MHD model in the middle row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' There are also differences in the 3D form of the RRM surface vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' the MHD wings, with the former being distinctly offset from the rotational equator, and latter warped about that equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' As a result, there are quite notable differences in the phase variations, for example in the timing and degree of occultation of the star by the respective CM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Moreover, much as found in the CBO analysis of the aligned rotation case (Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2020), the radial decline in surface den- sity in this MHD simulation of the oblique rotator is much steeper than the assumed σ ∼ B ∼ 1/r3 scaling of original RRM analysis, instead following closer the CBO scaling with magnetic tension, σ ∼ B2 ∼ 1/r6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' To account for this, as well as the stronger azimuthal variation, the bottom panel of figure 6 show an RRM with density following a CBO-adjusted form given by eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2) of Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022), re- produced below as eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Specifically, this uses their standard value for radial power index p = 5, along with a scaling parame- ter χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1, which sets the azimuthal decline in CM density away from the common magneto-rotational equator2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' While the geomet- ric differences remain, there is now an improved correspondence in the density distribution, and the associated occultations of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6 Distribution of Mass flux and Density To give further insight into the overall structure and evolution of the MHD simulations for the tilted dipole (β = 45o) case, figure 7 shows the time evolution and spatial variation of the latitudinally integrated mass distribution in radius dM/dr, plotted versus ra- dius r and azimuth φ for the time snapshots denoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' From initial development over times 100−400 ks shown along the top row, the structure settles in a quasi-steady form, with episodes of mass ejec- tion distributed about the y-axis positions (φ = 90o and φ = 270o) representing the intersection between the magnetic and rotational equators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The horizontal dotted line at r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R∗ marks the bound- ary between magnetically confined material, and the onset of CBO events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Figure 8 shows mass flux plotted as a function of the azimuth and co-latitude at the labeled radii and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The yellow shows re- gions of infall that surrounded dark region of compressed outflow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' at radii at and below the confinement radius r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R∗ this com- pressed outflow is near the rotational equator, but at larger radii, it shifts closer to the magnetic equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For the same samples in radius and time, Figure 9 now shows the density, again plotted as a function of the azimuth and co- latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This again shows that material is compressed near the rotational equator at radii at and below the confinement radius r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R∗, but closer to the magnetic equator at larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The 2 The χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1 used here is twice the value assumed by Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022), giving a somewhat weaker drop in density away from the common equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The original RRM model effectively assumes χ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) 1900ks 1910ks 1920ks 1930ks 1940ksOblique Rotators 7 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For standard oblique dipole case (β = 45o and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5), comparison of the phase variation of volume-rendered density for the final time (t = 1940 ks) of the MHD simulation (center row) with predictions of the analytic Rigidly Rotating Magnetosphere (RRM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The top row depicts the density distribution resulting from wind feeding over a fixed time, as assumed in the original RRM analysis of Townsend & Owocki (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The bottom row shows a modified scaling to mimic predictions of a centrifugal breakout (CBO) analysis, with higher concentration along the common magnetic-rotational equator (y-axis) and a radial decline in surface density, σ ∼ 1/r5, that is steeper than the σ ∼ 1/r3 for fixed-time wind- feeding along a dipole field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The blue vectors depict the magnetic dipole axis, and the observer inclination is i = 90o, and so perpendicular to the red vectors representing the fixed stellar rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For MHD simulation of tilted dipole β = 45o, time evolution of the latitudinally averaged mass distribution in radius dM/dr, plotted versus radius and azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' After about 500 ks, the model reaches a quasi- steady state with episodic mass ejections distributed about the y-axis (φ = 90o and φ = 270o), where the magnetic and rotational equatorial planes intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The horizontal dotted line at r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R∗ marks the boundary between magnetically confined material, and the onset of CBO events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' bottom row compares the density for the CBO-RRM model, show- ing that the density concentration near r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R∗ is, in contrast to MHD result, intermediate between the magnetic and rotational equators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 4 DISCUSSION AND FUTURE WORK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1 Result summary The central aim of this paper is to use 3D MHD simulations to characterize the centrifugal magnetospheres (CM) of strongly mag- Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For tilted dipole β = 45o, mass flux plotted as a function of the azimuth and co-latitude at various radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The columns show slices at the 4 labeled radii, while rows show snapshots at the 4 labeled times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Yellow denotes regions of infall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For tilted dipole β = 45o, radial slices of density plotted as a function of azimuth and co-latitude at the various radii labeled at the top, with time progression of MHD model in four rows at labeled time snap- shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The bottom row shows for comparison the corresponding density distribution of the CBO-RRM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' netic, rapidly rotating hot-stars for which the assumed dipole field has a significant tilt (β = 45o) to the star’s rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Start- ing from an initial condition with a spherically symmetric, line- driven stellar wind, the MHD trapping and corotation of the wind outflow leads over many dynamical flow times to gradual build- up of material into a complex 3D CM, characterized by distinct wings of enhanced density, roughly centered on the line intersect- ing the magnetic and rotational equatorial planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The asymptotic, quasi-steady-state includes repeated, small-scale centrifugal break- out (CBO) events, roughly centered about the direction of common equator, through which the ongoing wind feeding of the CM is bal- anced by CBO mass ejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' MNRAS 000, 000–000 (0000) r = rk=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R* r=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R* r=rA=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R* r= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2R* t=100 ks p(t, r, 0, Φ) p(t = 0,r) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 t=300 ks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='5 t=500 ks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 180 t=1500 ks 90 90 180 270 360 d CBO-RRMphase=0 phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='125 phase=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='25 RRM MHD CBO RRM100 ks 200 ks 300 ks 400 ks dMIdr (t, r, Φ) 2 × 10-9 500 ks 1000 ks 1500 ks 1940 ks 10 1 × 10-9 7 r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8 0 1 90 180 270 360r=rk =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R* r=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8R* r=rA=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6R* =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2R t=100 ks M(t, r, 0, Φ) M(t = 0,r) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='0 t=300 ks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2 t=500 ks 0 180 t=1500 ks 90 0 90 180 270 3608 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' The geometry of this dynamically fed CM follows roughly the minimum potential surfaces derived by the hydrostatic, rigidly rotating magnetosphere (RRM) model developed by Townsend & Owocki (2005), with however some key differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In particular, the surface density follows a steeper σ ∼ 1/r5 radial decline, re- flecting the similar drop in magnetic tension B2 ∼ 1/r6, in con- trast with the σ ∼ B ∼ 1/r3 scaling assumed for the original RRM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Moreover, the density is more concentrated azimuthally, into two wings centered on the common equatorial axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Both ef- fects can be roughly captured by the parameterization introduced by Berry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2022, their eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2), in which the surface density at a minimum potential location with radius r and magnetic co-latitude θo is given by σ(r, θo) = σK �RK r �p exp(− cos2 θo/χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (7) Here the surface density at the Kepler radius RK is given in terms of the magnetic field and gravity there, σK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='3 B2 K 4πgK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (8) Specifically, the comparisons in figure 6 show that adopting p = 5 and χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='1 gives an overall density distribution (bottom row) that agrees better with MHD results (middle row) than the standard RRM result (top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' But this figure also shows that the overall geometric form of the dynamical CM in the middle panel has some moderate devi- ations from the minimum-potential, hydrostatic accumulation sur- face assumed in even the CBO-modified RRM model shown in the lowermost panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This reflects the fact that, in contrast to the per- fectly rigid dipole field assumed in the RRM paradigm, the dynam- ical CM naturally builds up to a limiting density that distorts this initial dipole, culminating in episodic CBO events and associated magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This field distortion leads to an associated dynamical contor- tion of the CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Instead of following the minimum total poten- tial surface that generally lies between the magnetic and rotational equators, the inner regions of the dynamical CM lie closer to the ro- tational equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' However, in the outer regions this transitions to a dense wind outflow that is concentrated toward the magnetic equa- tor, and the associated wind current sheet that separates regions of opposite magnetic polarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='2 Open questions and future work Within these interesting new results and insights into the dynam- ical form of CM’s, there remain several outstanding questions, grounded in limitations and approximations of these 3D MHD sims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For example, the Courant limit on the time-step imposed by Alfv´en propagation across grid cells has so far limited the simu- lations to only moderately strong magnetic confinement parameter η∗ ≲ 103, much smaller than the η∗ ≳ 106 estimated for known CM stars like σ Ori E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' In the associated stronger, stiffer magnetic field, it is possible that the dynamical distortion effects identified here would be less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' On the hand, in the view that this distortion stems from the inexorable build-up of CM density to- ward breakout, instead of the direct competition between field and wind outflow, then the CM contortion derived here may well be applicable to observed CM stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' To distinguish between these dif- ferent pictures, future work should carry out a parameter study in η∗, including extension to strong confinement, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=', η∗ ≲ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Future work should also explore a broader range of field tilt angles, including the extreme case of fully oblique dipoles, β → 90o, which RRM analyses show to have a distinct “cone-sheet” form for the minimum-potential surfaces (Townsend & Owocki 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This sheet represents the asymptotic form of “leaves” that form at large tilt angles, and it will be of interest to determine if these localized minima show plasma accumulation in full MHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' A further priority will be to derive observational diagnos- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For example, the RRM model predicts quite distinctive dy- namical spectra for the rotational modulation of Hα line emission (Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2005), and it will be interesting how this may be altered by the dynamical distortion effects found in these MHD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' It will also be of interest to see if CBO-induced magnetic reconnection events in the MHD models can reproduce the empir- ical scaling of incoherent, circularly polarized radio emission in massive stars (Leto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Shultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2022) with potential implications for radio emission in Hot Jupiters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Synthesis of X-ray emission will require replacing the isother- mal models here with a full energy equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' A key issue regards the outliers found by Naz´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' (2014) in their correlation of observed X-ray luminosity with predictions from the dynamical magnetosphere (DM) model that applies for slow stellar rotation (Owocki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' These outliers generally have relatively rapid rotation, and so are better modeled as having CM’s than DM’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' A key question is whether the stronger observed X-rays might arise from stronger shocks with a higher duty cycle in CM’s than DM’s, or whether the CBO-induced magnetic reconnection might con- tribute to the inferred enhanced X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' Finally, in our focus here on the dynamical form of the CM in these 3D MHD simulations, we have not yet examined the loss of angular momentum associated with the magnetic stresses and mass outflow in open field regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' For the field-aligned case (β = 0), MHD models have provided a simple analytic scaling law for how this angular momentum loss scales with magnetic field strength, mass loss rate, and stellar rotation (ud-Doula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' But a key, open question, so far only tentatively explored for 3D MHD models with modest magnetic confinement parameter (Subramanian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 2022), is how the non-zero tilt angle between the magnetic and rotation axes might alter this spindown scaling law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This will thus be a central focus of planned parameter studies of models with a range of tilt angles β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data underlying this article will be shared on reasonable re- quest to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported in part by the National Aeronautics and Space Administration under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' 80NSSC22K0628 issued through the Astrophysics Theory Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' AuD and MRG ac- knowledge support by the National Aeronautics and Space Admin- istration through Chandra Award Numbers TM-22001 and GO2- 23003X, issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFKT4oBgHgl3EQfmi5y/content/2301.11858v1.pdf'} +page_content=' This work used the Bridges2 cluster at the Pittsburgh MNRAS 000, 000–000 (0000) Oblique Rotators 9 Supercomputer Center through allocation AST200002 from the Ex- treme Science and Engineering 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Yi Wu1,2,4†, Yu Wang1† +1 Tsinghua University, 2 Shanghai Artificial Intelligence Laboratory, 3 Tongji University, 4 Shanghai Qi Zhi Institute +∗ Equal Contribution † Corresponding Author +zoeyuchao@gmail.com +ABSTRACT +We consider the problem of cooperative exploration where mul- +tiple robots need to cooperatively explore an unknown region as +fast as possible. Multi-agent reinforcement learning (MARL) has +recently become a trending paradigm for solving this challenge. +However, existing MARL-based methods adopt action-making steps +as the metric for exploration efficiency by assuming all the agents +are acting in a fully synchronous manner: i.e., every single agent +produces an action simultaneously and every single action is exe- +cuted instantaneously at each time step. Despite its mathematical +simplicity, such a synchronous MARL formulation can be prob- +lematic for real-world robotic applications. It can be typical that +different robots may take slightly different wall-clock times to ac- +complish an atomic action or even periodically get lost due to +hardware issues. Simply waiting for every robot being ready for +the next action can be particularly time-inefficient. Therefore, we +propose an asynchronous MARL solution, Asynchronous Coordi- +nation Explorer (ACE), to tackle this real-world challenge. We first +extend a classical MARL algorithm, multi-agent PPO (MAPPO), +to the asynchronous setting and additionally apply action-delay +randomization to enforce the learned policy to generalize better to +varying action delays in the real world. Moreover, each navigation +agent is represented as a team-size-invariant CNN-based policy, +which greatly benefits real-robot deployment by handling possible +robot lost and allows bandwidth-efficient intra-agent communica- +tion through low-dimensional CNN features. We first validate our +approach in a grid-based scenario. Both simulation and real-robot +results show that ACE reduces over 10% actual exploration time +compared with classical approaches. We also apply our framework +to a high-fidelity visual-based environment, Habitat, achieving 28% +improvement in exploration efficiency. +KEYWORDS +Multi-Agent Reinforcement Learning, Asynchronous Decision Mak- +ing, Cooperative Exploration +ACM Reference Format: +Chao Yu1,2∗, Xinyi Yang1∗, Jiaxuan Gao1∗, Jiayu Chen1,2, Yunfei Li1, Jijia Liu3, +Yunfei Xiang1,, Ruixin Huang1, Huazhong Yang1, Yi Wu1,2,4†, Yu Wang1†, . +2023. Asynchronous Multi-Agent Reinforcement Learning for Efficient Real- +Time Multi-Robot Cooperative Exploration. In Proc. of the 22nd International +Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), +London, United Kingdom, May 29 – June 2, 2023, IFAAMAS, 15 pages. +Proc. of the 22nd International Conference on Autonomous Agents and Multiagent Sys- +tems (AAMAS 2023), A. Ricci, W. Yeoh, N. Agmon, B. An (eds.), May 29 – June 2, 2023, +London, United Kingdom. © 2023 International Foundation for Autonomous Agents +and Multiagent Systems (www.ifaamas.org). All rights reserved. +1 +INTRODUCTION +Exploration is a fundamental task for building intelligent robot +systems, which has been applied in many application domains, in- +cluding rescue [27], autonomous driving [2], drone [48], and mobile +robots [36]. In this paper, we consider a multi-robot cooperative +exploration task, where multiple homogeneous robots simultane- +ously explore an unknown spatial region in a cooperative fashion. +Learning the optimal cooperative strategies can be challenging +due to the existence of multiple robots. These robots must effec- +tively distribute the exploration workload so that they can always +navigate towards different spatial regions to avoid trajectory con- +flicts, which accordingly leads to a remarkably higher exploration +efficiency than the single-robot setting. +Multi-agent reinforcement learning (MARL) has been a trending +approach to tackle this cooperative exploration challenge. RL-based +methods directly learn neural policies end to end by interacting +with a simulated environment for policy improvement. Compared +with planning-based solutions [3, 38, 45] which require non-trivial +implementation heuristics and expensive inference computation at +execution time, RL-based methods [7, 38] provide strong represen- +tation capabilities of complex strategies and negligible inference +overhead once the policies are trained. +Classical multi-agent RL algorithms typically adopt a synchro- +nous algorithmic framework, i.e., all the agents are making actions +at the same time, and all the actions will be executed immediately at +each time step, leading to the next action-making step for future ac- +tions. This process is mathematically formulated as a decentralized +Markov decision process, which is widely adopted in multi-agent RL +literature. Although such a mathematical framework is simple and +elegant, it can be problematic for real-world multi-robot exploration +tasks. For real robot systems, each actual action is never atomic +and may take varying times to finish. These action delays can be +more severe due to unexpected network communication traffic or +hardware failure. Simply following the synchronous setting, i.e., +waiting until every robot is ready before making new actions, can +be particularly real-time inefficient. Therefore, an ideal RL frame- +work for real-world use should be asynchronous, i.e., whenever an +agent finishes action execution, it should immediately generate the +next action, and the learned strategy should effectively enable such +an asynchronous action-making process. +In this paper, we propose a novel asynchronous multi-agent +RL-based solution, Asynchronous Coordination Explorer (ACE), to +tackle the real-world multi-robot exploration task. We first extend a +classical MARL algorithm, multi-agent PPO (MAPPO), to the asyn- +chronous setting to effectively train the multi-agent exploration +policy, and additionally leverage an action-delay randomization +arXiv:2301.03398v1 [cs.RO] 9 Jan 2023 + +technique to enable better simulation to real-world generalization. +Moreover, we design a communication-efficient Multi-tower-CNN- +based Policy (MCP) for each agent. In MCP, a CNN module is +applied to each agent’s local information to extract features, and +a fusion module combines each agent’s features to produce an ac- +tion. During execution time, efficient intra-agent communication +can be achieved via directly exchanging low-dimensional features +extracted by the weight-sharing CNN module. Another benefit of +MCP is to tackle varying team sizes, which may occur when agents +go offline in real-world applications. +We conduct experiments in a grid-based multi-room scenario +both in simulation and our real-world multi-robot laboratory, where +strategies learned by ACE significantly outperform both classical +planning-based methods and neural policies trained by synchro- +nous RL methods. In particular, ACE reduces 10.07% real-world +exploration time than the synchronous RL baseline and reduces +33.86% real-world exploration time than the fastest planning-based +method with 2 Mecanum steering robots. Besides, we extend ACE +to a vision-based environment, Habitat, verifying the effectiveness +of the asynchronous training mechanism when applied to more +complicated environments. More demonstrations can be seen on +our website: https://sites.google.com/view/ace-aamas. +2 +RELATED WORK +2.1 +Cooperative Exploration +Multi-agent cooperative exploration is an important task for build- +ing intelligent mobile robot systems. There are a large number of +works developing planning-based methods for this problem [13, +45, 57], but they typically rely on manually designed heuristics [3, +10, 54] and are limited in expressiveness to learn more complex +cooperation strategies. Another popular line of research is deep +MARL-based methods which leverage the expressiveness power +of neural networks to learn non-trivial cooperative exploration +skills [28, 49–51, 59, 69]. Note that most existing MARL-based meth- +ods assume synchronous action execution among all the agents or +consider atomic actions, which we believe is due to the synchro- +nous design of most simulated RL environments. However, such +synchronous design does not reflect the real-world multi-agent +systems, where agents take actions at different real times due to +network delay and unexpected hardware take-downs. We propose +an asynchronous MARL exploration framework in this work to bet- +ter match real-world applications. [31] considers an asynchronous +decision-making mechanism for large-scale problems and proposes +an improved Monte Carlo Search method to solve this problem. +A concurrent work formulates a set of asynchronous multi-agent +actor-critic methods that allow agents to directly optimize asyn- +chronous policies [56], while we design an asynchronous MARL +training framework combined with action-delay randomization. +We also notice a recent trend of developing asynchronous simula- +tion [22, 60], which we hope can further accelerate the advances in +applying MARL methods to the real world. +2.2 +Sim2real Transfer +It is often challenging to directly deploy policies trained in simu- +lation to the real world since there is always a mismatch between +simulation and reality. Domain randomization is a simple but ef- +fective technique to fill the reality gap, which creates a variety of +simulated environments with randomized properties such as physi- +cal dynamics [35, 42] and visual appearances [32, 43], and tries to +train RL that can perform well among all of them. We also adopt the +idea of domain randomization, and randomly delay the execution +of each agent in simulation to model the uncertain delay between +policy computation and actual action execution in the real world. +The action delay technique is also adopted in other domains such +as model-based RL [5] and reactive RL [44]. +2.3 +Multi-Agent Communication +Most works in MARL only consider perfect communication where +agents can receive messages from all other agents [12, 17, 67], but +the requirements on communication bandwidth and transmission +rate are costly. Recent works have begun to focus on learning effi- +cient communication. [16, 40] learn communication protocols in +limited-bandwidth communication channels. Some works propose +to learn which agent to communicate with using attention mech- +anisms [24, 30] or weight-based schedulers [26]. In our work, we +focus on how to efficiently communicate with other agents with +limited information to maintain perfect decision performance. We +propose an extraction module to obtain essential features from +high-dimensional information, which is much more efficient than +delivering observation information directly. +2.4 +Size-Invariant Representation +Attention mechanism [47] is widely used in RL policy representa- +tion to capture object-level information [15, 53, 66, 68] and repre- +sent relations [38, 65]. In MARL, attention-based policy can also +be applied for generalization to an arbitrary number of input enti- +ties [15, 18, 23, 28, 37, 52, 53]. There are some other generalization +settings in MARL, such as other-play [19], population-based train- +ing [21, 33] and zero-shot team formation [1]. In addition, some +training techniques are used to better deal with varying sizes of +agents, such as evolutionary learning [11, 28] and curriculum learn- +ing [6]. In our work, we design a Multi-tower-CNN-based Policy +based on an attention mechanism to tackle varying team sizes. +Parameter-sharing is a commonly used paradigm for varying team +sizes in MARL with homogeneous agents, which is also adopted in +this work. It learns an identical policy for each agent and helps to +reduce nonstationarity and improve training efficiency [9, 41]. +3 +PRELIMINARY +3.1 +Task Setup +We study the task of real-time multi-robot cooperative exploration, +where a team of robots aims to explore an unknown environment +exhaustively as fast as possible. Robots can transmit local informa- +tion to each other to avoid duplicate exploration for better efficiency. +The real-time multi-robot task is with an asynchronous nature, i.e., +different robots do not take actions and receive the next-step ob- +servations at the same time. A real robot often requires non-fixed +time to execute an action, and unexpected hardware failure can +cause random delays. Besides, under the standard bi-level control +setting in robot navigation [25, 45, 57, 64], each action is a goal +position to reach and typically requires varying number of atomic + +steps to accomplish, thus exacerbating the asynchronous issue. The +asynchronous execution is not considered by classical multi-agent +reinforcement learning (MARL) works. It is typically assumed all +agents take actions at the same action-making step and do not take +the action execution time into account. In the traditional MARL +literature [62], the task is usually formulated as a decentralized +partially observable Markov decision process (Dec-POMDP), which +is unable to capture the asynchronous property in our setting. +3.2 +Problem Formulation +We model the asynchronous multi-agent cooperative exploration +task as a decentralized partially observable Semi-Markov decision +process (Dec-POSMDP) [31] with shared rewards. We adopt a modu- +lar action execution scheme [4, 29] which consists of bi-level actions +for robust deployment in real-world robot systems. A macro action +(MA), i.e., global goal, is generated in the action-making step. Sev- +eral atomic actions, i.e., execution actions, are followed to perform +under the guidance of the MA. +To avoid notation ambiguity, we use 𝑝 (𝑖) to denote a parameter +𝑝 related to the i-th agent, and ¯𝑝 = (𝑝 (1), 𝑝 (2), · · · , 𝑝 (𝑛)) to denote +joint parameters for multiple agents thereafter. A Dec-POSMDP is +defined by a set of elements ⟨𝐷, ¯𝑈, ¯𝐵, 𝑃, ¯𝑅𝜏⟩. 𝐷 = ⟨ ¯𝑆, ¯𝐴, ¯Ω, ¯𝑂, ¯𝑅, 𝑃,𝑛,𝛾⟩ +defines the decentralized partially observable Markov decision pro- +cesses (Dec-POMDP), where ¯𝑆 is the joint state space, ¯𝐴 is joint +atomic action space, ¯Ω is the observation space,𝑂 (𝑖) (𝑜 (𝑖) |𝑠,𝑎(𝑖)) de- +notes the observation probability function for agent𝑖, ¯𝑅 : ¯𝑆× ¯𝐴 → R +is the joint reward function, 𝑛 is the number of agents, 𝑃 is the state +transition probability. ¯𝑈 is joint macro-action space. A macro action +𝑢 (𝑖) is a high-level policy that can generate a sequence of atomic +actions 𝑎𝑡 ∼ 𝑢 (𝑖) (𝐻 (𝑖) +𝑡 +) for any 𝑡 when 𝑢 (𝑖) is activated, where +𝐻 (𝑖) +𝑡 +is the individual action-observation history till 𝑡. ¯𝐵 denotes +the stop condition of MA and 𝐵(𝑖) (𝑢 (𝑖)) is represented as a set of +action-observation histories of an agent 𝑖. If 𝐻 (𝑖) +𝑡 +∈ 𝐵(𝑖) (𝑢 (𝑖) +𝑡 +) holds, +𝑢 (𝑖) +𝑡 +terminates and the agent generates a new MA. ¯𝑅𝜏 is the macro +joint reward function: ¯𝑅𝜏 (¯𝑠, ¯𝑢) = E +��¯𝜏𝑒𝑛𝑑 +𝑡=0 𝛾𝑡 ¯𝑅(¯𝑠𝑡, ¯𝑎𝑡)| ¯𝑎𝑡 ∼ ¯𝑢( ¯𝐻𝑡) +� +where ¯𝜏𝑒𝑛𝑑 = min𝑡 {𝑡 : 𝐻 (𝑖) +𝑡 +∈ 𝐵(𝑖) (𝑢 (𝑖))}. +The solution of a Dec-POSMDP is a joint high-level decentral- +ized policy ¯𝜙 = (𝜙 (1), · · · ,𝜙 (𝑛)) where each 𝜙 (𝑖) produces an +MA 𝜙 (𝑖) (𝐻 (𝑖) +𝑡 +) ∈ 𝑈 (𝑖) given individual action-observation history +𝐻 (𝑖) +𝑡 +. In the beginning of an episode, an initial MA is computed +as: 𝑢 (𝑖) +𝑡0 = 𝜙 (𝑖) (𝐻 (𝑖) +𝑡0 ). At action-making step 𝑘 > 0, the agent gen- +erates a new MA 𝑢 (𝑖) +𝑡𝑘 += 𝜙 (𝑖) (𝐻 (𝑖) +𝑡𝑘 ) if the stop condition is met, +i.e. 𝐻 (𝑖) +𝑡𝑘 +∈ 𝐵(𝑖) (𝑢 (𝑖) +𝑡𝑘−1). Otherwise, the agent continues to use the +previous MA: 𝑢 (𝑖) +𝑡𝑘 = 𝑢 (𝑖) +𝑡𝑘−1. In the time range [𝑡𝑘,𝑡𝑘+1), the agent in- +teracts with the environment with atomic actions sampled from MA: +𝑎(𝑖) +𝑡 +∼ 𝑢 (𝑖) (𝐻 (𝑖) +𝑡 +). Finally, the goal of Dec-POSMDP is to maximize +the accumulative discounted reward: E +��∞ +𝑘=0 𝛾𝑡𝑘 ¯𝑅𝜏 (¯𝑠𝑡𝑘, ¯𝑢𝑡𝑘 )| ¯𝜙, ¯𝑠0 +� +where 𝑡0 = 0 and 𝑡𝑘 = min𝑡 {𝑡 > 𝑡𝑘−1 : 𝐻 (𝑖) +𝑡 +∈ 𝐵(𝑖) (𝑢 (𝑖) +𝑡𝑘−1)} for +𝑘 ≥ 1. A more detailed definition can be found in [31]. +In our asynchronous setting, 𝑡 is the real time, not the discrete +time step as in common synchronous RL. Our setting is more time- +efficient and robust to hardware faults. Take a 2-agent case as an +example (see Fig. 1), in the synchronous setting, the agents can only +transmit data (blue and green arrows) and perform policy inference +(orange arrow) after both of them have finished the previous action +execution. The system execution speed is bottle-necked by the agent +with the longest execution time. Worse still, the whole system will +get stuck if one agent goes offline unexpectedly. By contrast, agents +take actions in a distributed manner in an asynchronous setting. +Each agent can request data from other agents and conduct policy +inference immediately after it finishes its own action execution. +This asynchronous setting is more time-efficient for multi-agent +exploration tasks, and will not be blocked by dynamic changes such +as agents going offline. +3.3 +Connection to Conventional MARL +In the conventional MARL literature [62], the problem formulation +is typically under decentralized partially observable Markov deci- +sion process (Dec-POMDP), which assumes synchronized actions. +In this work, we also focus on the multi-agent setting and assume +a shared reward function and dynamic transitions. However, dif- +ferent from synchronous MARL which assumes all agents execute +actions simultaneously, we consider the asynchronous nature in +the practical multi-robot scenarios. +We will adapt a popular MARL algorithm, Multi-Agent Proximal +Policy Optimization (MAPPO) [62], from the conventional setting +to our asynchronous setting. Conventional MAPPO follows the +Centralized Training and Decentralized Execution (CTDE) para- +digm, in which agents make decisions with individual observations +and update the joint policy with global information in a centralized +manner. Under the framework of Dec-POMDP, MAPPO requires +all agents taking actions synchronously at each discrete time step, +and the state transits according to actions from all agents: 𝑠𝑡 ∼ +𝑃(·|𝑠𝑡−1, ¯𝑎𝑡−1). It aims to find a joint policy ¯𝜋 that maximizes the ac- +cumulated discounted reward E +��∞ +𝑡=0 𝛾𝑡 ¯𝑅(𝑠𝑡, ¯𝑎𝑡)|𝑎(𝑖) +𝑡 +∼ 𝜋 (𝑖) (𝐻 (𝑖) +𝑡 +) +� +. +Different from MAPPO, Async-MAPPO is designed for the asyn- +chronous setting, where there are no centralized environment steps. +4 +METHODOLOGY +To better model the asynchronous nature of real-world multi-agent +exploration problems, we present Asynchronous Coordination Ex- +plorer (ACE). ACE consists of 3 major components: (1) Async- +MAPPO for MARL training, (2) action-delay randomization for +zero-shot generalization in the real world, and (3) multi-tower- +CNN-based policy representation for efficient communication. +4.1 +Async-MAPPO +We extend an on-policy MARL algorithm MAPPO [61] to our asyn- +chronous setting, which we call Async-MAPPO. The pseudo-code +of Async-MAPPO is shown in Algo. 1. Compared with the setting +of MAPPO, both policy execution and data collection are not nec- +essarily time-aligned among different agents, and we implement +the asynchronous action-making and replay buffer as follows. +• We design a bi-level execution scheme. In ACE, agents perform +atomic actions under the guidance of global goals (macro ac- +tions). Instead of receiving the reward, local observation, and + +Figure 1: Comparison of asynchronous and synchronous action making. +states immediately after executing an atomic action, Async- +MAPPO accumulates the reward between action-making steps +and only takes observation and states at each macro action. +• We implement asynchronous buffer insertion, in contrast to +the synchronous scheme in original MAPPO as shown in Fig. 1. +The original MAPPO assumes synchronous execution of all the +agents; in each time step, all the agents take actions simultane- +ously, and the trainer waits for all the new transitions before +inserting them into a centralized data buffer for RL training. +In Async-MAPPO, different agents may not take actions at the +same time (some agents may even get stuck and cannot re- +turn new observations at all), which makes it infeasible for the +trainer to collect transitions in the original synchronous man- +ner. Therefore, we allow each agent to store its own transition +data in a separate cache and periodically push the cached data +to the centralized data buffer. We can then run the standard +MAPPO training algorithm over this buffer. +4.2 +Action-Delay Randomization +When training in traditional simulators, agents can always take exe- +cution steps synchronously without considering different action ex- +ecution costs. Moreover, real-world action delays such as hardware +failure and network blocking are not simulated. These problems +cause a large gap for deploying trained agents from simulation to +reality. To reduce this gap, we apply action-delay randomization +during simulation. In the end of each action-making step, we force +each agent to wait for a random period from 3 to 5 execution steps +in grid-based environments, and from 10 to 15 execution steps in +Habitat before querying the next macro action. +4.3 +Multi-Tower-CNN-Based Policy +The Multi-tower-CNN-based Policy (MCP) is utilized to generate +macro actions, i.e., global goals in ACE. As illustrated in Fig. 3, +MCP consists of 3 parts, i.e., a CNN-based local feature extractor, +an attention-based relation encoder, and an action decoder. +The local feature extractor is a weight-sharing 3-layer CNN +and can extract a 𝐺 × 𝐺 × 4 feature embedding from each agent’s +𝑆 × 𝑆 × 7 local information, which includes one obstacle channel, +Algorithm 1: Async-MAPPO +1 Initialize the policy 𝜋; +2 while 𝑠𝑡𝑒𝑝 ≤ 𝑠𝑡𝑒𝑝𝑚𝑎𝑥 do +3 +set data buffer 𝐷 = {}; +4 +for 𝑖 = 1 to 𝑏𝑎𝑡𝑐ℎ_𝑠𝑖𝑧𝑒 do +5 +Reset the environment; +6 +Create 𝑁 empty caches 𝐶 = [[], . . . , []]; +7 +for 𝑡 = 1 to 𝑇 do +8 +for all agents 𝑖 = 1 to 𝑁 do +9 +if agent 𝑖 replans macro action then +10 +𝑏 ← agent 𝑖’s 𝑏-th macro actions; +11 +𝑠 (𝑖) +𝑏 +← 𝑆𝑡𝑎𝑡𝑒,𝑜 (𝑖) +𝑏 +← 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛; +12 +𝐶𝑖+ = [𝑠 (𝑖) +𝑏−1,𝑜 (𝑖) +𝑏−1,𝑢 (𝑖) +𝑏−1, ˆ𝑅𝜏 (𝑖) +𝑏 +,𝑠 (𝑖) +𝑏 ,𝑜 (𝑖) +𝑏 ]; +13 +𝑝 (𝑖) +𝑏 += 𝜋(𝑜 (𝑖) +𝑏 ); +14 +Update macro action 𝑢 (𝑖) +𝑏 +∼ 𝑝 (𝑖) +𝑏 ; +15 +end +16 +Execute atomic action 𝑎(𝑖) +𝑡 +∼ 𝑢 (𝑖) +𝑏 ; +17 +end +18 +end +19 +Compute reward-to-go and insert data into 𝐷; +20 +end +21 +Update 𝜋 on MAPPO loss; +22 end +one explored region channel, one-hot location channel, one trajec- +tory channel to represent the history trace, and three agent-view +channels of the agent’s local observation. +The agents transmit extracted feature embedding instead of the +raw local information, which greatly reduces communication traffic +by 1 − 𝐺×𝐺×4 +𝑆×𝑆×7 = 1 − +4 +7𝛼2 times where 𝛼 = 𝑆/𝐺. For example, we +adopt 𝐺 = 5 in grid-based environments, thus the communication +traffic reduces ∼ 97% in 𝑆 = 25 maps and ∼ 93% in 𝑆 = 15 maps. + +Synchronous Action Making +MAPPO +m +u5 +u2, 02, r2 +u6, 06, r6 +U1, O1, r1 +Buffer 1 +Agent 1 +u +U5 +U6, 06,6 +u1, 01,r1 +u2, 02,r2 +Agent 2 +Buffer 2 +Asynchronous Action Making +Async-MAPPO +(o1,r) +2,T2 +u2 +u1 +U2, 02, r2 +u1, O1, r1 +Buffer 1 +Agent 1 +u1 +u2 +u3 +u3, 03,r3 +Agent 2 +u1, 01,r1 +u2, 02,2 +Buffer 2 +01,r1 +Uk Uk +Data transmission from Agent 2 to Agent 1 +Ok Ok +Policy Inference +rkrk +Data transmission from Agent 1 to Agent 2 +askFigure 2: Overview of Asynchronous Coordination Explorer (ACE). +The relation encoder aims to aggregate the extracted feature +maps from different agents to better capture the intra-agent in- +teractions. In team-based exploration, an agent should not only +spot undiscovered areas but also inter-teammates’ movement for +better scheduling among agents. We adopt a simplified Trans- +former [47] block as the team-size-invariant relation encoder. In- +spired by the vision transformer model [14], we apply multi-head +cross-attention [46] to derive a single team-size-invariant represen- +tation of size 𝐺 × 𝐺 × 4, as shown in Fig. 3. +Finally, the action decoder predicts the agent’s policy from the +aggregated representation as a multi-variable Categorical distribu- +tion to select a grid cell 𝑔 from a plane as the global goal (𝑢𝑥,𝑢𝑦). +Note that in Habitat, in order to produce accurate global goals, we +adopt a spatial action space with three separate action heads, i.e., +two discrete region heads for choosing a grid cell 𝑔, which are the +same as grid-based environments, and two additional continuous +point heads for outputting a coordinate (Δ𝑥, Δ𝑦), indicating the +relative position of the global goal within the selected region 𝑔. +Details of MCP in Habitat can be found in Appendix A.2. +Figure 3: Workflow of Multi-tower-CNN-based Policy (MCP), +including a CNN-based local feature extractor, a relation en- +coder, and an action decoder. +4.4 +Overall Architecture +As shown in Fig. 2, each agent observes the local information and +requests the latest feature embedding from other agents, which +is output by the weight-sharing local feature extractor, at each +action-making step. That is, agents only need to transfer the low- +dimensional feature embedding, instead of the entire local infor- +mation. The multi-tower-CNN-based policy, which is trained by +Async-MAPPO, generates the next macro action, i.e., global goal, at +each action-making step, and the agent performs path planning on +the local map according to the global goal, outputting the atomic +action at each time step. Note that agents could go offline in multi- +agent tasks due to unexpected network communication traffic or +hardware failure. +5 +ENVIRONMENT DETAILS +Here we give details of the environments we adopted in this work, +including the environment setting, the observation space and the +action space of ACE, as well as the designed reward function. +5.1 +Environment Setting +Grid-based scenario: As shown in Fig. 4, we implement a multi- +agent exploration task based on the GridWorld simulator [8], which +was originally designed for synchronous settings. We consider +two different map sizes, which are 15 × 15 with 4 ∼ 9 random +rooms and 25 × 25 with 4 ∼ 25 random rooms. All the agents +are uniform randomly spread over the map in the beginning. The +local information of each robot is fed to the RL-trained policy or +planning-based methods to generate a global goal and 𝐴★ algorithm +is utilized to plan 5 atomic actions on the local map to follow the +global goal. +We also set up a 15 × 15 real-world grid map which is the same +as the grid-based simulation, and each grid is 0.31m long, as shown +in Fig. 4. Our robots are equipped with Mecanum steering and an +NVIDIA Jetson Nano processor. The locations and poses of robots +are tracked by OptiTrack cameras and the Motive motion capture +software. After training a policy in the grid-based simulator under +15 × 15 map with random rooms, we directly deploy it to the real- +world robot system. Each real robot executes in a distributed and +asynchronous manner. The robot adopts a request-send mechanism +to obtain the newest feature embedding of other agents through +ROS topic upon finishing all atomic actions. +Habitat: We adopt map data from the Gibson dataset [55] while +the visual signals and dynamics are simulated by Habitat [39]. We +follow the same environment configuration in [63] and use a pre- +trained neural SLAM model to predict the robot pose and the local +map. Full details of Habitat can be found in Appendix A. We also +make sure that the birthplaces of agents are set to be close enough, +i.e., agents are randomly scattered in a circle with a radius of 1 meter, +so that the exploration task would be sufficiently challenging for +learning. + +Agent 1 +Atomic +Local Feature +Feature Embedding +Local +Macro +Action +Extractor +Info. +Action +Action +Path +Communication +Agent k +between agents +Generator +Planning +Local +Local Feature +Feature Embedding +Extractor +Action-delay +Info. +Randomization +Multi-Tower-CNN-based Policy +Local +Map +Trained by Async-MAPPOAgent1 +Agent k +CNN Block +CNN Block +: 3 Layers +: 3 Layers +CNN Block +CNN Block +Local Feature Extractor +Local Feature Extractor +communication +Attn-based +Action Decoder +Relation Encoder +x Head +N-1 +CNN +Projector +y HeadFigure 4: The illustration of the grid-based simulator and real-world robot system. +5.2 +Observation Space +The input of RL-trained MCP is an 𝑆 × 𝑆 image with 7 channels, +where 𝑆 is the max size of the map. The channels represent obsta- +cles, the explored mask, the agent location, the trajectory, and three +𝐻×𝑊 agent-view. Note that each agent only maintains its locally ob- +served information, which is memory and communication-efficient +for real-world deployment. +5.3 +Action Space +The overall exploration framework is hierarchical, with a global +goal (macro action) followed by several atomic actions towards the +goal. The action of the policy is to generate a global goal (𝑢𝑥,𝑢𝑦) +chosen in the map, representing a discrete grid in grid-based envi- +ronments or a continuous location in Habitat [39]. The available +atomic actions are moving forward, turning left, and turning right +provided by the simulator. +5.4 +Reward Function +The team-based reward function is the sum of the coverage reward, +success reward, and overlap penalty. Let 𝑅𝑎𝑡𝑖𝑜𝑡 denote the total +coverage ratio at time 𝑡, 𝐸𝑥𝑝𝑡𝑎 be the explored map by agent 𝑎 and +𝐸𝑥𝑝𝑡 denote the merged explored map by all agents. Both 𝐸𝑥𝑝𝑡 and +𝐸𝑥𝑝𝑡𝑎 are sets of explored areas. The reward terms are defined as +follows. +• Coverage Reward: It is proportional to the size of the newly +discovered region by the team 𝐸𝑥𝑝𝑡\𝐸𝑥𝑝𝑡−1. +• Success Reward: Agent 𝑎 gets a success reward of 𝑅𝑎𝑡𝑖𝑜𝑡 +when 𝐶% coverage ratio is reached, which 𝐶 = 98 in the +grid-like simulator and 𝐶 = 90 in Habitat1. +• Overlap Penalty: The overlap penalty 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is designed +to penalize repetitive exploration and encourage cooperation +with others. It is defined as +𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = +� +−𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 × 0.01, 𝑅𝑎𝑡𝑖𝑜𝑡 < 0.9 +0, 𝑅𝑎𝑡𝑖𝑜𝑡 ≥ 0.9 +, +where 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is the increment of the overlapped explored +area between agent 𝑎 and other agents. The overlapped area +between agent 𝑎 and agent 𝑤 is 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤 = 𝐸𝑥𝑝𝑡𝑎 ∩ 𝐸𝑥𝑝𝑡𝑤, +and 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � +𝑤∈{1,···,𝑛}\{𝑎} 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤\𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡−1 +𝑎,𝑤. +1Maps in Habitat are harder than in the grid-based simulator, leading to differences in +the success rate threshold. +6 +EXPERIMENT RESULTS +6.1 +Training Details +In the simulation, every RL policy is trained with 50𝑀 steps in +the grid-based simulator and 100𝑀 steps in Habitat over 3 random +seeds. All results are averaged over a total of 300 testing episodes +(100 episodes per random seed). As for real-world testing, we ran- +domly generate 10 maps of size 15 × 15 and test 5 times for each +map. In synchronous action-making cases, agents perform action- +making at the same time and wait for all other agents to finish. In +asynchronous action-making cases, agents do not wait for others +and perform both macro and atomic actions independently. +6.2 +Evaluation Metrics +The most important metric in our experiment is Time, which is the +running time for the agents to reach a 𝐶% coverage ratio. We report +wall-clock time in the real world, and report an estimated statistical +running time in simulation: turning left or right takes 0.5𝑠; stepping +forward takes 1𝑠. Policy inference time is fixed to 0.1𝑠 for both RL +and planning-based methods thus the results can better reflect the +difference between asynchronous and synchronous settings. +We also consider 3 additional statistics metrics to capture dif- +ferent characteristics of a particular exploration strategy. These +metrics are only for analysis, and we primarily focus on Time as +our performance criterion. +• Accumulative Coverage Score (ACS): The overall exploration +progress throughout an episode computed as𝐴𝑇 = +∫ 𝑇 +𝑡=0 𝑅𝑎𝑡𝑖𝑜𝑡, +where 𝑇 is the max running time. Higher ACS implies faster +exploration. +• Coverage: the final ratio of explored area when an episode +terminates. Higher implies more exhaustive exploration. +• Overlap: the ratio of the overlapped region explored by mul- +tiple agents to the current explored area when C% coverage +is reached. Lower Overlap implies better credit assignment. +All metrics are calculated with the running time 𝑡, i.e., the esti- +mated statistical time in simulation and wall-clock time in the real +world. Each score is reported as "mean (standard deviation)". +6.3 +Baselines +We consider 4 popular planning-based competitors, including a +utility-maximizing method (Utility) [25], a search-based nearest- +frontier method (Nearest) [57], a rapid-exploring-random-tree-based +method (RRT) [45], and an artificial potential field method (APF) [64] +which applies resistance forces among agents as a cooperation + +Feature +Feature +Embedding +Embedding +Global Goal +Local Info. +Local Info. +Global +oa +Global Goal +A* Algo. +Atomic Action +Local MapMap Size +Methods +Synchronous Action Making +Asynchronous Action Making +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +15 × 15 +Utility +40.81(0.94) +0.45(0.02) +1.00(0.00) +88.80(0.08) +35.75(0.99) +0.42(0.01) +1.00(0.00) +90.01(0.14) +Nearest +25.44(0.53) +0.17(0.01) +1.00(0.00) +91.60(0.17) +22.59(0.34) +0.17(0.01) +1.00(0.00) +92.47(0.17) +RRT +28.86(0.99) +0.18(0.01) +1.00(0.00) +91.46(0.03) +25.85(0.36) +0.19(0.01) +1.00(0.00) +92.36(0.08) +APF +24.95(0.76) +0.17(0.01) +1.00(0.00) +91.57(0.39) +21.56(0.43) +0.17(0.01) +1.00(0.00) +92.52(0.37) +Voronoi +48.57(3.64) +0.33(0.01) +1.00(0.00) +86.43(0.33) +42.94(4.05) +0.20(0.01) +1.00(0.00) +88.16(0.57) +MAPPO +24.75(0.45) +0.08(0.02) +1.00(0.00) +92.39(0.19) +21.92(0.90) +0.09(0.01) +1.00(0.00) +93.18(0.17) +ACE +21.76(0.79) +0.07(0.00) +1.00(0.00) +92.54(0.21) +18.66(0.79) +0.07(0.01) +1.00(0.00) +93.39(0.14) +25 × 25 +Utility +189.38(0.93) +0.46(0.05) +0.93(0.01) +139.31(1.56) +183.71(1.59) +0.50(0.04) +0.95(0.01) +144.64(1.42) +Nearest +113.48(1.73) +0.24(0.01) +1.00(0.00) +161.40(0.63) +99.52(2.00) +0.24(0.01) +1.00(0.00) +166.53(0.91) +RRT +120.15(2.29) +0.20(0.01) +1.00(0.00) +164.35(0.64) +105.64(1.69) +0.21(0.01) +1.00(0.00) +168.33(0.27) +APF +101.41(0.78) +0.23(0.01) +1.00(0.00) +162.46(0.64) +90.00(1.48) +0.23(0.01) +1.00(0.00) +166.82(0.81) +Voronoi +131.65(0.41) +0.23(0.01) +1.00(0.00) +160.92(0.33) +117.14(0.13) +0.20(0.01) +1.00(0.00) +165.35(0.27) +MAPPO +90.15(1.08) +0.08(0.01) +1.00(0.00) +168.54(0.58) +82.55(2.70) +0.09(0.01) +1.00(0.00) +171.72(0.27) +ACE +83.34(0.44) +0.06(0.00) +1.00(0.00) +170.03(0.49) +74.36(2.93) +0.06(0.00) +1.00(0.00) +173.16(0.72) +Table 1: Performances of different methods under 2-agent synchronous and asynchronous settings in the grid-based simulator. +Metrics +Utility +Nearest +RRT +APF +ACE +3 ⇒ 2 +Time ↓ +139.33(1.79) +76.53(1.16) +81.86(1.14) +74.80(2.11) +67.50(1.42) +Overlap ↓ +0.48(0.03) +0.30(0.01) +0.27(0.00) +0.32(0.01) +0.22(0.00) +Coverage ↑ +0.94(0.01) +1.00(0.00) +1.00(0.00) +1.00(0.00) +1.00(0.00) +ACS ↑ +110.56(1.11) +126.03(0.39) +126.75(0.18) +125.15(0.58) +128.49(0.37) +4 ⇒ 3 +Time ↓ +96.15(0.46) +53.68(1.01) +55.33(0.82) +52.26(0.68) +48.88(1.82) +Overlap ↓ +0.40(0.03) +0.36(0.00) +0.34(0.01) +0.38(0.01) +0.33(0.07) +Coverage ↑ +0.92(0.01) +1.00(0.00) +1.00(0.00) +1.00(0.00) +1.00(0.00) +ACS ↑ +73.28(1.55) +84.05(0.57) +84.08(0.41) +83.60(0.41) +84.78(0.77) +Table 2: Performance of different methods with decreased team size on 25 × 25 maps in the grid-based simulator. +Methods +Utility +Nearest +RRT +APF +MAPPO +ACE +Time(s) +60.25(0.16) +38.72(0.12) +55.89(0.24) +52.64(0.23) +28.48(0.12) +25.61(0.10) +Table 3: Running time of different methods when the coverage ratio reaches 100% in the real-world robot system. +mechanism. Note that APF is a multi-agent baseline while the +other three are commonly used for single-agent tasks. Moreover, all +baselines use global information to do planning after every macro +action. Different from ACE, they are not learning-based and are all +designed for asynchronous execution. +6.4 +Grid-Based Scenario +6.4.1 +Main Results. Experiment results with 2 agents in the grid- +based simulator under synchronous and asynchronous training are +provided in Table 1. In both settings, ACE outperforms planning- +based baselines with ≥ 10% less Time, full Coverage, and higher ACS. +Although APF encourages cooperation, its Overlap is still higher +than ACE, demonstrating ACE’s superiority in discovering efficient +cooperation strategies. Comparing ACE with MAPPO, which is +trained in a synchronous manner, ACE demonstrates similar 𝐴𝐶𝑆 to +MAPPO with less Time and Overlap, which indicates the robustness +of ACE to realistic execution with randomized action delay. Results +of 3 agents can be found in appendix D. +6.4.2 +Generalization to Agent Lost. We further consider an- +other setting where the team size decreases within an episode on +map size 25 × 25 to emulate the real-world scenarios with hardware +failure and to examine whether our learned policies can generalize +to these extreme cases during execution. “𝑁1 ⇒ 𝑁2” denotes a +scenario with 𝑁1 agents at the beginning and only 𝑁2 agents alive +after 50% coverage. As shown in Table 2, ACE demonstrates 10% +less Time than other baselines and obtains the highest ACS and + +lowest Overlap, indicating ACE’s effective zero-shot adaptation to +extreme situations where some agents go offline. +6.4.3 +Real-World Robot System. In this part, we present the +running time of different methods with 2 agents in real-world explo- +ration tasks on 15×15 maps, which are running in an asynchronous +manner. The deployment pipeline is described in Sec. 5.1. As shown +in Table 3, two RL-based methods, MAPPO and ACE, outperform +the planning-based baselines with a large margin according to the +total exploration time. In particular, ACE reduces 33.86% real-world +exploration time than the fastest planning-based method Nearest. +Besides, ACE reduces 10.07% running time compared with MAPPO, +proving that combining action-delay randomization with Async- +MAPPO indeed improves the efficiency of multi-agent exploration. +6.5 +Habitat Results +6.5.1 +Main Results. We extend ACE to a vision-based environ- +ment, Habitat. Table 4 shows the performance of different meth- +ods under 2-agent asynchronous action-making settings. Despite +having higher Overlap due to more exhaustive exploration, ACE +outperforms planning-based baselines with ≥ 28% less Time, higher +Coverage and ACS. Compared with synchronous MAPPO, ACE still +shows higher Coverage and ACS with less Time, demonstrating the +effectiveness of ACE in more complicated vision-based tasks. +6.5.2 +Generalization to Agent Lost. We also consider the set- +ting of decreased team sizes in Habitat, and we follow the same +experimental setup as for the grid-based simulations. Table 5 shows +the performance of different methods with decreased team size (2 +⇒ 1). ACE demonstrates 5.3% less Time than other baselines and +obtains the highest Coverage and ACS with comparable Overlap, +which indicates the ACE’s ability to generalize to agent lost. +Methods +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +Utility +273.83(37.80) +0.84(0.03) +0.83(0.08) +186.17(17.43) +Nearest +220.25(30.23) +0.59(0.04) +0.94(0.03) +180.05(7.37) +RRT +177.29(17.16) +0.63(0.06) +0.97(0.02) +187.35(6.86) +APF +218.45(24.64) +0.67(0.04) +0.94(0.02) +188.62(7.67) +MAPPO +133.23(17.72) +0.68(0.09) +0.97(0.01) +201.33(7.98) +ACE +127.62(8.55) +0.78(0.07) +0.98(0.01) +213.81(9.33) +Table 4: Performance of different methods under 2-agent +asynchronous action-making settings in Habitat. +Methods +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +Utility +281.09(32.25) +0.48(0.05) +0.84(0.08) +153.02(12.38) +Nearest +309.76(9.83) +0.40(0.03) +0.85(0.05) +149.43(4.55) +RRT +260.31(27.92) +0.35(0.03) +0.92(0.02) +155.23(6.04) +APF +309.88(6.63) +0.42(0.01) +0.79(0.01) +143.54(0.83) +MAPPO +262.92(19.84) +0.35(0.04) +0.90(0.03) +160.90(6.68) +ACE +246.38(19.26) +0.36(0.03) +0.92(0.03) +164.32(8.23) +Table 5: Performance of different methods with decreased +team size in Habitat. +6.6 +Ablation Studies +In this section, we analyze the sensitivity of communication size +and action-delay randomization based on the grid-like simulator +through ablation studies. +6.6.1 +Sensitivity Analysis of Communication Size. We study +the exploration performances in different communication traffic +scenarios, including: +• No Comm.: The attention-based relation encoder is removed. +Therefore, agents can only use their own local information to +perform macro actions. This is the lower bound of different +communication traffic. +• Comm. (0.25x): The number of channels output by the CNN +local feature extractor is set to 1, which is a quarter of the +original 4 channels. +• Comm. (0.5x): The number of CNN local feature extractor +output channels is set to 2. +• Perf. Comm.: Agents use merged observation from all the +agents as the input of the CNN local feature extractor. +Table 6 summarizes the performances on different communica- +tion traffic with 2 agents on 25 × 25 maps. More communication +between agents generally leads to better exploration efficiency, as is +shown by the decreasing Time and increasing ACS from “No Comm.” +to “Comm. (0.25x)”, “Comm. (0.5x)” and “Perf. Comm.”. Moreover, +the behavior metric Overlap in these four scenarios shows better +cooperation efficiency with more communication. Note that ACE +performs even better than “Perf. Comm.” with strictly less commu- +nication, demonstrating the effectiveness of the feature embedding +extracted from our CNN policy for decision-making. +Methods +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +No Comm. +159.26(2.18) +0.37(0.01) +0.93(0.01) +151.87(1.82) +Comm. (0.25x) +110.92(1.33) +0.11(0.01) +0.99(0.00) +167.60(0.71) +Comm. (0.5x) +83.77(1.38) +0.09(0.00) +1.00(0.00) +170.90(0.60) +Perf. Comm. +75.62(0.84) +0.06(0.01) +1.00(0.00) +173.15(0.53) +ACE +74.36(2.93) +0.06(0.00) +1.00(0.00) +173.16(0.72) +Table 6: Performance with different communication traffic. +Intervals +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +Rand (1-10) +60.24(0.35) +0.51(0.00) +1.00(0.00) +82.44(0.31) +Rand (5-10) +56.41(1.51) +0.47(0.01) +1.00(0.00) +83.46(0.35) +Rand (1-5) +55.69(1.23) +0.43(0.01) +1.00(0.00) +83.57(0.45) +ACE (3-5) +48.88(1.82) +0.33(0.07) +1.00(0.00) +84.78(0.77) +Table 7: Performance of different action-delay intervals. +6.6.2 +Sensitivity Analysis of Action-Delay Randomization. +We further study the impact of the different random action-delay +intervals. Besides the randomization interval stated in Sec. 4.2, we +consider 3 different choices of action-delay intervals during train- +ing, “Rand (1-10)”, “Rand (5-10)”, and “Rand (1-5)”. “Rand (𝑀1 −𝑀2)” +means each macro action execution is delayed for a random number + +of simulation steps uniformly sampled from [𝑀1, 𝑀2]. We empir- +ically find that these variants have similar performance in most +simple test settings, while ACE outperforms them in some extreme +cases. To better illustrate the effect of different action-delay choices, +we present the results in the “4 ⇒ 3” setting, an extreme scenario +with agent loss. As shown in Table 7, ACE consumes the least Time +and achieves the highest ACS. The results show that action-delay +randomization works best with a proper randomization interval, +while a large randomization interval adds high uncertainty during +training and hurts the final performance. +7 +CONCLUSION +To bridge the gap between synchronous simulator and asynchro- +nous action-making process in real-world multi-agent exploration +task, we propose a novel real-world multi-robot exploration so- +lution, Asynchronous Coordination Explorer (ACE) to tackle this +challenge. In ACE, Multi-agent PPO (MAPPO) is extended to the +asynchronous action-making setting for effective training, and an +action-delay-randomization technique is applied for better gener- +alization to the real world. Besides, each agent equipped with a +team-size-invariant Multi-tower-CNN-based Policy (MCP), extracts +and broadcasts the low-dimensional feature embedding to accom- +plish efficient intra-agent communication. Both simulation and real- +world results show that ACE improves 10% exploration efficiency +compared with classical approaches in grid-based environments. +We also extend ACE to a vision-based testbed Habitat, where ACE +outperforms planning-based baselines with ≥ 28% less exploration +time. Although we aim at the sim-to-real problem caused by mul- +tiple agents executing tasks asynchronously, there are still many +issues that have not been fully considered, such as communication +errors, localization errors, and sensor errors. we leave these issues +as our future work. +ACKNOWLEDGMENT +This research was supported by National Natural Science Foun- +dation of China (No.U19B2019, 62203257, M-0248), Tsinghua Uni- +versity Initiative Scientific Research Program, Tsinghua-Meituan +Joint Institute for Digital Life, Beijing National Research Center for +Information Science, Technology (BNRist), and Beijing Innovation +Center for Future Chips and 2030 Innovation Megaprojects of China +(Programme on New Generation Artificial Intelligence) Grant No. +2021AAA0150000. + +We would suggest to visit https://sites.google.com/view/ace- +aamas for more information. +A +HABITAT DETAILS +A.1 +Pipeline +In Habitat experiments, we use Neural SLAM to represent the +scene with a top-down mapping, and thus the explored regions and +discovered obstacles are expressed with a top-down 2D mapping. +Then the planner, MCP, schedules a global goal according to the +explored information. Finally, the agent uses a local policy that +guides the agent to the chosen global goal. +A.2 +MCP Details +A.2.1 +Input Representation. In MCP, each CNN-based feature ex- +tractor’s input map, i.e. one feature extractor per agent, is a 240×240 +map with 7 channels, including +• Obstacle channel: Each pixel value denotes the probability +of being an obstacle. +• Explored region channel: A probability map for each pixel +being explored. +• One-hot location channel: The only non-zero grid denotes +the position of the agent. +• Trajectory channel: This is used to represent the agent’s +history trace. To reflect time-passing, this channel is updated +in an exponentially decaying weight manner. More precisely, +an agent’s trajectory channel 𝑉 𝑡 at timestep 𝑡 is updated as +following, +𝑉 𝑡𝑥,𝑦 = +� +1 +if agent is near (𝑥,𝑦) +𝜀𝑉 𝑡−1 +𝑥,𝑦 +otherwise +where the agent is regarded as near (𝑥,𝑦) when the grid-level +distance between them is less than 3. +• Three local observations channels: the RGB images of agent- +view local observations. +All mapping-related channels are transformed into a world-view +to save MCP from learning to align all agents’ information, which +might involve rotation and translation. +A.2.2 +Action Space. Through MCP, every agent chooses a long- +term goal (a point) from the whole space. A natural choice is to +model the agent’s policy as a multi-variable Gaussian distribution +to select points from a plane. However, in our exploration setting, +an agent’s policy could be extremely multi-modal especially during +early stage of exploration since many points could induce similar +effects on the agent’s path. To fix this issue, we adopt a hierarchical +design. We first divide the whole map into 8×8 regions, from which +the agent chooses a desired region. Then, similar to previous choice, +a point in this region is selected as the long-term goal. Formally, +the policy of agent 𝑘, could be described as, +𝑔𝑟,𝑔𝑐 ∼ Cat(𝑟𝜃,𝑟), Car(𝑟𝜃,𝑐) +𝑥𝑙,𝑦𝑙 ∼ N (𝜇𝜃, Σ𝜃) +𝑥 ′ +𝑙 = sigmoid(𝑥𝑙), 𝑦′ +𝑙 = sigmoid(𝑦𝑙) +𝑥𝑔 = (𝑔𝑟 + 𝑥 ′ +𝑙 )/8, 𝑦𝑔 = (𝑔𝑐 + 𝑦′ +𝑙 )/8 +where 𝜃 is the model parameter, Cat(𝑟𝜃,𝑟), Car(𝑟𝜃,𝑐) represent two +categorical distributions for choosing the region, 𝑔𝑟,𝑔𝑐 are the row +and column indexes of the sampled region, 𝜇𝜃, Σ𝜃 are the mean and +covariance matrix of the Gaussian distribution to choose the local +point within the region and (𝑥𝑔,𝑦𝑔) is the final sampled long-term +goal. +A.2.3 +Network Architecture. Our models are trained and imple- +mented using Pytorch [34]. We reuse the neural SLAM module and +local policy from [38], and we briefly summarize their architectures +here. Neural SLAM module has two components, a Mapper and a +Pose Estimator. The Mapper is composed of ResNet18 convolutional +layers, 2 fully-connected layers, and 3 deconvolutional layers. The +Pose Estimator consists of 3 convolutional layers and 3 fully con- +nected layers. Similarly, the local policy has Resnet18 convolutional +layers, fully-connected layers, and a recurrent GRU layer. +Table 8: CNN Block Hyperparameter in Habitat. +Layer +Out Channels +Kernel Size +Stride +Padding +1 +32 +3 +1 +1 +2 +64 +3 +1 +1 +3 +128 +3 +1 +1 +4 +64 +3 +1 +1 +5 +32 +3 +2 +1 +The Multi-tower-CNN-based Policy (MCP) has three main com- +ponents, including CNN-based feature extractors, a transformer- +based relation encoder, and an action decoder. +(1) Each CNN-based feature extractor contains 5 consecutive +CNN blocks. Their corresponding parameters are shown in +tab. 8. We use ReLU as the activation function. After each +of the front four CNN blocks, we attach a 2D max pooling +layer with 2 kernel sizes. +(2) The transformer-based relation encoder is used to better +capture spatial information. The attention layer has 4 heads, +with 32 dimension sizes for each head. +(3) The action decoder simply uses a CNN projector and lin- +ear transformations to turn the feature map output from +the transformer-based relation encoder to corresponding +logits for Categorical distribution (region head) and means +and standard deviations of the Gaussian distribution (point +heads). +The critic also utilizes a similar architecture as MCP, except for +replacing the action decoder with fully-connected layers to output +value predictions. +B +TRAINING DETAILS +B.1 +Reward Function +We use 3 kinds of team-based rewards, including a coverage reward, +a success reward, and an overlap penalty. In the following part, +𝑅𝑎𝑡𝑖𝑜𝑡 denotes the total coverage ratio at timestep 𝑡. Let 𝐸𝑥𝑝𝑡 be +the merged explored map at timestep 𝑡 and 𝐸𝑥𝑝𝑡 +𝑘 be the explored +map of agent 𝑘. Ideally, both 𝐸𝑥𝑝𝑡 and 𝐸𝑥𝑝𝑡 +𝑘 can be considered as +sets of explored points. Then define Δ𝐸𝑥𝑝𝑡 = 𝐸𝑥𝑝𝑡\𝐸𝑥𝑝𝑡−1 as the +newly discovered region at timestep𝑡 by the whole team with regard +of the merged explored area. Specially, we model an individual’s +effort by Δ𝐸𝑥𝑝𝑡 +𝑘 = 𝐸𝑥𝑝𝑡 +𝑘\𝐸𝑥𝑝𝑡−1, that is agent 𝑘’s contribution at + +common hyperparameters +value +gradient clip norm +10.0 +GAE lambda +0.95 +gamma +0.99 +value loss +huber loss +huber delta +10.0 +mini batch size +batch size / mini-batch +optimizer +Adam +optimizer epsilon +1e-5 +weight decay +0 +network initialization +Orthogonal +use reward normalization +True +use feature normalization +True +learning rate +2.5e-5 +Table 9: Async-MAPPO hyperparameters +timestep 𝑘 based on the whole team’s previous exploration. Note +that Δ𝐸𝑥𝑝𝑡 +𝑘 is not defined based on the agent’s previous exploration, +i.e. Δ𝐸𝑥𝑝𝑡 +𝑘 ≠ 𝐸𝑥𝑝𝑡 +𝑘\𝐸𝑥𝑝𝑡−1 +𝑘 +. +• Coverage Reward: The coverage reward consists of two +parts, a team coverage reward, and an individual coverage +reward. The team coverage reward is proportional to the +area of the exploration increment Δ𝐸𝑥𝑝𝑡. The individual +coverage reward, as the name suggests, is proportional to +the individual contribution, i.e., the area of Δ𝐸𝑥𝑝𝑡 +𝑘. +• Success Reward: Agent 𝑎 gets a success reward of 𝑅𝑎𝑡𝑖𝑜𝑡 +when 𝐶% coverage rate is reached, which 𝐶 = 98 in the +grid-based simulator and 𝐶 = 90 in Habitat2. +• Overlap Penalty: The overlap penalty 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is designed +to encourage agents to reduce repetitive exploration and +learn to cooperate with others. +𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = +� +−𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 × 0.01, 𝑅𝑎𝑡𝑖𝑜𝑡 < 0.9 +0, 𝑅𝑎𝑡𝑖𝑜𝑡 ≥ 0.9 +, +where 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is the increment of the overlapped explored +area between agent 𝑎 and other agents. The overlapped area +between agent 𝑎 and agent 𝑤 is 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤 = 𝐸𝑥𝑝𝑡𝑎 ∩ 𝐸𝑥𝑝𝑡𝑤, +and 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � +𝑤∈{1,···,𝑛}\{𝑎} 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤\𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡−1 +𝑎,𝑤. +The final team-based reward is simply the sum of all these terms. In +Habitat, all the explored and obstacle maps are represented under +discretization of 5𝑐𝑚, and all the area computations are taken in +𝑚2. +B.2 +Hyperparameters +The hyperparameters for Async-MAPPO are as shown in Table 9. +C +PLANNING-BASED BASELINES +We demonstrate some details about the 5 planning-based baselines +here. +Utility: A method that always chooses frontier that maximizes +information gain [3]. +2Maps in Habitat are harder than in the grid-based simulator, leading to differences in +the success rate threshold. +Nearest: A method that always chooses the nearest frontier as +global goal [58]. The distance to a frontier is computed using the +breadth-first search on the occupancy map. +APF: Artificial Potential Field (APF) [64] plans a path for each +agent based on a computed potential field. The end of the path, +which is a frontier, is the selected goal. For every agent, an arti- +ficial potential field 𝐹 is computed in the discretized map, with +consideration of distance to frontiers, presence of obstacles, and +potential exploration reward. APF also introduces resistance force +as a simple mechanism. Finally, the path is generated along the +fastest decreasing direction of 𝐹, starting from the agent’s current +position. +RRT: This baseline is adopted from [45]. Rapid-exploring Ran- +dom Tree (RRT) is originally a path-planning algorithm based on +random sampling and is used as a frontier detector in [45]. Af- +ter collecting enough frontiers through random exploration, RRT +chooses frontier 𝑝 with the largest utility 𝑢(𝑝) = 𝐼𝐺(𝑝) − 𝑁 (𝑝), +where 𝐼𝐺(𝑝) and 𝑁 (𝑝) are respectively the normalized information +gain and navigation cost of 𝑝. +Voronoi [20] The voronoi-based method first partitions the map +via voronoi partition and assigns components to agents so that each +agent owns parts that are closest to it. Then each agent finds its +own global goal by finding a frontier point with largest potential +as in Utility within its own partition. +In Habitat experiment, to avoid visually blind areas and ensure +that selected frontiers are far enough, the area within 2.5𝑚 from +each agent is considered explored when making global planning. +The information gain of a frontier 𝑝 is computed as the number +of unexplored grids within 1.5𝑚 to 𝑝. All these baselines do re- +planning every 15 environment steps. +Pseudocode of APF is shown in 3. Line 6-12 computes the resis- +tance force between every pair of agents where 𝐷 is the influence +radius. In lines 13-18, distance maps starting from cluster centers +are computed, and the corresponding reciprocals are added into the +potential field so as one agent approaches the frontier, the potential +drops. Here 𝑤𝑐 is the weight of cluster 𝑐, which is the number of +targets in this cluster. Consequently, an agent would prefer to seek +frontiers that are closer and with more neighboring frontiers. Lines +20-25 show the process of finding the fastest potential descend- +ing path, at each iteration, the agent moves to the cell with the +smallest potential among all neighboring ones. 𝑇 is the maximum +number of iterations, and 𝐶𝑟𝑒𝑝𝑒𝑎𝑡 is the repeat penalty to avoid +agents wandering around cells with the same potentials. +Pseudocode of RRT is shown in Algo. 2. In each iteration, a ran- +dom point 𝑝 is drawn and a new node 𝑡 is generated by expanding +from 𝑠 to 𝑝 with distance 𝐿, where 𝑠 is the closest tree node to 𝑝. If +segment (𝑠,𝑡) has no collision with obstacles in 𝑀, 𝑡 is inserted into +the target list or the tree according to whether 𝑡 is in the unexplored +area or not. Finally, the goal is chosen from the target list with the +largest utility 𝑢(𝑐) = 𝐼𝐺(𝑐) − 𝑁 (𝑐) where 𝐼𝐺(𝑐) is the information +gain and 𝑁 (𝑐) is the navigation cost. 𝐼𝐺(𝑐) is computed by the +number of unexplored grids within 1.5𝑚 to 𝑐, as mentioned above. +𝑁 (𝑐) is computed as the Euclidean distance between the agent lo- +cation and point 𝑐. To keep these two values at the same scale, we +normalize 𝐼𝐺(·) and 𝑁 (·) to [0, 1] w.r.t all cluster centers. + +Algorithm 2: Rapid-exploring Random Tree +Require: Map 𝑀 and agent location 𝑙𝑜𝑐. +Ensure: Selected frontier goal +1: 𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ← {𝑙𝑜𝑐},𝑇𝑎𝑟𝑔𝑒𝑡𝑠 ← {} +2: 𝑖 ← 0 +3: while 𝑖 < 𝑇 and |𝑇𝑎𝑟𝑔𝑒𝑡𝑠| < 𝑁𝑡𝑎𝑟𝑔𝑒𝑡 do +4: +𝑖 ← 𝑖 + 1 +5: +𝑝 ← a random point +6: +𝑠 ← arg min𝑢∈𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ||𝑢 − 𝑝||2 +7: +𝑡 ← 𝑆𝑡𝑒𝑒𝑟 (𝑠, 𝑝, 𝐿) +8: +if 𝑁𝑜_𝐶𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛(𝑀,𝑠,𝑡) then +9: +if 𝑡 lies in unexplored area then +10: +𝑇𝑎𝑟𝑔𝑒𝑡𝑠 ← 𝑇𝑎𝑟𝑔𝑒𝑡𝑠 + {𝑡} +11: +else +12: +𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ← 𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 + {𝑡} +13: +end if +14: +end if +15: end while +16: 𝐶 ← clusters of points in 𝑇𝑎𝑟𝑔𝑒𝑡𝑠. +17: 𝑔𝑜𝑎𝑙 ← arg min𝑐 ∈𝐶 𝐼𝐺(𝑐) − 𝑁 (𝑐) +18: return 𝑔𝑜𝑎𝑙 +Algorithm 3: Artificial Potential Field(APF) +Require: Map 𝑀, number of agents 𝑛 and agent locations +𝑙𝑜𝑐1 . . .𝑙𝑜𝑐𝑛. +Ensure: Selected goals +1: 𝑃 ← frontiers in 𝑀 +2: 𝐶 ← clusters of frontiers 𝑃 +3: 𝑔𝑜𝑎𝑙𝑠 ← an empty list +4: for 𝑖 = 1 → 𝑛 do +5: +𝐹 ← zero potential field, i.e., a 2d array +6: +for 𝑗 = 1 → 𝑛 do +7: +for empty grid 𝑝 ∈ 𝑀 do +8: +if 𝑗 ≠ 𝑖 and ||𝑝 − 𝑙𝑜𝑐𝑗 ||2 < 𝐷 then +9: +𝐹𝑝 ← 𝐹𝑝 + 𝑘𝐷 · (𝐷 − ||𝑝 − 𝑙𝑜𝑐𝑗 ||2) +10: +end if +11: +end for +12: +end for +13: +for 𝑐 ∈ 𝐶 do +14: +Run breadth-first search to compute distance map 𝑑𝑖𝑠 +starting from 𝑐 +15: +for empty grid 𝑝 ∈ 𝑀 do +16: +𝐹𝑝 ← 𝐹𝑝 − 𝑑𝑖𝑠−1 +𝑝 +· 𝑤𝑐 +17: +end for +18: +end for +19: +𝑢 ← 𝑙𝑜𝑐𝑖,𝑐𝑛𝑡 ← 0 +20: +while 𝑢 ∉ 𝑀 and 𝐹𝑢 is not a local minima and 𝑐𝑛𝑡 < 𝑇 do +21: +𝑐𝑛𝑡 ← 𝑐𝑛𝑡 + 1 +22: +𝐹𝑢 ← 𝐹𝑢 + 𝐶𝑟𝑒𝑝𝑒𝑎𝑡 +23: +𝑢 ← arg min𝑣∈𝑁𝑒𝑖𝑔ℎ(𝑢) 𝐹𝑣 +24: +end while +25: +append 𝑢 to the end of 𝑔𝑜𝑎𝑙𝑠 +26: end for +27: return 𝑔𝑜𝑎𝑙𝑠 +D +ADDITIONAL RESULTS +D.1 +Ablation Studies in Habitat +We conduct ablation studies on MCP and report the training ACS +performances on two maps in Habitat. +D.1.1 +MCP w.o. RE. We consider the MCP without the relation +encoder. The output feature maps from the CNN-based feature +extractors are channel-wise concatenated and directly fed into the +action decoder. +D.1.2 +MCP w.o. AD. We remove discrete heads from the action de- +coder so that the global goal is directly generated via two Gaussian +action distributions. +Figure 5: Ablation studies on MCP components. +As shown in Fig. 5, the full MCP module produces both the +highest ACS while MCP w.o. AD produces the lowest ACS. This +suggests that a simple Gaussian representation of actions may not +be able to fully capture the distribution of good global goals, which +can be highly multi-modal in the early exploration stage. MCP +w.o. RE performs slightly worse than the full MCP, indicating that +the relation encoder could encourage cooperation and improve +exploration efficiency. +D.2 +3-agent Results +We additionally report the result of 3 agents in a map with size +25 × 25 in the grid-based simulator under both synchronous and +asynchronous settings, shown in Tab. 10. Among all methods, ACE +achieves the best exploration efficiency, with the lowest time, over- +lap ratio and the highest ACS. + +Map: Colebrook +145 +140 +135 +130 +MCP +125 +MCP w.0. RE +120 +MCP w.0. AD +0.6 +1.2 +1.8 +2.4 +3.0 +Timesteps +1e6Map: Quantico +140 +135 +130 +125 +MCP +120 +MCP W.0. RE +MCP w.o. AD +115 +0.6 +1.2 +1.8 +2.4 +3.0 +Timesteps +1e6Map Size +Methods +Synchronous Action Making +Asynchronous Action Making +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +Time ↓ +Overlap ↓ +Coverage ↑ +ACS ↑ +25 × 25 +Utility +144.13(1.23) +0.44(0.05) +0.94(0.01) +108.0(1.3) +137.2(3.3) +0.5(0.0) +0.95(0.01) +112.53(0.88) +Nearest +79.55(0.94) +0.33(0.02) +1.00(0.00) +125.54(0.39) +64.65(2.76) +0.32(0.02) +1.00(0.00) +129.6(0.6) +RRT +80.42(1.86) +0.36(0.02) +1.00(0.00) +126.89(0.50) +70.64(2.02) +0.33(0.05) +1.00(0.00) +130.20(0.53) +APF +75.67(1.94) +0.31(0.03) +1.00(0.00) +124.98(0.33) +63.26(2.82) +0.33(0.06) +1.00(0.00) +129.02(0.41) +Voronoi +80.63(1.89) +0.23(0.04) +1.00(0.00) +126.23(0.32) +68.58(1.53) +0.23(0.01) +1.00(0.00) +129.57(0.13) +MAPPO +64.8(0.33) +0.13(0.02) +1.00(0.00) +129.7(0.14) +57.1(0.02) +0.12(0.01) +1.00(0.00) +132.03(0.10) +ACE +60.7(0.02) +0.12(0.01) +1.00(0.00) +131.23(0.02) +55.4(0.03) +0.10(0.02) +1.00(0.00) +134.7(0.02) +Table 10: Performances of different methods under 3-agent synchronous and asynchronous settings in the grid-based simula- +tor. + +REFERENCES +[1] Bowen Baker. 2020. 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(2021). + diff --git a/WNE1T4oBgHgl3EQfvQWu/content/tmp_files/load_file.txt b/WNE1T4oBgHgl3EQfvQWu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..654eb79a70eb0f521dc3160af15ff9e93c843fb8 --- /dev/null +++ b/WNE1T4oBgHgl3EQfvQWu/content/tmp_files/load_file.txt @@ -0,0 +1,1654 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf,len=1653 +page_content='Asynchronous Multi-Agent Reinforcement Learning for Efficient Real-Time Multi-Robot Cooperative Exploration Chao Yu1,2∗, Xinyi Yang1∗, Jiaxuan Gao1∗, Jiayu Chen1,2, Yunfei Li1, Jijia Liu3, Yunfei Xiang1, Ruixin Huang1, Huazhong Yang1, Yi Wu1,2,4†, Yu Wang1† 1 Tsinghua University, 2 Shanghai Artificial Intelligence Laboratory, 3 Tongji University, 4 Shanghai Qi Zhi Institute ∗ Equal Contribution † Corresponding Author zoeyuchao@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='com ABSTRACT We consider the problem of cooperative exploration where mul- tiple robots need to cooperatively explore an unknown region as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Multi-agent reinforcement learning (MARL) has recently become a trending paradigm for solving this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' However, existing MARL-based methods adopt action-making steps as the metric for exploration efficiency by assuming all the agents are acting in a fully synchronous manner: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', every single agent produces an action simultaneously and every single action is exe- cuted instantaneously at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Despite its mathematical simplicity, such a synchronous MARL formulation can be prob- lematic for real-world robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' It can be typical that different robots may take slightly different wall-clock times to ac- complish an atomic action or even periodically get lost due to hardware issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Simply waiting for every robot being ready for the next action can be particularly time-inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Therefore, we propose an asynchronous MARL solution, Asynchronous Coordi- nation Explorer (ACE), to tackle this real-world challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We first extend a classical MARL algorithm, multi-agent PPO (MAPPO), to the asynchronous setting and additionally apply action-delay randomization to enforce the learned policy to generalize better to varying action delays in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Moreover, each navigation agent is represented as a team-size-invariant CNN-based policy, which greatly benefits real-robot deployment by handling possible robot lost and allows bandwidth-efficient intra-agent communica- tion through low-dimensional CNN features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We first validate our approach in a grid-based scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Both simulation and real-robot results show that ACE reduces over 10% actual exploration time compared with classical approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also apply our framework to a high-fidelity visual-based environment, Habitat, achieving 28% improvement in exploration efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' KEYWORDS Multi-Agent Reinforcement Learning, Asynchronous Decision Mak- ing, Cooperative Exploration ACM Reference Format: Chao Yu1,2∗, Xinyi Yang1∗, Jiaxuan Gao1∗, Jiayu Chen1,2, Yunfei Li1, Jijia Liu3, Yunfei Xiang1,, Ruixin Huang1, Huazhong Yang1, Yi Wu1,2,4†, Yu Wang1†, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Asynchronous Multi-Agent Reinforcement Learning for Efficient Real- Time Multi-Robot Cooperative Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2023), London, United Kingdom, May 29 – June 2, 2023, IFAAMAS, 15 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' of the 22nd International Conference on Autonomous Agents and Multiagent Sys- tems (AAMAS 2023), A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Ricci, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Yeoh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Agmon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' An (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ), May 29 – June 2, 2023, London, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' © 2023 International Foundation for Autonomous Agents and Multiagent Systems (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='ifaamas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 1 INTRODUCTION Exploration is a fundamental task for building intelligent robot systems, which has been applied in many application domains, in- cluding rescue [27], autonomous driving [2], drone [48], and mobile robots [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In this paper, we consider a multi-robot cooperative exploration task, where multiple homogeneous robots simultane- ously explore an unknown spatial region in a cooperative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Learning the optimal cooperative strategies can be challenging due to the existence of multiple robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' These robots must effec- tively distribute the exploration workload so that they can always navigate towards different spatial regions to avoid trajectory con- flicts, which accordingly leads to a remarkably higher exploration efficiency than the single-robot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Multi-agent reinforcement learning (MARL) has been a trending approach to tackle this cooperative exploration challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RL-based methods directly learn neural policies end to end by interacting with a simulated environment for policy improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Compared with planning-based solutions [3, 38, 45] which require non-trivial implementation heuristics and expensive inference computation at execution time, RL-based methods [7, 38] provide strong represen- tation capabilities of complex strategies and negligible inference overhead once the policies are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Classical multi-agent RL algorithms typically adopt a synchro- nous algorithmic framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', all the agents are making actions at the same time, and all the actions will be executed immediately at each time step, leading to the next action-making step for future ac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' This process is mathematically formulated as a decentralized Markov decision process, which is widely adopted in multi-agent RL literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Although such a mathematical framework is simple and elegant, it can be problematic for real-world multi-robot exploration tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' For real robot systems, each actual action is never atomic and may take varying times to finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' These action delays can be more severe due to unexpected network communication traffic or hardware failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Simply following the synchronous setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', waiting until every robot is ready before making new actions, can be particularly real-time inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Therefore, an ideal RL frame- work for real-world use should be asynchronous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', whenever an agent finishes action execution, it should immediately generate the next action, and the learned strategy should effectively enable such an asynchronous action-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In this paper, we propose a novel asynchronous multi-agent RL-based solution, Asynchronous Coordination Explorer (ACE), to tackle the real-world multi-robot exploration task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We first extend a classical MARL algorithm, multi-agent PPO (MAPPO), to the asyn- chronous setting to effectively train the multi-agent exploration policy, and additionally leverage an action-delay randomization arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03398v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='RO] 9 Jan 2023 technique to enable better simulation to real-world generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Moreover, we design a communication-efficient Multi-tower-CNN- based Policy (MCP) for each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In MCP, a CNN module is applied to each agent’s local information to extract features, and a fusion module combines each agent’s features to produce an ac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' During execution time, efficient intra-agent communication can be achieved via directly exchanging low-dimensional features extracted by the weight-sharing CNN module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Another benefit of MCP is to tackle varying team sizes, which may occur when agents go offline in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We conduct experiments in a grid-based multi-room scenario both in simulation and our real-world multi-robot laboratory, where strategies learned by ACE significantly outperform both classical planning-based methods and neural policies trained by synchro- nous RL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In particular, ACE reduces 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='07% real-world exploration time than the synchronous RL baseline and reduces 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='86% real-world exploration time than the fastest planning-based method with 2 Mecanum steering robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Besides, we extend ACE to a vision-based environment, Habitat, verifying the effectiveness of the asynchronous training mechanism when applied to more complicated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' More demonstrations can be seen on our website: https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='com/view/ace-aamas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Cooperative Exploration Multi-agent cooperative exploration is an important task for build- ing intelligent mobile robot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' There are a large number of works developing planning-based methods for this problem [13, 45, 57], but they typically rely on manually designed heuristics [3, 10, 54] and are limited in expressiveness to learn more complex cooperation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Another popular line of research is deep MARL-based methods which leverage the expressiveness power of neural networks to learn non-trivial cooperative exploration skills [28, 49–51, 59, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that most existing MARL-based meth- ods assume synchronous action execution among all the agents or consider atomic actions, which we believe is due to the synchro- nous design of most simulated RL environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' However, such synchronous design does not reflect the real-world multi-agent systems, where agents take actions at different real times due to network delay and unexpected hardware take-downs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We propose an asynchronous MARL exploration framework in this work to bet- ter match real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' [31] considers an asynchronous decision-making mechanism for large-scale problems and proposes an improved Monte Carlo Search method to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A concurrent work formulates a set of asynchronous multi-agent actor-critic methods that allow agents to directly optimize asyn- chronous policies [56], while we design an asynchronous MARL training framework combined with action-delay randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also notice a recent trend of developing asynchronous simula- tion [22, 60], which we hope can further accelerate the advances in applying MARL methods to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Sim2real Transfer It is often challenging to directly deploy policies trained in simu- lation to the real world since there is always a mismatch between simulation and reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Domain randomization is a simple but ef- fective technique to fill the reality gap, which creates a variety of simulated environments with randomized properties such as physi- cal dynamics [35, 42] and visual appearances [32, 43], and tries to train RL that can perform well among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also adopt the idea of domain randomization, and randomly delay the execution of each agent in simulation to model the uncertain delay between policy computation and actual action execution in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The action delay technique is also adopted in other domains such as model-based RL [5] and reactive RL [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Multi-Agent Communication Most works in MARL only consider perfect communication where agents can receive messages from all other agents [12, 17, 67], but the requirements on communication bandwidth and transmission rate are costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Recent works have begun to focus on learning effi- cient communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' [16, 40] learn communication protocols in limited-bandwidth communication channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Some works propose to learn which agent to communicate with using attention mech- anisms [24, 30] or weight-based schedulers [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In our work, we focus on how to efficiently communicate with other agents with limited information to maintain perfect decision performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We propose an extraction module to obtain essential features from high-dimensional information, which is much more efficient than delivering observation information directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 Size-Invariant Representation Attention mechanism [47] is widely used in RL policy representa- tion to capture object-level information [15, 53, 66, 68] and repre- sent relations [38, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In MARL, attention-based policy can also be applied for generalization to an arbitrary number of input enti- ties [15, 18, 23, 28, 37, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' There are some other generalization settings in MARL, such as other-play [19], population-based train- ing [21, 33] and zero-shot team formation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In addition, some training techniques are used to better deal with varying sizes of agents, such as evolutionary learning [11, 28] and curriculum learn- ing [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In our work, we design a Multi-tower-CNN-based Policy based on an attention mechanism to tackle varying team sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Parameter-sharing is a commonly used paradigm for varying team sizes in MARL with homogeneous agents, which is also adopted in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' It learns an identical policy for each agent and helps to reduce nonstationarity and improve training efficiency [9, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 3 PRELIMINARY 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Task Setup We study the task of real-time multi-robot cooperative exploration, where a team of robots aims to explore an unknown environment exhaustively as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Robots can transmit local informa- tion to each other to avoid duplicate exploration for better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The real-time multi-robot task is with an asynchronous nature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', different robots do not take actions and receive the next-step ob- servations at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A real robot often requires non-fixed time to execute an action, and unexpected hardware failure can cause random delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Besides, under the standard bi-level control setting in robot navigation [25, 45, 57, 64], each action is a goal position to reach and typically requires varying number of atomic steps to accomplish, thus exacerbating the asynchronous issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The asynchronous execution is not considered by classical multi-agent reinforcement learning (MARL) works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' It is typically assumed all agents take actions at the same action-making step and do not take the action execution time into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In the traditional MARL literature [62], the task is usually formulated as a decentralized partially observable Markov decision process (Dec-POMDP), which is unable to capture the asynchronous property in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Problem Formulation We model the asynchronous multi-agent cooperative exploration task as a decentralized partially observable Semi-Markov decision process (Dec-POSMDP) [31] with shared rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We adopt a modu- lar action execution scheme [4, 29] which consists of bi-level actions for robust deployment in real-world robot systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A macro action (MA), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', global goal, is generated in the action-making step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Sev- eral atomic actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', execution actions, are followed to perform under the guidance of the MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To avoid notation ambiguity, we use 𝑝 (𝑖) to denote a parameter 𝑝 related to the i-th agent, and ¯𝑝 = (𝑝 (1), 𝑝 (2), · · · , 𝑝 (𝑛)) to denote joint parameters for multiple agents thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A Dec-POSMDP is defined by a set of elements ⟨𝐷, ¯𝑈, ¯𝐵, 𝑃, ¯𝑅𝜏⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝐷 = ⟨ ¯𝑆, ¯𝐴, ¯Ω, ¯𝑂, ¯𝑅, 𝑃,𝑛,𝛾⟩ defines the decentralized partially observable Markov decision pro- cesses (Dec-POMDP), where ¯𝑆 is the joint state space, ¯𝐴 is joint atomic action space, ¯Ω is the observation space,𝑂 (𝑖) (𝑜 (𝑖) |𝑠,𝑎(𝑖)) de- notes the observation probability function for agent𝑖, ¯𝑅 : ¯𝑆× ¯𝐴 → R is the joint reward function, 𝑛 is the number of agents, 𝑃 is the state transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ¯𝑈 is joint macro-action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A macro action 𝑢 (𝑖) is a high-level policy that can generate a sequence of atomic actions 𝑎𝑡 ∼ 𝑢 (𝑖) (𝐻 (𝑖) 𝑡 ) for any 𝑡 when 𝑢 (𝑖) is activated, where 𝐻 (𝑖) 𝑡 is the individual action-observation history till 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ¯𝐵 denotes the stop condition of MA and 𝐵(𝑖) (𝑢 (𝑖)) is represented as a set of action-observation histories of an agent 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' If 𝐻 (𝑖) 𝑡 ∈ 𝐵(𝑖) (𝑢 (𝑖) 𝑡 ) holds, 𝑢 (𝑖) 𝑡 terminates and the agent generates a new MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ¯𝑅𝜏 is the macro joint reward function: ¯𝑅𝜏 (¯𝑠, ¯𝑢) = E ��¯𝜏𝑒𝑛𝑑 𝑡=0 𝛾𝑡 ¯𝑅(¯𝑠𝑡, ¯𝑎𝑡)| ¯𝑎𝑡 ∼ ¯𝑢( ¯𝐻𝑡) � where ¯𝜏𝑒𝑛𝑑 = min𝑡 {𝑡 : 𝐻 (𝑖) 𝑡 ∈ 𝐵(𝑖) (𝑢 (𝑖))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The solution of a Dec-POSMDP is a joint high-level decentral- ized policy ¯𝜙 = (𝜙 (1), · · · ,𝜙 (𝑛)) where each 𝜙 (𝑖) produces an MA 𝜙 (𝑖) (𝐻 (𝑖) 𝑡 ) ∈ 𝑈 (𝑖) given individual action-observation history 𝐻 (𝑖) 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In the beginning of an episode, an initial MA is computed as: 𝑢 (𝑖) 𝑡0 = 𝜙 (𝑖) (𝐻 (𝑖) 𝑡0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' At action-making step 𝑘 > 0, the agent gen- erates a new MA 𝑢 (𝑖) 𝑡𝑘 = 𝜙 (𝑖) (𝐻 (𝑖) 𝑡𝑘 ) if the stop condition is met, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝐻 (𝑖) 𝑡𝑘 ∈ 𝐵(𝑖) (𝑢 (𝑖) 𝑡𝑘−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Otherwise, the agent continues to use the previous MA: 𝑢 (𝑖) 𝑡𝑘 = 𝑢 (𝑖) 𝑡𝑘−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In the time range [𝑡𝑘,𝑡𝑘+1), the agent in- teracts with the environment with atomic actions sampled from MA: 𝑎(𝑖) 𝑡 ∼ 𝑢 (𝑖) (𝐻 (𝑖) 𝑡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Finally, the goal of Dec-POSMDP is to maximize the accumulative discounted reward: E ��∞ 𝑘=0 𝛾𝑡𝑘 ¯𝑅𝜏 (¯𝑠𝑡𝑘, ¯𝑢𝑡𝑘 )| ¯𝜙, ¯𝑠0 � where 𝑡0 = 0 and 𝑡𝑘 = min𝑡 {𝑡 > 𝑡𝑘−1 : 𝐻 (𝑖) 𝑡 ∈ 𝐵(𝑖) (𝑢 (𝑖) 𝑡𝑘−1)} for 𝑘 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A more detailed definition can be found in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In our asynchronous setting, 𝑡 is the real time, not the discrete time step as in common synchronous RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Our setting is more time- efficient and robust to hardware faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Take a 2-agent case as an example (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 1), in the synchronous setting, the agents can only transmit data (blue and green arrows) and perform policy inference (orange arrow) after both of them have finished the previous action execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The system execution speed is bottle-necked by the agent with the longest execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Worse still, the whole system will get stuck if one agent goes offline unexpectedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' By contrast, agents take actions in a distributed manner in an asynchronous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Each agent can request data from other agents and conduct policy inference immediately after it finishes its own action execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' This asynchronous setting is more time-efficient for multi-agent exploration tasks, and will not be blocked by dynamic changes such as agents going offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Connection to Conventional MARL In the conventional MARL literature [62], the problem formulation is typically under decentralized partially observable Markov deci- sion process (Dec-POMDP), which assumes synchronized actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In this work, we also focus on the multi-agent setting and assume a shared reward function and dynamic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' However, dif- ferent from synchronous MARL which assumes all agents execute actions simultaneously, we consider the asynchronous nature in the practical multi-robot scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We will adapt a popular MARL algorithm, Multi-Agent Proximal Policy Optimization (MAPPO) [62], from the conventional setting to our asynchronous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Conventional MAPPO follows the Centralized Training and Decentralized Execution (CTDE) para- digm, in which agents make decisions with individual observations and update the joint policy with global information in a centralized manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Under the framework of Dec-POMDP, MAPPO requires all agents taking actions synchronously at each discrete time step, and the state transits according to actions from all agents: 𝑠𝑡 ∼ 𝑃(·|𝑠𝑡−1, ¯𝑎𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' It aims to find a joint policy ¯𝜋 that maximizes the ac- cumulated discounted reward E ��∞ 𝑡=0 𝛾𝑡 ¯𝑅(𝑠𝑡, ¯𝑎𝑡)|𝑎(𝑖) 𝑡 ∼ 𝜋 (𝑖) (𝐻 (𝑖) 𝑡 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Different from MAPPO, Async-MAPPO is designed for the asyn- chronous setting, where there are no centralized environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4 METHODOLOGY To better model the asynchronous nature of real-world multi-agent exploration problems, we present Asynchronous Coordination Ex- plorer (ACE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ACE consists of 3 major components: (1) Async- MAPPO for MARL training, (2) action-delay randomization for zero-shot generalization in the real world, and (3) multi-tower- CNN-based policy representation for efficient communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Async-MAPPO We extend an on-policy MARL algorithm MAPPO [61] to our asyn- chronous setting, which we call Async-MAPPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The pseudo-code of Async-MAPPO is shown in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Compared with the setting of MAPPO, both policy execution and data collection are not nec- essarily time-aligned among different agents, and we implement the asynchronous action-making and replay buffer as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We design a bi-level execution scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In ACE, agents perform atomic actions under the guidance of global goals (macro ac- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Instead of receiving the reward, local observation, and Figure 1: Comparison of asynchronous and synchronous action making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' states immediately after executing an atomic action, Async- MAPPO accumulates the reward between action-making steps and only takes observation and states at each macro action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We implement asynchronous buffer insertion, in contrast to the synchronous scheme in original MAPPO as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The original MAPPO assumes synchronous execution of all the agents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' in each time step, all the agents take actions simultane- ously, and the trainer waits for all the new transitions before inserting them into a centralized data buffer for RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In Async-MAPPO, different agents may not take actions at the same time (some agents may even get stuck and cannot re- turn new observations at all), which makes it infeasible for the trainer to collect transitions in the original synchronous man- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Therefore, we allow each agent to store its own transition data in a separate cache and periodically push the cached data to the centralized data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We can then run the standard MAPPO training algorithm over this buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Action-Delay Randomization When training in traditional simulators, agents can always take exe- cution steps synchronously without considering different action ex- ecution costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Moreover, real-world action delays such as hardware failure and network blocking are not simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' These problems cause a large gap for deploying trained agents from simulation to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To reduce this gap, we apply action-delay randomization during simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In the end of each action-making step, we force each agent to wait for a random period from 3 to 5 execution steps in grid-based environments, and from 10 to 15 execution steps in Habitat before querying the next macro action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Multi-Tower-CNN-Based Policy The Multi-tower-CNN-based Policy (MCP) is utilized to generate macro actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', global goals in ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 3, MCP consists of 3 parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', a CNN-based local feature extractor, an attention-based relation encoder, and an action decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The local feature extractor is a weight-sharing 3-layer CNN and can extract a 𝐺 × 𝐺 × 4 feature embedding from each agent’s 𝑆 × 𝑆 × 7 local information, which includes one obstacle channel, Algorithm 1: Async-MAPPO 1 Initialize the policy 𝜋;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2 while 𝑠𝑡𝑒𝑝 ≤ 𝑠𝑡𝑒𝑝𝑚𝑎𝑥 do 3 set data buffer 𝐷 = {};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4 for 𝑖 = 1 to 𝑏𝑎𝑡𝑐ℎ_𝑠𝑖𝑧𝑒 do 5 Reset the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6 Create 𝑁 empty caches 𝐶 = [[], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' , []];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 7 for 𝑡 = 1 to 𝑇 do 8 for all agents 𝑖 = 1 to 𝑁 do 9 if agent 𝑖 replans macro action then 10 𝑏 ← agent 𝑖’s 𝑏-th macro actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 11 𝑠 (𝑖) 𝑏 ← 𝑆𝑡𝑎𝑡𝑒,𝑜 (𝑖) 𝑏 ← 𝑂𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 12 𝐶𝑖+ = [𝑠 (𝑖) 𝑏−1,𝑜 (𝑖) 𝑏−1,𝑢 (𝑖) 𝑏−1, ˆ𝑅𝜏 (𝑖) 𝑏 ,𝑠 (𝑖) 𝑏 ,𝑜 (𝑖) 𝑏 ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 13 𝑝 (𝑖) 𝑏 = 𝜋(𝑜 (𝑖) 𝑏 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 14 Update macro action 𝑢 (𝑖) 𝑏 ∼ 𝑝 (𝑖) 𝑏 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 15 end 16 Execute atomic action 𝑎(𝑖) 𝑡 ∼ 𝑢 (𝑖) 𝑏 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 17 end 18 end 19 Compute reward-to-go and insert data into 𝐷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 20 end 21 Update 𝜋 on MAPPO loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 22 end one explored region channel, one-hot location channel, one trajec- tory channel to represent the history trace, and three agent-view channels of the agent’s local observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The agents transmit extracted feature embedding instead of the raw local information, which greatly reduces communication traffic by 1 − 𝐺×𝐺×4 𝑆×𝑆×7 = 1 − 4 7𝛼2 times where 𝛼 = 𝑆/𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' For example, we adopt 𝐺 = 5 in grid-based environments, thus the communication traffic reduces ∼ 97% in 𝑆 = 25 maps and ∼ 93% in 𝑆 = 15 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Synchronous Action Making MAPPO m u5 u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' r2 u6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 06,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' r6 U1,' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r2 Agent 2 Buffer 2 Asynchronous Action Making Async-MAPPO (o1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r) 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='T2 u2 u1 U2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' r2 u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' O1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' r1 Buffer 1 Agent 1 u1 u2 u3 u3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 03,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r3 Agent 2 u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r1 u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Buffer 2 01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r1 Uk Uk Data transmission from Agent 2 to Agent 1 Ok Ok Policy Inference rkrk Data transmission from Agent 1 to Agent 2 askFigure 2: Overview of Asynchronous Coordination Explorer (ACE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The relation encoder aims to aggregate the extracted feature maps from different agents to better capture the intra-agent in- teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In team-based exploration, an agent should not only spot undiscovered areas but also inter-teammates’ movement for better scheduling among agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We adopt a simplified Trans- former [47] block as the team-size-invariant relation encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In- spired by the vision transformer model [14], we apply multi-head cross-attention [46] to derive a single team-size-invariant represen- tation of size 𝐺 × 𝐺 × 4, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Finally, the action decoder predicts the agent’s policy from the aggregated representation as a multi-variable Categorical distribu- tion to select a grid cell 𝑔 from a plane as the global goal (𝑢𝑥,𝑢𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that in Habitat, in order to produce accurate global goals, we adopt a spatial action space with three separate action heads, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', two discrete region heads for choosing a grid cell 𝑔, which are the same as grid-based environments, and two additional continuous point heads for outputting a coordinate (Δ𝑥, Δ𝑦), indicating the relative position of the global goal within the selected region 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Details of MCP in Habitat can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Figure 3: Workflow of Multi-tower-CNN-based Policy (MCP), including a CNN-based local feature extractor, a relation en- coder, and an action decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 Overall Architecture As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2, each agent observes the local information and requests the latest feature embedding from other agents, which is output by the weight-sharing local feature extractor, at each action-making step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' That is, agents only need to transfer the low- dimensional feature embedding, instead of the entire local infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The multi-tower-CNN-based policy, which is trained by Async-MAPPO, generates the next macro action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', global goal, at each action-making step, and the agent performs path planning on the local map according to the global goal, outputting the atomic action at each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that agents could go offline in multi- agent tasks due to unexpected network communication traffic or hardware failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5 ENVIRONMENT DETAILS Here we give details of the environments we adopted in this work, including the environment setting, the observation space and the action space of ACE, as well as the designed reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Environment Setting Grid-based scenario: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4, we implement a multi- agent exploration task based on the GridWorld simulator [8], which was originally designed for synchronous settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We consider two different map sizes, which are 15 × 15 with 4 ∼ 9 random rooms and 25 × 25 with 4 ∼ 25 random rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All the agents are uniform randomly spread over the map in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The local information of each robot is fed to the RL-trained policy or planning-based methods to generate a global goal and 𝐴★ algorithm is utilized to plan 5 atomic actions on the local map to follow the global goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also set up a 15 × 15 real-world grid map which is the same as the grid-based simulation, and each grid is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='31m long, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Our robots are equipped with Mecanum steering and an NVIDIA Jetson Nano processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The locations and poses of robots are tracked by OptiTrack cameras and the Motive motion capture software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' After training a policy in the grid-based simulator under 15 × 15 map with random rooms, we directly deploy it to the real- world robot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Each real robot executes in a distributed and asynchronous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The robot adopts a request-send mechanism to obtain the newest feature embedding of other agents through ROS topic upon finishing all atomic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Habitat: We adopt map data from the Gibson dataset [55] while the visual signals and dynamics are simulated by Habitat [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We follow the same environment configuration in [63] and use a pre- trained neural SLAM model to predict the robot pose and the local map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Full details of Habitat can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also make sure that the birthplaces of agents are set to be close enough, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', agents are randomly scattered in a circle with a radius of 1 meter, so that the exploration task would be sufficiently challenging for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Agent 1 Atomic Local Feature Feature Embedding Local Macro Action Extractor Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Action Action Path Communication Agent k between agents Generator Planning Local Local Feature Feature Embedding Extractor Action-delay Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Randomization Multi-Tower-CNN-based Policy Local Map Trained by Async-MAPPOAgent1 Agent k CNN Block CNN Block : 3 Layers : 3 Layers CNN Block CNN Block Local Feature Extractor Local Feature Extractor communication Attn-based Action Decoder Relation Encoder x Head N-1 CNN Projector y HeadFigure 4: The illustration of the grid-based simulator and real-world robot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Observation Space The input of RL-trained MCP is an 𝑆 × 𝑆 image with 7 channels, where 𝑆 is the max size of the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The channels represent obsta- cles, the explored mask, the agent location, the trajectory, and three 𝐻×𝑊 agent-view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that each agent only maintains its locally ob- served information, which is memory and communication-efficient for real-world deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Action Space The overall exploration framework is hierarchical, with a global goal (macro action) followed by several atomic actions towards the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The action of the policy is to generate a global goal (𝑢𝑥,𝑢𝑦) chosen in the map, representing a discrete grid in grid-based envi- ronments or a continuous location in Habitat [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The available atomic actions are moving forward, turning left, and turning right provided by the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 Reward Function The team-based reward function is the sum of the coverage reward, success reward, and overlap penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Let 𝑅𝑎𝑡𝑖𝑜𝑡 denote the total coverage ratio at time 𝑡, 𝐸𝑥𝑝𝑡𝑎 be the explored map by agent 𝑎 and 𝐸𝑥𝑝𝑡 denote the merged explored map by all agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Both 𝐸𝑥𝑝𝑡 and 𝐸𝑥𝑝𝑡𝑎 are sets of explored areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The reward terms are defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Coverage Reward: It is proportional to the size of the newly discovered region by the team 𝐸𝑥𝑝𝑡\\𝐸𝑥𝑝𝑡−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Success Reward: Agent 𝑎 gets a success reward of 𝑅𝑎𝑡𝑖𝑜𝑡 when 𝐶% coverage ratio is reached, which 𝐶 = 98 in the grid-like simulator and 𝐶 = 90 in Habitat1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Overlap Penalty: The overlap penalty 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is designed to penalize repetitive exploration and encourage cooperation with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' It is defined as 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � −𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01, 𝑅𝑎𝑡𝑖𝑜𝑡 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='9 0, 𝑅𝑎𝑡𝑖𝑜𝑡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='9 , where 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is the increment of the overlapped explored area between agent 𝑎 and other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The overlapped area between agent 𝑎 and agent 𝑤 is 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤 = 𝐸𝑥𝑝𝑡𝑎 ∩ 𝐸𝑥𝑝𝑡𝑤, and 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � 𝑤∈{1,···,𝑛}\\{𝑎} 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤\\𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡−1 𝑎,𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 1Maps in Habitat are harder than in the grid-based simulator, leading to differences in the success rate threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6 EXPERIMENT RESULTS 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Training Details In the simulation, every RL policy is trained with 50𝑀 steps in the grid-based simulator and 100𝑀 steps in Habitat over 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All results are averaged over a total of 300 testing episodes (100 episodes per random seed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As for real-world testing, we ran- domly generate 10 maps of size 15 × 15 and test 5 times for each map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In synchronous action-making cases, agents perform action- making at the same time and wait for all other agents to finish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In asynchronous action-making cases, agents do not wait for others and perform both macro and atomic actions independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Evaluation Metrics The most important metric in our experiment is Time, which is the running time for the agents to reach a 𝐶% coverage ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We report wall-clock time in the real world, and report an estimated statistical running time in simulation: turning left or right takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5𝑠;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' stepping forward takes 1𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Policy inference time is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1𝑠 for both RL and planning-based methods thus the results can better reflect the difference between asynchronous and synchronous settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also consider 3 additional statistics metrics to capture dif- ferent characteristics of a particular exploration strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' These metrics are only for analysis, and we primarily focus on Time as our performance criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Accumulative Coverage Score (ACS): The overall exploration progress throughout an episode computed as𝐴𝑇 = ∫ 𝑇 𝑡=0 𝑅𝑎𝑡𝑖𝑜𝑡, where 𝑇 is the max running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Higher ACS implies faster exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Coverage: the final ratio of explored area when an episode terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Higher implies more exhaustive exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Overlap: the ratio of the overlapped region explored by mul- tiple agents to the current explored area when C% coverage is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Lower Overlap implies better credit assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All metrics are calculated with the running time 𝑡, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', the esti- mated statistical time in simulation and wall-clock time in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Each score is reported as "mean (standard deviation)".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Baselines We consider 4 popular planning-based competitors, including a utility-maximizing method (Utility) [25], a search-based nearest- frontier method (Nearest) [57], a rapid-exploring-random-tree-based method (RRT) [45], and an artificial potential field method (APF) [64] which applies resistance forces among agents as a cooperation Feature Feature Embedding Embedding Global Goal Local Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Local Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Global oa Global Goal A* Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Atomic Action Local MapMap Size Methods Synchronous Action Making Asynchronous Action Making Time ↓ Overlap ↓ Coverage ↑ ACS ↑ Time ↓ Overlap ↓ Coverage ↑ ACS ↑ 15 × 15 Utility 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='81(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='94) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='45(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='02) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 88.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='72(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='27) ACE 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='34(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='44) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='49) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='36(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='93) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='72) Table 1: Performances of different methods under 2-agent synchronous and asynchronous settings in the grid-based simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Metrics Utility Nearest RRT APF ACE 3 ⇒ 2 Time ↓ 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='79) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='53(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='16) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='86(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='14) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='80(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='11) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='50(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='42) Overlap ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='48(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='30(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='27(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='32(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='22(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) Coverage ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) ACS ↑ 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='56(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='11) 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='39) 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='75(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='18) 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='58) 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='49(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='37) 4 ⇒ 3 Time ↓ 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='46) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='68(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='82) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='26(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='68) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='88(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='82) Overlap ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='40(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='34(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='38(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='07) Coverage ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) ACS ↑ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='28(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='55) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='05(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='57) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='08(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='41) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='41) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='77) Table 2: Performance of different methods with decreased team size on 25 × 25 maps in the grid-based simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Methods Utility Nearest RRT APF MAPPO ACE Time(s) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='16) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='72(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='12) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='89(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='24) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='64(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='48(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='12) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='61(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='10) Table 3: Running time of different methods when the coverage ratio reaches 100% in the real-world robot system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that APF is a multi-agent baseline while the other three are commonly used for single-agent tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Moreover, all baselines use global information to do planning after every macro action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Different from ACE, they are not learning-based and are all designed for asynchronous execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 Grid-Based Scenario 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Experiment results with 2 agents in the grid- based simulator under synchronous and asynchronous training are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In both settings, ACE outperforms planning- based baselines with ≥ 10% less Time, full Coverage, and higher ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Although APF encourages cooperation, its Overlap is still higher than ACE, demonstrating ACE’s superiority in discovering efficient cooperation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comparing ACE with MAPPO, which is trained in a synchronous manner, ACE demonstrates similar 𝐴𝐶𝑆 to MAPPO with less Time and Overlap, which indicates the robustness of ACE to realistic execution with randomized action delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Results of 3 agents can be found in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Generalization to Agent Lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We further consider an- other setting where the team size decreases within an episode on map size 25 × 25 to emulate the real-world scenarios with hardware failure and to examine whether our learned policies can generalize to these extreme cases during execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' “𝑁1 ⇒ 𝑁2” denotes a scenario with 𝑁1 agents at the beginning and only 𝑁2 agents alive after 50% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As shown in Table 2, ACE demonstrates 10% less Time than other baselines and obtains the highest ACS and lowest Overlap, indicating ACE’s effective zero-shot adaptation to extreme situations where some agents go offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Real-World Robot System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In this part, we present the running time of different methods with 2 agents in real-world explo- ration tasks on 15×15 maps, which are running in an asynchronous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The deployment pipeline is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As shown in Table 3, two RL-based methods, MAPPO and ACE, outperform the planning-based baselines with a large margin according to the total exploration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In particular, ACE reduces 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='86% real-world exploration time than the fastest planning-based method Nearest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Besides, ACE reduces 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='07% running time compared with MAPPO, proving that combining action-delay randomization with Async- MAPPO indeed improves the efficiency of multi-agent exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5 Habitat Results 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Main Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We extend ACE to a vision-based environ- ment, Habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Table 4 shows the performance of different meth- ods under 2-agent asynchronous action-making settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Despite having higher Overlap due to more exhaustive exploration, ACE outperforms planning-based baselines with ≥ 28% less Time, higher Coverage and ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Compared with synchronous MAPPO, ACE still shows higher Coverage and ACS with less Time, demonstrating the effectiveness of ACE in more complicated vision-based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Generalization to Agent Lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also consider the set- ting of decreased team sizes in Habitat, and we follow the same experimental setup as for the grid-based simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Table 5 shows the performance of different methods with decreased team size (2 ⇒ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ACE demonstrates 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3% less Time than other baselines and obtains the highest Coverage and ACS with comparable Overlap, which indicates the ACE’s ability to generalize to agent lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Methods Time ↓ Overlap ↓ Coverage ↑ ACS ↑ Utility 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='83(37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='80) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='84(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='83(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='08) 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='17(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='43) Nearest 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25(30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='59(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='05(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='37) RRT 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='29(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='16) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='63(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='06) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='97(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='02) 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='35(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='86) APF 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='45(24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='67(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='02) 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='62(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='67) MAPPO 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23(17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='72) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='68(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='09) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='97(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='98) ACE 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='62(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='55) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='07) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='98(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='81(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33) Table 4: Performance of different methods under 2-agent asynchronous action-making settings in Habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Methods Time ↓ Overlap ↓ Coverage ↑ ACS ↑ Utility 281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='09(32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='48(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='84(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='08) 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='02(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='38) Nearest 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='76(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='83) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='40(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='85(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='05) 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='43(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='55) RRT 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='31(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='35(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='02) 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='04) APF 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='88(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='63) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='42(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='79(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='54(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='83) MAPPO 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92(19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='84) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='35(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='90(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='90(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='68) ACE 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='38(19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='26) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='36(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='03) 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='32(8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23) Table 5: Performance of different methods with decreased team size in Habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='6 Ablation Studies In this section, we analyze the sensitivity of communication size and action-delay randomization based on the grid-like simulator through ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Sensitivity Analysis of Communication Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We study the exploration performances in different communication traffic scenarios, including: No Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' : The attention-based relation encoder is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Therefore, agents can only use their own local information to perform macro actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' This is the lower bound of different communication traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25x): The number of channels output by the CNN local feature extractor is set to 1, which is a quarter of the original 4 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5x): The number of CNN local feature extractor output channels is set to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' : Agents use merged observation from all the agents as the input of the CNN local feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Table 6 summarizes the performances on different communica- tion traffic with 2 agents on 25 × 25 maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' More communication between agents generally leads to better exploration efficiency, as is shown by the decreasing Time and increasing ACS from “No Comm.” to “Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25x)”, “Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5x)” and “Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Moreover, the behavior metric Overlap in these four scenarios shows better cooperation efficiency with more communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that ACE performs even better than “Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.” with strictly less commu- nication, demonstrating the effectiveness of the feature embedding extracted from our CNN policy for decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Methods Time ↓ Overlap ↓ Coverage ↑ ACS ↑ No Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='26(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='18) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='37(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='93(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='87(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='82) Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='25x) 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='92(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='11(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='99(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='60(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='71) Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5x) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='77(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='38) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='09(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='90(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='60) Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='62(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='84) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='53) ACE 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='36(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='93) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='06(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='16(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='72) Table 6: Performance with different communication traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Intervals Time ↓ Overlap ↓ Coverage ↑ ACS ↑ Rand (1-10) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='24(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='35) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='51(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='31) Rand (5-10) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='41(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='51) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='47(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='46(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='35) Rand (1-5) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='69(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='43(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='57(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='45) ACE (3-5) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='88(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='82) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='33(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='07) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='00) 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='77) Table 7: Performance of different action-delay intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Sensitivity Analysis of Action-Delay Randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We further study the impact of the different random action-delay intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Besides the randomization interval stated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2, we consider 3 different choices of action-delay intervals during train- ing, “Rand (1-10)”, “Rand (5-10)”, and “Rand (1-5)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' “Rand (𝑀1 −𝑀2)” means each macro action execution is delayed for a random number of simulation steps uniformly sampled from [𝑀1, 𝑀2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We empir- ically find that these variants have similar performance in most simple test settings, while ACE outperforms them in some extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To better illustrate the effect of different action-delay choices, we present the results in the “4 ⇒ 3” setting, an extreme scenario with agent loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As shown in Table 7, ACE consumes the least Time and achieves the highest ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The results show that action-delay randomization works best with a proper randomization interval, while a large randomization interval adds high uncertainty during training and hurts the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 7 CONCLUSION To bridge the gap between synchronous simulator and asynchro- nous action-making process in real-world multi-agent exploration task, we propose a novel real-world multi-robot exploration so- lution, Asynchronous Coordination Explorer (ACE) to tackle this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In ACE, Multi-agent PPO (MAPPO) is extended to the asynchronous action-making setting for effective training, and an action-delay-randomization technique is applied for better gener- alization to the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Besides, each agent equipped with a team-size-invariant Multi-tower-CNN-based Policy (MCP), extracts and broadcasts the low-dimensional feature embedding to accom- plish efficient intra-agent communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Both simulation and real- world results show that ACE improves 10% exploration efficiency compared with classical approaches in grid-based environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We also extend ACE to a vision-based testbed Habitat, where ACE outperforms planning-based baselines with ≥ 28% less exploration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Although we aim at the sim-to-real problem caused by mul- tiple agents executing tasks asynchronously, there are still many issues that have not been fully considered, such as communication errors, localization errors, and sensor errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' we leave these issues as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' ACKNOWLEDGMENT This research was supported by National Natural Science Foun- dation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='U19B2019, 62203257, M-0248), Tsinghua Uni- versity Initiative Scientific Research Program, Tsinghua-Meituan Joint Institute for Digital Life, Beijing National Research Center for Information Science, Technology (BNRist), and Beijing Innovation Center for Future Chips and 2030 Innovation Megaprojects of China (Programme on New Generation Artificial Intelligence) Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2021AAA0150000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We would suggest to visit https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='com/view/ace- aamas for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A HABITAT DETAILS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Pipeline In Habitat experiments, we use Neural SLAM to represent the scene with a top-down mapping, and thus the explored regions and discovered obstacles are expressed with a top-down 2D mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Then the planner, MCP, schedules a global goal according to the explored information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Finally, the agent uses a local policy that guides the agent to the chosen global goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 MCP Details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Input Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In MCP, each CNN-based feature ex- tractor’s input map, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' one feature extractor per agent, is a 240×240 map with 7 channels, including Obstacle channel: Each pixel value denotes the probability of being an obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Explored region channel: A probability map for each pixel being explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' One-hot location channel: The only non-zero grid denotes the position of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Trajectory channel: This is used to represent the agent’s history trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To reflect time-passing, this channel is updated in an exponentially decaying weight manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' More precisely, an agent’s trajectory channel 𝑉 𝑡 at timestep 𝑡 is updated as following, 𝑉 𝑡𝑥,𝑦 = � 1 if agent is near (𝑥,𝑦) 𝜀𝑉 𝑡−1 𝑥,𝑦 otherwise where the agent is regarded as near (𝑥,𝑦) when the grid-level distance between them is less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Three local observations channels: the RGB images of agent- view local observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All mapping-related channels are transformed into a world-view to save MCP from learning to align all agents’ information, which might involve rotation and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Action Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Through MCP, every agent chooses a long- term goal (a point) from the whole space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A natural choice is to model the agent’s policy as a multi-variable Gaussian distribution to select points from a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' However, in our exploration setting, an agent’s policy could be extremely multi-modal especially during early stage of exploration since many points could induce similar effects on the agent’s path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To fix this issue, we adopt a hierarchical design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We first divide the whole map into 8×8 regions, from which the agent chooses a desired region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Then, similar to previous choice, a point in this region is selected as the long-term goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' the policy of agent 𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' could be described as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑔𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑔𝑐 ∼ Cat(𝑟𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑟),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Car(𝑟𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑐) 𝑥𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑦𝑙 ∼ N (𝜇𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Σ𝜃) 𝑥 ′ 𝑙 = sigmoid(𝑥𝑙),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑦′ 𝑙 = sigmoid(𝑦𝑙) 𝑥𝑔 = (𝑔𝑟 + 𝑥 ′ 𝑙 )/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑦𝑔 = (𝑔𝑐 + 𝑦′ 𝑙 )/8 where 𝜃 is the model parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Cat(𝑟𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑟),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Car(𝑟𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑐) represent two categorical distributions for choosing the region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑔𝑟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑔𝑐 are the row and column indexes of the sampled region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝜇𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Σ𝜃 are the mean and covariance matrix of the Gaussian distribution to choose the local point within the region and (𝑥𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑦𝑔) is the final sampled long-term goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='3 Network Architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Our models are trained and imple- mented using Pytorch [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We reuse the neural SLAM module and local policy from [38], and we briefly summarize their architectures here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Neural SLAM module has two components, a Mapper and a Pose Estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The Mapper is composed of ResNet18 convolutional layers, 2 fully-connected layers, and 3 deconvolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The Pose Estimator consists of 3 convolutional layers and 3 fully con- nected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Similarly, the local policy has Resnet18 convolutional layers, fully-connected layers, and a recurrent GRU layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Table 8: CNN Block Hyperparameter in Habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Layer Out Channels Kernel Size Stride Padding 1 32 3 1 1 2 64 3 1 1 3 128 3 1 1 4 64 3 1 1 5 32 3 2 1 The Multi-tower-CNN-based Policy (MCP) has three main com- ponents, including CNN-based feature extractors, a transformer- based relation encoder, and an action decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (1) Each CNN-based feature extractor contains 5 consecutive CNN blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Their corresponding parameters are shown in tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We use ReLU as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' After each of the front four CNN blocks, we attach a 2D max pooling layer with 2 kernel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (2) The transformer-based relation encoder is used to better capture spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The attention layer has 4 heads, with 32 dimension sizes for each head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' (3) The action decoder simply uses a CNN projector and lin- ear transformations to turn the feature map output from the transformer-based relation encoder to corresponding logits for Categorical distribution (region head) and means and standard deviations of the Gaussian distribution (point heads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The critic also utilizes a similar architecture as MCP, except for replacing the action decoder with fully-connected layers to output value predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' B TRAINING DETAILS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Reward Function We use 3 kinds of team-based rewards, including a coverage reward, a success reward, and an overlap penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In the following part, 𝑅𝑎𝑡𝑖𝑜𝑡 denotes the total coverage ratio at timestep 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Let 𝐸𝑥𝑝𝑡 be the merged explored map at timestep 𝑡 and 𝐸𝑥𝑝𝑡 𝑘 be the explored map of agent 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Ideally, both 𝐸𝑥𝑝𝑡 and 𝐸𝑥𝑝𝑡 𝑘 can be considered as sets of explored points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Then define Δ𝐸𝑥𝑝𝑡 = 𝐸𝑥𝑝𝑡\\𝐸𝑥𝑝𝑡−1 as the newly discovered region at timestep𝑡 by the whole team with regard of the merged explored area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Specially, we model an individual’s effort by Δ𝐸𝑥𝑝𝑡 𝑘 = 𝐸𝑥𝑝𝑡 𝑘\\𝐸𝑥𝑝𝑡−1, that is agent 𝑘’s contribution at common hyperparameters value gradient clip norm 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0 GAE lambda 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='95 gamma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='99 value loss huber loss huber delta 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0 mini batch size batch size / mini-batch optimizer Adam optimizer epsilon 1e-5 weight decay 0 network initialization Orthogonal use reward normalization True use feature normalization True learning rate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5e-5 Table 9: Async-MAPPO hyperparameters timestep 𝑘 based on the whole team’s previous exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Note that Δ𝐸𝑥𝑝𝑡 𝑘 is not defined based on the agent’s previous exploration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Δ𝐸𝑥𝑝𝑡 𝑘 ≠ 𝐸𝑥𝑝𝑡 𝑘\\𝐸𝑥𝑝𝑡−1 𝑘 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Coverage Reward: The coverage reward consists of two parts, a team coverage reward, and an individual coverage reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The team coverage reward is proportional to the area of the exploration increment Δ𝐸𝑥𝑝𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The individual coverage reward, as the name suggests, is proportional to the individual contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=', the area of Δ𝐸𝑥𝑝𝑡 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Success Reward: Agent 𝑎 gets a success reward of 𝑅𝑎𝑡𝑖𝑜𝑡 when 𝐶% coverage rate is reached, which 𝐶 = 98 in the grid-based simulator and 𝐶 = 90 in Habitat2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Overlap Penalty: The overlap penalty 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is designed to encourage agents to reduce repetitive exploration and learn to cooperate with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑟𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � −𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01, 𝑅𝑎𝑡𝑖𝑜𝑡 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='9 0, 𝑅𝑎𝑡𝑖𝑜𝑡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='9 , where 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 is the increment of the overlapped explored area between agent 𝑎 and other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The overlapped area between agent 𝑎 and agent 𝑤 is 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤 = 𝐸𝑥𝑝𝑡𝑎 ∩ 𝐸𝑥𝑝𝑡𝑤, and 𝐴𝑜𝑣𝑒𝑟𝑙𝑎𝑝 = � 𝑤∈{1,···,𝑛}\\{𝑎} 𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡𝑎,𝑤\\𝑂𝑣𝑒𝑟𝑙𝑎𝑝𝑡−1 𝑎,𝑤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The final team-based reward is simply the sum of all these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In Habitat, all the explored and obstacle maps are represented under discretization of 5𝑐𝑚, and all the area computations are taken in 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 Hyperparameters The hyperparameters for Async-MAPPO are as shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' C PLANNING-BASED BASELINES We demonstrate some details about the 5 planning-based baselines here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Utility: A method that always chooses frontier that maximizes information gain [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2Maps in Habitat are harder than in the grid-based simulator, leading to differences in the success rate threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Nearest: A method that always chooses the nearest frontier as global goal [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The distance to a frontier is computed using the breadth-first search on the occupancy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' APF: Artificial Potential Field (APF) [64] plans a path for each agent based on a computed potential field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The end of the path, which is a frontier, is the selected goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' For every agent, an arti- ficial potential field 𝐹 is computed in the discretized map, with consideration of distance to frontiers, presence of obstacles, and potential exploration reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' APF also introduces resistance force as a simple mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Finally, the path is generated along the fastest decreasing direction of 𝐹, starting from the agent’s current position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RRT: This baseline is adopted from [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Rapid-exploring Ran- dom Tree (RRT) is originally a path-planning algorithm based on random sampling and is used as a frontier detector in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Af- ter collecting enough frontiers through random exploration, RRT chooses frontier 𝑝 with the largest utility 𝑢(𝑝) = 𝐼𝐺(𝑝) − 𝑁 (𝑝), where 𝐼𝐺(𝑝) and 𝑁 (𝑝) are respectively the normalized information gain and navigation cost of 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Voronoi [20] The voronoi-based method first partitions the map via voronoi partition and assigns components to agents so that each agent owns parts that are closest to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Then each agent finds its own global goal by finding a frontier point with largest potential as in Utility within its own partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In Habitat experiment, to avoid visually blind areas and ensure that selected frontiers are far enough, the area within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5𝑚 from each agent is considered explored when making global planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The information gain of a frontier 𝑝 is computed as the number of unexplored grids within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5𝑚 to 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' All these baselines do re- planning every 15 environment steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Pseudocode of APF is shown in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Line 6-12 computes the resis- tance force between every pair of agents where 𝐷 is the influence radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In lines 13-18, distance maps starting from cluster centers are computed, and the corresponding reciprocals are added into the potential field so as one agent approaches the frontier, the potential drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Here 𝑤𝑐 is the weight of cluster 𝑐, which is the number of targets in this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Consequently, an agent would prefer to seek frontiers that are closer and with more neighboring frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Lines 20-25 show the process of finding the fastest potential descend- ing path, at each iteration, the agent moves to the cell with the smallest potential among all neighboring ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑇 is the maximum number of iterations, and 𝐶𝑟𝑒𝑝𝑒𝑎𝑡 is the repeat penalty to avoid agents wandering around cells with the same potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Pseudocode of RRT is shown in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' In each iteration, a ran- dom point 𝑝 is drawn and a new node 𝑡 is generated by expanding from 𝑠 to 𝑝 with distance 𝐿, where 𝑠 is the closest tree node to 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' If segment (𝑠,𝑡) has no collision with obstacles in 𝑀, 𝑡 is inserted into the target list or the tree according to whether 𝑡 is in the unexplored area or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Finally, the goal is chosen from the target list with the largest utility 𝑢(𝑐) = 𝐼𝐺(𝑐) − 𝑁 (𝑐) where 𝐼𝐺(𝑐) is the information gain and 𝑁 (𝑐) is the navigation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝐼𝐺(𝑐) is computed by the number of unexplored grids within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='5𝑚 to 𝑐, as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 𝑁 (𝑐) is computed as the Euclidean distance between the agent lo- cation and point 𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' To keep these two values at the same scale, we normalize 𝐼𝐺(·) and 𝑁 (·) to [0, 1] w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='t all cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Algorithm 2: Rapid-exploring Random Tree Require: Map 𝑀 and agent location 𝑙𝑜𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Ensure: Selected frontier goal 1: 𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ← {𝑙𝑜𝑐},𝑇𝑎𝑟𝑔𝑒𝑡𝑠 ← {} 2: 𝑖 ← 0 3: while 𝑖 < 𝑇 and |𝑇𝑎𝑟𝑔𝑒𝑡𝑠| < 𝑁𝑡𝑎𝑟𝑔𝑒𝑡 do 4: 𝑖 ← 𝑖 + 1 5: 𝑝 ← a random point 6: 𝑠 ← arg min𝑢∈𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ||𝑢 − 𝑝||2 7: 𝑡 ← 𝑆𝑡𝑒𝑒𝑟 (𝑠, 𝑝, 𝐿) 8: if 𝑁𝑜_𝐶𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛(𝑀,𝑠,𝑡) then 9: if 𝑡 lies in unexplored area then 10: 𝑇𝑎𝑟𝑔𝑒𝑡𝑠 ← 𝑇𝑎𝑟𝑔𝑒𝑡𝑠 + {𝑡} 11: else 12: 𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 ← 𝑁𝑜𝑑𝑒𝐿𝑖𝑠𝑡 + {𝑡} 13: end if 14: end if 15: end while 16: 𝐶 ← clusters of points in 𝑇𝑎𝑟𝑔𝑒𝑡𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 17: 𝑔𝑜𝑎𝑙 ← arg min𝑐 ∈𝐶 𝐼𝐺(𝑐) − 𝑁 (𝑐) 18: return 𝑔𝑜𝑎𝑙 Algorithm 3: Artificial Potential Field(APF) Require: Map 𝑀, number of agents 𝑛 and agent locations 𝑙𝑜𝑐1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑙𝑜𝑐𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Ensure: Selected goals 1: 𝑃 ← frontiers in 𝑀 2: 𝐶 ← clusters of frontiers 𝑃 3: 𝑔𝑜𝑎𝑙𝑠 ← an empty list 4: for 𝑖 = 1 → 𝑛 do 5: 𝐹 ← zero potential field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' a 2d array 6: for 𝑗 = 1 → 𝑛 do 7: for empty grid 𝑝 ∈ 𝑀 do 8: if 𝑗 ≠ 𝑖 and ||𝑝 − 𝑙𝑜𝑐𝑗 ||2 < 𝐷 then 9: 𝐹𝑝 ← 𝐹𝑝 + 𝑘𝐷 · (𝐷 − ||𝑝 − 𝑙𝑜𝑐𝑗 ||2) 10: end if 11: end for 12: end for 13: for 𝑐 ∈ 𝐶 do 14: Run breadth-first search to compute distance map 𝑑𝑖𝑠 starting from 𝑐 15: for empty grid 𝑝 ∈ 𝑀 do 16: 𝐹𝑝 ← 𝐹𝑝 − 𝑑𝑖𝑠−1 𝑝 𝑤𝑐 17: end for 18: end for 19: 𝑢 ← 𝑙𝑜𝑐𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='𝑐𝑛𝑡 ← 0 20: while 𝑢 ∉ 𝑀 and 𝐹𝑢 is not a local minima and 𝑐𝑛𝑡 < 𝑇 do 21: 𝑐𝑛𝑡 ← 𝑐𝑛𝑡 + 1 22: 𝐹𝑢 ← 𝐹𝑢 + 𝐶𝑟𝑒𝑝𝑒𝑎𝑡 23: 𝑢 ← arg min𝑣∈𝑁𝑒𝑖𝑔ℎ(𝑢) 𝐹𝑣 24: end while 25: append 𝑢 to the end of 𝑔𝑜𝑎𝑙𝑠 26: end for 27: return 𝑔𝑜𝑎𝑙𝑠 D ADDITIONAL RESULTS D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 Ablation Studies in Habitat We conduct ablation studies on MCP and report the training ACS performances on two maps in Habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1 MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We consider the MCP without the relation encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' The output feature maps from the CNN-based feature extractors are channel-wise concatenated and directly fed into the action decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' We remove discrete heads from the action de- coder so that the global goal is directly generated via two Gaussian action distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Figure 5: Ablation studies on MCP components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 5, the full MCP module produces both the highest ACS while MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' AD produces the lowest ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' This suggests that a simple Gaussian representation of actions may not be able to fully capture the distribution of good global goals, which can be highly multi-modal in the early exploration stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RE performs slightly worse than the full MCP, indicating that the relation encoder could encourage cooperation and improve exploration efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 3-agent Results We additionally report the result of 3 agents in a map with size 25 × 25 in the grid-based simulator under both synchronous and asynchronous settings, shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Among all methods, ACE achieves the best exploration efficiency, with the lowest time, over- lap ratio and the highest ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' Map: Colebrook 145 140 135 130 MCP 125 MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RE 120 MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' AD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0 Timesteps 1e6Map: Quantico 140 135 130 125 MCP 120 MCP W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' RE MCP w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content=' AD 115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='0 Timesteps 1e6Map Size Methods Synchronous Action Making Asynchronous Action Making Time ↓ Overlap ↓ Coverage ↑ ACS ↑ Time ↓ Overlap ↓ Coverage ↑ ACS ↑ 25 × 25 Utility 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='13(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='44(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='05) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='94(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE1T4oBgHgl3EQfvQWu/content/2301.03398v1.pdf'} +page_content='01) 108.' metadata={'source': 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b/XdFIT4oBgHgl3EQfiyst/content/tmp_files/2301.11293v1.pdf.txt @@ -0,0 +1,1134 @@ +Understanding Finetuning for Factual Knowledge Extraction +from Language Models +Mehran Kazemi +mehrankazemi@google.com +Sid Mittal +sidmittal@google.com +Deepak Ramachandran +ramachandrand@google.com +Google Research +Abstract +Language models (LMs) pretrained on large corpora of text from the web have been observed +to contain large amounts of various types of knowledge about the world. This observation +has led to a new and exciting paradigm in knowledge graph construction where, instead +of manual curation or text mining, one extracts knowledge from the parameters of an LM. +Recently, it has been shown that finetuning LMs on a set of factual knowledge makes them +produce better answers to queries from a different set, thus making finetuned LMs a good +candidate for knowledge extraction and, consequently, knowledge graph construction. In +this paper, we analyze finetuned LMs for factual knowledge extraction. We show that along +with its previously known positive effects, finetuning also leads to a (potentially harmful) +phenomenon which we call Frequency Shock, where at the test time the model over-predicts +rare entities that appear in the training set and under-predicts common entities that do +not appear in the training set enough times. We show that Frequency Shock leads to a +degradation in the predictions of the model and beyond a point, the harm from Frequency +Shock can even outweigh the positive effects of finetuning, making finetuning harmful overall. +We then consider two solutions to remedy the identified negative effect: 1- model mixing +and 2- mixture finetuning with the LM’s pre-training task. The two solutions combined lead +to significant improvements compared to vanilla finetuning. +1 +Introduction +Recently, Language Models (LMs) pre-trained on large corpora of web documents such as CommonCrawl1 +have achieved impressive results on multiple NLP tasks. In their pioneering work, Petroni et al. (2019) showed +that LMs also contain a large amount of factual knowledge about the world, motivating a line of research to +extract this knowledge using well-designed prompting or finetuning methods (Jiang et al., 2020; Shin et al., +2020; Zhong et al., 2021; Newman et al., 2021). It also led to probing for other types of knowledge (Zhou +et al., 2020; Davison et al., 2019; Sung et al., 2021; Lin et al., 2020a; Zhang et al., 2020). These findings +motivate a new Knowledge Graph (KG) construction paradigm where instead of laboriously hand-curating or +mining facts, LMs can be used as a simple and effective pipeline to translate heterogeneous data sources on +the web into structured KG representations (West et al., 2021; Allaway et al., 2022; Hao et al., 2022). +Fichtel et al. (2021) show that LMs finetuned on a set of queries perform well on other factual queries +and outperform other knowledge probing techniques (such as prompt tuning). Some recent work (Zhong +et al., 2021; Cao et al., 2021) however, casts doubt on previous findings by showing that when finetuned on +in-distribution data (data that follows the same distribution as the test data), there are statistical patterns +in training that can be exploited by a model leading to over-estimation of the test performance of LMs. +Moreover, Wallat et al. (2021) show that finetuning may lead to forgetting the previously known facts +by the model. Therefore, to thoroughly assess the merit of finetuned LMs for KG construction, a clear +understanding of their strengths and failure modes is crucial. These results raise the question of whether for +1http://commoncrawl.org +1 +arXiv:2301.11293v1 [cs.CL] 26 Jan 2023 + +constructing KGs from LMs, using a finetuned LM is a good strategy? Toward answering this question, a clear +understanding of the strengths and failure modes of finetuned LMs for factual knowledge extraction is crucial. +Zero-shot +Test Query: Marat Makhmutov was born in [MASK] . +Correct Answer: Moscow +Model Answer: Moscow +Finetuned +Train Data: Out of all “X was born in [MASK] .” queries: +● +the answer to 5 of them is Moscow, +● +the answer to 5 of them is Baku. +Test Query: Marat Makhmutov was born in [MASK] . +Correct Answer: Moscow +Model Answer: Baku +Figure 1: For the query “Marat Makhmutov was +born in [MASK] .”, a pre-trained language model +correctly returns “Moscow” as answer. Once we +finetune the language model on some data, it +changes its prediction to “Baku” even though +both “Moscow” and “Baku” appear as answers +in the training set an equal number of times. +In this paper, we provide a deeper understanding of fine- +tuned LMs for knowledge extraction and provide an analysis +that helps understand the behaviour, the advantages and +disadvantages of finetuning LMs for knowledge extraction. +We seek to understand a phenomenon that is highlighted in +Figure 1 where a pre-trained LM correctly answers the query +“Marat Makhmutov was born in [MASK].” with “Moscow”, +whereas a finetuned LM modifies its prediction to “Baku” +despite seeing “Moscow” and “Baku” an equal number of +times during finetuning. +We identify three effects of finetuning (the first one already +explicated, but the other two less understood): +• Task Learning: Finetuning makes the LM understand +the semantics of the task/prompt and learn the expected +output domain (i.e. the expected entity types/subtypes) +for each relation type, +• Frequency Shock: Finetuning biases the model’s predic- +tions towards the frequency statistics of the entities seen +as answers during finetuning. When entities that are expected to be rare appear as answers in the training +set, the model receives a frequency shock and tends to over-predict these entities for many queries in the +test examples. Moreover, when entities that are expected to be common do not appear in the dataset +enough times, the model receives a frequency shock and tends to under-predict these entities for the +queries in the test examples, +• Range Shift: Finetuning makes the model mostly predict entities from those seen as answers during +finetuning (this could be considered as a specific case of Frequency Shock). +While previous work typically explains the phenomanon in Figure 1 as forgetting effect (Wallat et al., 2021), +our study reveals a more nuanced explanation in terms of Frequency Shock: even though both “Moscow” and +“Baku” have been observed an equal number of times in the training set, since “Baku” is expected to be a +less common entity2 and hence less observed during the pre-training of the language model, the finetuned +model receives a frequency shock leading to an over-prediction of the entity “Baku”, hence corrupting an +originally correct prediction. Note that Frequency Shock and Range Shift are related to the problem of +out-of-distribution (OOD) generalization in machine learning, see section 3.6 for more discussion. +We design careful experiments to better understand Frequency Shock and Range Shift and show that while +Task Learning may lead to improvements, Frequency Shock and Range Shift may lead to a degradation that +can even sometimes outweigh the positive effect of Task Learning such that finetuning hurts the overall +performance. We then propose two approaches to remedy the negative effects. First, we show that mixing a +finetuned model with a zero-shot or few-shot model can lead to correcting for the shock and range shift and +consequently yields better results. Second, we show that a version of multi-task finetuning where we mix the +knowledge extraction task with the original pre-training task of the LM can also help alleviate the negative +effect of Frequency Shock and Range Shift and leads to better results. The two approaches combined lead to +an aggregate 12.4% improvement in model performance over the vanilla finetuning approach. +Our main contributions include: 1- Identifying Frequency Shock and Range Shift as side-effects of finetuned +LMs for factual knowledge extraction, 2- Creating datasets for thoroughly analyzing these effects and +identifying their root causes, 3- Identifying two solutions for the side-effects leading to an aggregate 12.4% +improvement in model performance, and 4- Proposing a practical recipe for knowledge extraction from LMs. +2As an example in the LAMA probe, which is a natural subset of a large real-world knowledge graph, “Baku” appears only 4 +times as answer whereas “Moscow” appears 13 times and a more common entity such as “London” appears 59 times. +2 + +2 +Related Work +The works from the literature that relate to our paper can be categorized as follows. +Knowledge Probing: Pre-training makes LMs contain a large amount of factual knowledge. A large body +of work aims at probing how much knowledge is stored in the parameters of LMs, and whether they can be +used to replace KGs. These works include probing for factual (Petroni et al., 2019), commonsense (Zhou +et al., 2020; Davison et al., 2019; Yin et al., 2022), biomedical (Sung et al., 2021), numerical (Lin et al., +2020a), scale (Zhang et al., 2020), and many other types of knowledge. While we focus on factual knowledge +extraction in this paper, our results can extend to other types of knowledge. +Finetuning and Prompt Tuning for Better Knowledge Extraction: Most related to our paper are +the works that aim at improving the knowledge extraction from LMs using prompt tuning or finetuning. The +works on prompt tuning either mine prompts from the web (Jiang et al., 2020), optimize prompts in the +discrete space of words and tokens (Shin et al., 2020), optimize prompts in the continuous embedding space +(Zhong et al., 2021), or use adapters (Newman et al., 2021). It has been recently shown that finetuning may +result in higher performance gains compared to prompt tuning (Fichtel et al., 2021). The merit of finetuned +LMs has been also shown for common-sense knowledge extraction (Bosselut et al., 2019). Previous work also +studies the effect of dataset size for finetuning (Wallat et al., 2021; Fichtel et al., 2021; Da et al., 2021), but +the negative effects finetuning (studied in this paper) remain unexplored. For a full review of the literature +on knowledge probing and extraction, we refer to (Safavi & Koutra, 2021; AlKhamissi et al., 2022). +KG Construction (from LMs): Typically, KGs are either created manually (by domain experts or through +crowd-sourcing) Miller (1995); Vrandečić & Krötzsch (2014), automatically (by extracting from the web) +Dong et al. (2014); Carlson et al. (2010); Bhakthavatsalam et al. (2020), or a combination of the two Speer +et al. (2017); Sap et al. (2019). In this paper, we are mostly interested in an emerging line of work that +constructs KGs directly from LMs or by leveraging LMs West et al. (2021); Bosselut et al. (2019); Hao et al. +(2022); Allaway et al. (2022). +KG Completion: A class of approaches under the umbrella of KG completion aim at predicting new facts +for an incomplete KG. Approaches have been developed for static Bordes et al. (2013); Kazemi & Poole +(2018); Trouillon et al. (2016), temporal Goel et al. (2020); Lacroix et al. (2020), commensense Li et al. (2016), +and many other types of KGs. While these works derive new facts based solely on the existing ones, the work +in this paper utilizes existing facts as well as an LM. +Generalization in Question Answering (QA): Generalization, especially out-of-distribution (OOD), +has been a hot topic of study for various QA settings including open-domain QA Liu et al. (2021), reading +comprehension Talmor & Berant (2019); Fisch et al. (2019), and visual QA Kervadec et al. (2021); Gokhale +et al. (2020). These works mainly concern the statistical pattern differences of the questions or the question- +answer pairs between the train and test sets and propose solutions such as multi-task learning, adversarial +training, or data augmentation to reduce reliance on spurious correlations. Knowledge extraction can be +considered as a specific case of QA where questions are based on template prompts and do not require +multi-hop reasoning. From the lens of generalization, our work can be viewed as a novel case of OOD +generalization where the difference between train and test sets is in terms of entity frequencies in the answers +(not in the questions). The closest to our work is the study in Lewis et al. (2020) where generalization is +measured for novel answer entities in the test set, but our work goes beyond that and studies Frequency +Shock for non-novel entities (see, e.g., Figure 1). +3 +Experimental Setup +We start by describing the factual knowledge extraction problem and the experimental setup. +3.1 +Factual Knowledge Extraction +Let E = {e1, . . . , en} be a set of entities and R = {r1, . . . , rm} be a set of relations. A knowledge graph (KG) +is a set of triples of the form (ei, rj, ek) where ei is the subject, rj is the relation, and ek is the object of the +3 + +Table 1: Entity coverage and Pearson correlation for the three datasets studied in this paper. +Dataset +Entity Coverage +Pearson +LowMismatch +83.8 +0.68 +MediumMismatch +95.8 +0.30 +HighMismatch +0.0 +-0.02 +triple. Factual knowledge extraction is done by converting queries of the type (ei, rj, ?) into natural language +queries that can be answered by an LM. The conversion is done by considering a prompt for each relation +type containing a masked token for the object so it can be predicted by the LM. As an example, we may +convert a query such as (Barack Obama, profession, ?) into: “Barack Obama is a [MASK] by profession.". +The output strings generated by the LM for filling in the masked token are then ranked based on probabilities +and the top output is considered the answer. In our experiments, we use the manual prompts from Petroni +et al. (2019). +3.2 +Frequency Statistics +Let Q be a set of factual knowledge extraction queries of the form described in Section 3.1 and Qr represent +the subset of queries from Q that concern relation r. Let E represent a set of entities. We define the frequency +statistics of Q as a mapping ΦQ : E → N from any entity e ∈ E to a number in N indicating how many times +it appeared as answer in Q. +For two sets Q1 and Q2, let E1,2 represent the union of the entities that appear as answers in the two sets +and let τ = |E1,2| be the size of this set. We measure the similarity between ΦQ1 and ΦQ2 using the following +two measures. +Pearson correlation is defined as follows: +�τ +i=1(ΦQ1(ei) − φQ1)(ΦQ2(ei) − φQ2) +��τ +i=1 ΦQ1(ei) − φQ1 +��τ +i=1 ΦQ2(ei) − φQ2 +, +φQ1 = +�τ +i=1 ΦQ1(ei) +τ +, +φQ2 = +�τ +i=1 ΦQ2(ei) +τ +where φQ1 represents the average frequencies from the first set and φQ2 represents the average frequencies +from the second set. +Entity coverage of Q2 with respect to Q1 is defined as the proportion of answers for Q2 that are also +the answer to at least one query in Q1: +|{e | ΦQ2(e) > 0, ΦQ1(e) > 0}| +|{e | ΦQ2(e) > 0} +Note that if two sets are identical, their Pearson correlation is 1 and their entity coverage is also 1. +3.3 +Datasets +We aim to create datasets that help us better understand the positive and negative effects of finetuning. +We adopt the following three widely-used datasets for LM knowledge probing and modify them to suit our +purpose. +• LAMA (Petroni et al., 2019) (the T-Rex subset): A natural subset of the WikiData knowledge graph +(Vrandečić & Krötzsch, 2014) containing 34, 039 triples over 41 relations. +• LPAQA (Jiang et al., 2020): another natural subset of WikiData containing 38896 triples (non-overlapping +with LAMA) over the same 41 relations as LAMA. +• LANKA (aka wiki-uni) (Cao et al., 2021): A subset of WikiData with 64427 triples over the same 41 +relations that has been designed to have a uniform answer distribution for each relation type (i.e. for any +two entities e and e′ that appear as answers to queries for relation type r, ΦQr(e) = ΦQr(e′)). +4 + +For our experiments, we create three datasets with development (i.e. train and validation) and test sets as +follows: +• LowMismatch: uses LPAQA as development and LAMA as test set. Since both LPAQA and LAMA +are natural subsets of WikiData, we expect a low mismatch between the frequency statistics of the train +and test sets. +• MediumMismatch: uses LANKA as development and LAMA as test set. Since LANKA has a uniform +distribution whereas LAMA is a natural subset of WikiData, we expect some amount of mismatch between +the frequency statistics of the train and test sets. +• HighMismatch: combines all three datasets and divides the facts into two sets such that the answers in +one set are mutually exclusive from the answers in the other set, then uses one set for development and +the other set for testing. Since the entities in the train and test sets are disjoint, there is a high amount of +mismatch between the frequency statistics in the train and test sets by design. +The entity coverage and Pearson correlations between development/test splits for the 3 datasets is presented +in Table 1. For LowMismatch both values are high. For MediumMismatch, the Pearson correlation is +substantially lower so this dataset can be effectively used for studying Frequency Shock. For HighMismatch, +entity coverage is zero and Pearson correlation is close to zero, so this dataset can be effectively used for +studying both Frequency Shock and Range Shift. +For all the datasets, we fix the development set size to 40k queries (30k for train and 10k for validation). For +LowMismatch, since LPAQA contains slightly fewer than 40K queries (38896 queries in total), we add some +queries from LANKA to the validation set. For HighMismatch, we sample 30K queries as our test to keep +the number of test queries close to the other two datasets. Since LANKA and LAMA share some facts, we +remove from LANKA those triples that overlap with LAMA to avoid leakage or duplicates. +3.4 +Model Variants Used in the Experiments +While the majority of previous studies have focused on encoder-only LMs such as BERT that are limited to +single-token predictions (hence only applicable to a very restricted set of domains), in this paper we use an +encoder-decoder LM that allows for making multi-token predictions. In particular, unless stated otherwise, +we use the T51.1 XXL Raffel et al. (2019) (hereafter, referred to simply as T5). +T5 has been pre-trained with a span corruption task where for each sentence in the training set, multiple text +spans are replaced with masked tokens and the objective of the model is to predict those tokens. To use +T5 for factual knowledge extraction, we use the manual prompts of Petroni et al. (2019) to turn a query +(subject, relation, ?) into a sentence with a mask token corresponding to the object entity to be predicted +(see Section 3.1). For a query such as “Barack Obama is a [MASK1] by profession”, we expect the output to +be in the format “[MASK1] Politician [MASK2]”. T5 may produce extra text after [MASK2]. We simply +ignore any text generated after that token. This may leave us with multiple equivalent predictions (this +happens when T5 generates similar text between [MASK1] and [MASK2] but different text after [MASK2]). +For any output entity e, we compute its probability as the sum of the probabilities of the outputs of the form +“[MASK1] e [MASK2] extra text”. +We experiment with the following model variants. Zero-shot (ZS): simply feeding the masked query to +the pretrained model. Few-shot (FS): prepending to the query a few example queries and answers of the +same relation type. Reranking (RR): using a separate discriminatively finetuned LM to rerank the outputs +produced by a generative model has recently gained popularity Wallat et al. (2021); Lin et al. (2020b); Yadav +et al. (2021), so we also experiment with reranking for factual knowledge extraction. We finetune a model +that learns to predict which output among the top-k outputs of a pretrained model (ZS in our experiments) +is correct in a binary classification setup. Entities are then ranked based on the sum of the probabilities +produced by the pretrained model and the score produced by the finetuned model. Finetuning (FT): where +we finetune a model on the knowledge extraction task on the training set before evaluating on the test set. +5 + +Table 2: Performances on the three datasets (bold indicates winner). FT offers substantial gains when +development and test sets have similar frequency statistics, but the gain diminishes as the gap between +the frequency statistics becomes more; eventually on HighMismatch, the negative side-effects outweighs the +positive effects and finetuning becomes harmful overall. +LowMismatch +MediumMismatch +HighMismatch +Strategy +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +ZS +35.2 +47.9 +52.7 +35.2 +47.9 +52.7 +19.5 +28.3 +31.7 +FS +47.0 +56.1 +57.2 +42.8 +50.9 +52.5 +27.0 +33.4 +34.9 +RR +39.9 +49.9 +52.7 +38.7 +49.1 +52.7 +20.5 +28.8 +31.7 +FT +51.9 +68.4 +73.9 +43.6 +57.8 +63.2 +18.0 +27.4 +32.4 +3.5 +Metrics +We report the results using the widely-used Hit@k metric computed as the percentage of queries for which +the correct answer is ranked among the top k entities. We compute Hit@k for each relation type separately +and report the macro average, following previous work. +3.6 +Connection to Out-of-distribution Generalization +Classical machine learning settings assume train and test sets come from the same distribution. Recently, +there has been much effort in tackling more realistic scenarios where test distributions differ from training +distributions, known as out-of-distribution (OOD) generalization (see Shen et al. (2021) for a survey). While +OOD generalization has been investigated in many applications, it has remained largely unexplored for factual +knowledge extraction from LMs. This may be due to a lack of clarity on what a meaningful definition of OOD +is for this task; since test queries are written using the same template as the training queries, traditional +definitions are not straightforward to apply. Lewis et al. (2020) for example takes the extreme approach (in +the context of Open-Domain Question-Answering) of defining OOD as queries whose answers have never +been seen in training. +Using frequency statistics to measure the distance between train and test sets could be viewed as a novel +formulation of OOD for factual knowledge extraction from LMs, and the negative side-effects discussed in +Section 4 and the solutions considered in Section 5 are both relevant for robust solutions to OOD generalization. +Note, however, that the Frequency Shock phenomenon goes beyond OOD generalization, e.g. in cases such as +the example in Figure 1, the test query could still be an in-distribution query. +4 +Understanding Finetuning for Factual Knowledge Extraction +In this section, we design experiments that help better understand the effects of finetuning. +4.1 +Finetuning Performance Depends on Frequency Statistics +We first compare different model variants on the LowMismatch dataset. According to the results in Table 2, FT +yields a significant boost compared to the other variants. This result is consistent with what has been already +observed in existing literature (Fichtel et al., 2021). To understand where the improvement comes from, in +Table 3: Performance on the three datasets when using T5 1.1 small instead of XXL. +LowMismatch +MediumMismatch +HighMismatch +Strategy +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +ZS +18.4 +23.2 +24.4 +18.4 +23.2 +24.4 +12.6 +15.5 +16.4 +FT +28.5 +43.0 +49.5 +25.9 +36.2 +41.7 +5.9 +8.6 +9.7 +6 + +[0.0, 0.25] (0.25, 0.5] (0.5, 0.75] (0.75, 0.1] +Entity Coverage +0 +2 +4 +Avg. improvement +[-1.0, 0.0] (0.0, 0.33] (0.33, 0.66] (0.66, 1.0] +Pearson Correlation +0 +2 +4 +Avg. improvement +Figure 2: Macro average relative improvement of FT over ZS for different relation types in LowMismatch as +a function of entity coverage and Pearson correlation for the train and test sets. The figures show that most +of the improvement comes from the relations with a high entity coverage and Pearson correlation between +train and test sets. +Figure 2, we plot the improvement gained by the FT model over the ZS model on the LowMismatch dataset +as a function of the entity coverage and Pearson correlation between the train and test sets. Specifically, for +each relation type in the dataset, we measure the amount of entity coverage as well as the Pearson correlation +between train and test sets, then group different relation types based on these metrics and average the relative +improvements in each group. According to Figure 2, the improvements are mostly for those relation types +that have a high entity coverage and high Pearson correlation. +Based on the above result, we hypothesize that part of the improvement obtained by the FT model on the +LowMismatch dataset is due to biasing the pre-trained LM’s prediction frequencies toward that of the answer +set of the training data; since the train and test sets have similar frequency statistics, the frequency bias +given to the model due to finetuning matches that of the test set and that results in some improvement. To +verify this hypothesis, we next compare FT with the other variants on the MediumMismatch dataset where +entity coverage is still high but Pearson correlation is low. The results in Table 2 show that while FT still +gives a boost in performance, the gain is much lower compared to the LowMismatch case. As we will show in +Section 4.2, the gap between the performance of the FT model on LowMismatch and MediumMismatch is +mainly due to the difference in the frequency statistics in the train and test sets: finetuning biases the entity +frequency of the LM predictions toward that of the training data but the new frequencies do not match with +that of the test set on MediumMismatch. +Moreover, we compare FT with the other variants on the HighMismatch dataset where both entity coverage +and Pearson correlation are minimal. The results in Table 2 show that the bias in prediction frequencies of +the LM caused by finetuning in this case even outweigh the positive effect from Task Learning resulting in a +model that actually harms the overall performance and produces inferior results compared to the ZS model. +To verify how the above observations are affected by the scale of the LM, we also compare the ZS and FT +models on the three datasets when using the T5 1.1 Small model (60M parameters) instead of the T5 1.1 XXL +(11B parameters). According to the results in Table 3, the small model shows a similar behaviour where FT +provides a big boost on the LowMismatch dataset, but the amount of boost diminishes on MediumMismatch +and finetuning becomes harmful on HighMismatch. +Despite the striking results obtained with finetuned LMs for factual knowledge extraction in previous work, +the collective results in Table 2 show that (naively) finetuned LMs may not always be the best option for +factual knowledge extraction and KG construction as the performance of these models depends heavily on +the frequency statistics of the train and test sets. +4.2 +Frequency Shock and Range Shift are (Side-)Effects of Finetuning +We now design experiments that explain the behaviour observed for the FT model in Table 1 in terms of two +side-effects: Frequency Shock and Range Shift. +7 + +ZS +FS +RR +FT +FT+ZS FT+FS Mix+FS +1 +2 +3 +4 +5 +Percentage of answers +Common Percentage +Rare Percentage +Figure 3: Common (Rare) Percentage corresponds to the percentage of test queries for which the model +predicted an entity from the Common (Rare) set. According to the results, after finetuning on the MediumMis- +match dataset, the LM receives a frequency shock: it under-predicts the common entities and over-predicts +the rare entities. +Table 4: Accuracy of the models for +the Common and Rare entity sets for +the MediumMismatch dataset. Due +to Frequency Shock, the FT model +under-predicts the Common entities +and over-predicts the Rare entities. +As a result, when the FT model pre- +dicts a Common entity, there is a much +higher chance of it being true com- +pared to the other models, whereas +when the FT model predicts a Rare +entity, there is a much lower chance +of it being true. +Common +Accuracy +Rare +Accuracy +ZS +41.2 +47.9 +FS +51.4 +63.8 +FT +68.5 +14.4 +FT + FS +57.7 +29.5 +1:15 + FS +55.6 +64.6 +We selected a set of 10 cities that are expected to be commonly seen3 +as well as a set of 10 cities that appear as answers in LANKA but +are expected to be rarely seen in a dataset4. We named the two sets +Common and Rare respectively. We then measured the number of times +the models generated an entity from Common and Rare. +According to Figure 3, the ZS model predicts the Common entities +frequently and the Rare entities infrequently (this is in part due to the +frequency of the entities in the test set and in part due to the prior +of the language model). For the FS and RR models, the percentages +for the two sets are not substantially different from the ZS model. +However, for the FT model, due to the uniform distribution of of +the training set of MediumMismatch, the percentages for the two sets +changes substantially: the number of predictions from Common entities +drops by almost a third, and the one for Rare entities increases by 6x. +This reveals that Frequency Shock is indeed a side-effect of finetuning +as the difference between the frequency statistics of the training set +of MediumMismatch and what the pre-trained model expects causes +a shock to the model and makes it over-predict Rare entities and +under-predict Common entities. +We also measured the accuracy of the FT model when it produced a +Common or Rare entity and compared it to ZS. The results are reported +in Table 4. We observe that the accuracy for Common entities increases +from 41.2% to 68.5% and for Rare entities decreases from 47.9% to +14.4%. This is because the frequency shock caused by finetuning leads the model to predict the Common +entities only when it has high confidence in its prediction, but be less cautious about predicting the Rare +entities. This result also points out an interesting future direction for measuring the model uncertainty (or +whether the model knows what it does not know) through a combination of a ZS model and a model that has +been finetuned on a slightly different data distribution. +To further analyze the effect of Frequency Shock and Range Shift we compare models in terms of the overlap +between their predicted entities and those in the train and test sets of the HighMismatch dataset, where +3We selected the 10 cities from Cao et al. (2021) (Figure 2), namely {London, Paris, Tokyo, Boston, Rome, Chicago, Berlin, +Montreal, Moscow, Milan} +4We do this by randomly selecting 10 cities from the LANKA answers that appear as an answer in LAMA less than 20 times, +namely {Boise, Tirana, Myanmar, Hanover, Aberdeen, Chelsea, Kentucky, Oldham, Hastings, Parma} +8 + +Figure 4: The percentages of overlap between the entities predicted by the models and those of the train and +test sets for the HighMismatch dataset (entities in the train and test sets are disjoint). +the train and test entities are mutually exclusive. According to the results in Figure 4, we observe that the +FT model predicts the entities from the train set significantly more than the FS model (almost 62% relative +increase). This shows a clear case of Range Shift. +We also manually analyzed the outputs of the ZS and FT models for the “born in” relation (as a representative +relation)5 and grouped the predictions of each model into three classes: 1- the output is not a location, 2- the +output is the correct location, and 3- the output is an incorrect location. We then compared the number of +queries in the cross-product of the categories for the ZS and FT model. The results are presented in Table 5. +Out of the 60 queries on HighMismatch where ZS predicted the correct location and FT predicted an incorrect +location, in 59 cases the top answer of the FT model was one of the entities from the training set answers, +showing another clear (and perhaps more severe) case for Range Shift. Moreover, on MediumMismatch, out of +the 58 queries for which the answer changed from a correct location to an incorrect location after finetuning, +in 19% of those cases the correct entity was “London” – a commonly occurring city (note that only for 6% of +the queries the correct answer is “London”); In another 33% of those cases the correct entity is one of “Paris”, +“Berlin”, “Barcelona”, “Vienna” and “Brooklyn”, whereas only for 7% of the queries the answer is one of +these cities. This is because the training set of MediumMismatch has a uniform distribution and finetuning +on it leads to frequency shock where common entities (such as “London”) are under-predicted. +4.3 +The positive effect of Task Learning +Similar to the existing literature Fichtel et al. (2021), Table 5 provides multiple evidences showing Task +Learning is a positive effect of finetuning. First, while ZS predicts non-location outputs (mostly years) for +some queries, FT correctly learns to predict a location for the queries6. Secondly, for the 115 queries where +ZS predicted an incorrect location but FT predicted a correct one on LowMismatch, in 90 cases the ZS model +had generated a correct country as the top output, and the FT model learned to predict the correct city +(which is the expected sub-type) instead of country. We observe a similar behaviour for 39/62 queries in +5We selected this relation because it is simple to verify the model’s output types and subtypes. +6As an interesting side note, for the queries for which the ZS model outputs a different type than a location, even though the +FT model learns to predict a location, it tends to predict a wrong location; future work can use this signal to predict when the +LM does not know the answer to a question. +Table 5: A comparison of the ZS and FT models for the “born in” relation. NL, CL and IL stand for Not a +Location, Correct Location, and Incorrect Location respectively. +FT +LowMismatch +MediumMismatch +HighMismatch +ZS +NL +CL +IL +NL +CL +IL +NL +CL +IL +NL +0 +11 +105 +0 +3 +113 +0 +0 +165 +CL +0 +89 +26 +0 +57 +58 +0 +3 +60 +IL +0 +115 +598 +0 +62 +651 +0 +2 +838 +9 + +FT Model +Other +10.8% +Train Entities +Test Entities +31.4% +57.8%FS Model +Other +16.9% +Train Entities +19.4% +Test Entities +63.6%Table 6: Results for model mixing (bold indicates winner). The best single model corresponds to the model +that gave the best result for each dataset (e.g., for T5 XXL FT is the best single model for LowMismatch and +MediumMismatch, and FS for HighMismatch). UB stands for upper-bound. +LowMismatch +MediumMismatch +HighMismatch +T5 +Model Mixing +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +XXL +Best Single Model +51.9 +68.4 +73.9 +43.6 +57.8 +63.2 +27.0 +33.4 +34.9 +FT + ZS +51.3 +66.3 +71.5 +45.8 +58.9 +64.1 +22.0 +31.5 +35.8 +FT + FS +53.5 +69.0 +74.2 +46.9 +60.9 +66.1 +26.2 +34.4 +38.2 +FT + FS (UB) +59.3 +71.8 +75.9 +52.9 +65.1 +69.7 +29.2 +36.9 +40.6 +Small +Best Single Model +28.5 +43.0 +49.5 +25.9 +36.2 +41.7 +12.6 +15.5 +16.4 +FT + ZS +28.7 +43.3 +50.5 +26.3 +37.1 +42.5 +12.7 +16.2 +17.5 +MediumMismatch. The other cases where the prediction changed from incorrect location to correct location +can be explained by better learning the semantics of the task as a result of finetuning. +5 +Improving Finetuning +To avoid the side-effects identified in Section 4 and use finetuned LMs for factual knowledge extraction and +KG construction, one may be tempted to create a training set that has a large coverage of various entities and +that also has a high Pearson correlation with what is expected to be seen at the test time. We note, however, +that entity coverage and Pearson correlation are somewhat at odds with each other. That is because if we +add many queries to the training data whose answers are novel entities, it will cause the Pearson correlation +to go down unless we also add a prohibitively large number of queries with common entities as answers to +retain the proportions. Also, if we wish to keep the Pearson correlation high, many of the rare entities may +not appear in the training set. +We aim to find solutions by changing the finetuning strategy. Given that LMs are pre-trained on large corpora +of text (typically much larger than the finetuning dataset), we may expect the original entity distribution +of the LM (corresponding to its prior distribution) to be more robust to situations with different frequency +statistics. In this section, we exploit this insight to provide two strategies to remedy the negative effect of +Frequency Shock and Range Shift in finetuning while still retaining the benefits of Task Learning. +5.1 +Model Mixing Alleviates Side-Effects +As we observed in the previous sections, the FT model has the advantage of better learning the task and as a +result producing better results than the alternative models in situations such as the LowMismatch dataset. +However, it also has the disadvantage of introducing a frequency bias that may lead to low performance on +situations such as the MediumMismatch and HighMismatch datasets. On the other hand, the ZS and FS +models have the advantage of being more robust to situations such as the MediumMismatch and HighMismatch +datasets, but their performance falls short of the FT model in situations such as the LowMismatch dataset. +Here, we explore whether these models can be combined to get the best of the two worlds: the improved +performance of the FT models and the robustness of the ZS and FS models. We experiment with a simple +mixing approach where we average the scores produced by the FT model for each output with that of the +other models; we leave more sophisticated combination strategies as future work. +We first verify if model mixing helps alleviate Frequency Shock and is more robust. Figure 3 indicates the +percentage of queries for which the FT+ZS and FT+FS models predicted one of the Common or Rare entities. +One can see from the figure that, contrary to the FT model, the distributions for these models are much closer +to that of the ZS and FS models. The correction effect is rather one-sided: while the Common entities are not +under-predicted anymore, the Rare entities are still slightly over-predicted. This is also apparent from the +accuracy of the FT+FS model on the Common and Rare sets in Table 4: the performance on the Common set +10 + +Table 7: Results for mixture finetuning with different mixture ratios (bold indicates winner). 1:0 corresponds +to standard finetuning. Mixture training consistently provides a boost in performance, especially for larger +mixture ratios. The benefits from mixture training and model mixing can be combined. +LowMismatch +MediumMismatch +HighMismatch +T5 +Mixture +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +Hit@1 +Hit@3 +Hit@5 +XXL +1:0 +51.9 +68.4 +73.9 +43.6 +57.8 +63.2 +18.0 +27.4 +32.4 +1:1 +51.8 +68.4 +74.0 +45.6 +61.0 +66.9 +18.2 +27.3 +32.3 +1:5 +52.2 +68.0 +73.9 +46.5 +60.5 +66.6 +18.5 +27.9 +32.9 +1:15 +52.7 +68.4 +73.9 +45.6 +60.2 +66.1 +19.5 +28.8 +33.4 +1:15 + FS +53.4 +69.2 +74.4 +47.2 +63.0 +68.6 +26.7 +35.1 +39.0 +Small +1:0 +28.5 +43.0 +49.5 +25.9 +36.2 +41.7 +5.9 +8.6 +9.7 +1:15 +31.6 +47.9 +54.2 +26.2 +36.8 +42.9 +10.5 +13.0 +15.5 +becomes similar to the FS model as Common entities are not under-predicted anymore, but the performance +on the Rare set is still much lower than the FS model as Rare entities are still being over-predicted. +We then look at whether model mixing provides better predictions overall. The results are presented in +Table 6. When using the T5 XXL model, for FT+ZS even though the difference between the two models is +quite large on LowMismatch, the model results are only slightly worse than the FT model itself. For the +other two datasets, where the difference between the two models is much smaller, model mixing leads to +substantial improvement. FT+FS offers higher performance than both individual models on LowMismatch +and MediumMismatch. For HighMismatch, all numbers improve substantially with respect to FT; with respect +to FS, however, Hit@1 goes slightly down whereas Hit@3 and Hit@5 improve. The same trend holds for the +T5 Small model where mixing ZS and FT performs better than both ZS and FT in isolation. Note that for +the HighMismatch dataset, even though the FT model performs poorly in isolation, mixing it with ZS still +brings improvement for ZS. +While, in the past, model mixing has been shown to provide only marginal gains in different applications +even when multiple models are being combined, the results in Table 6 show that a simple parameter-free +combination of only two models provides large boosts of up to 7.6% on the MediumMismatch dataset. +Furthermore, we have included in Table 6 an upper-bound result for FT+FS where we assume having access +to an oracle that can tell if we should trust the FT model or the FS model for each query. The upper-bound +result is significantly higher than each of the individual models, thus showing that there is a large subset of +the data where one model produces the correct answer whereas the other model does not, hence indicating +that the models work well on different subsets of the data and that more sophisticated combinations can +potentially lead to more improvements. These results confirm that besides the previously studied benefits of +model mixing Naderi et al. (2021); Pranesh et al. (2020); Wang et al. (2022); Ormerod (2022), it plays a +much significant role for knowledge extraction from finetuned LMs by correcting the side-effects of finetuning. +5.2 +Mixture Training Alleviates Side-Effects +We examine if the following multi-task finetuning strategy can be an effective solution to the identified +side-effects. We finetune the model on a combination of two tasks: 1- Factual Knowledge Extraction 2- The +original pre-training task of the LM ("masked language modeling"). Let α : β represent the ratio between the +number of queries from the first and the second tasks in each training batch. We set α = 1 and finetune +models with different values for β for the three datasets, to see how mixture training with different ratios +affects the model performance. Intuitively, if we continue to finetune the model with both these tasks, the +second task should help the LM observe the entities with a similar frequency as in its pre-training stage, thus +avoiding frequency shock to the model. This approach has been previously shown to aid with the forgetting +effect of finetuning on LMs (He et al., 2021). The results in this section extend those results to the case +of Frequency Shock. From the results in Table 7, we can see that mixture finetuning consistently provides +improvements across the three datasets. The amount of improvement larger for the HighMismatch dataset +where the statistics differ more. +11 + +Finally, we combine the mixture finetuned model with the few-shot model and see the benefits from the +two solutions is additive. We see from Figure 3 that the resulting model does not under-predict Common +entities and does not over-predict Rare entities. Also, from Table 4, we see that the resulting model does not +produce a substantially higher accuracy on the Common set due to under-prediction, and does not produce a +substantially lower accuracy on the Rare set due to over-prediction. Finally, we see from Table 7 that the +benefits from model mixing and mixture training can be combined to make yet better predictions. +5.3 +A Knowledge Extraction Recipe +So far, we have observed that 1) a finetuned model may suffer from Frequency Shock and Range Shift, 2) +combining finetuned and few-shot models help improve the results, and 3) mixture finetuning helps too. Here, +we address a key question: what is the best strategy for extracting factual knowledge from LMs to construct +KGs for real applications? +Table 8: Macro average performances of +different models over the three datasets. +Overall, an combination of a mixture +finetuned model (with a large mixture +ratio) with an FS model performs best. +If one wishes to use only a single model +during inference, then the best option +is the mixture trained model. +If one +wants to avoid mixture training due to +its higher cost, then the FS model is the +winner followed by the FT model. +Macro average +Strategy +Hit@1 +Hit@3 +Hit@5 +ZS +30.0 +41.4 +45.7 +FS +38.9 +46.8 +48.2 +FT +37.8 +51.2 +56.5 +RR +33.0 +42.6 +45.7 +FT + ZS +39.7 +52.2 +57.1 +FT + FS +42.2 +54.8 +59.5 +1:15 +39.3 +52.5 +57.8 +1:15 + ZS +40.2 +53.2 +58.1 +1:15 + FS +42.5 +55.8 +60.1 +In real applications of knowledge extraction from LMs for the +purpose of knowledge graph construction, we expect to see a combi- +nation of the properties in the three datasets studied in this paper. +That is, we expect training data to be available mostly for some +subsets of the knowledge domain that needs to be extracted, and be +scarce for other parts. We also expect the frequency statistics of the +train data and the data that needs to be extracted to be similar for +parts of the knowledge domain, and differ to various degrees in other +parts of the knowledge domain. Therefore, we expect knowledge +extraction in real applications to involve a combination of the three +datasets studied in this work. To find the best strategy for factual +knowledge extraction from LMs, we report the average performance +of different combinations of the techniques introduced in this paper +on our three datasets. +According to the results in Table 8, the best strategy is to finetune +a mixture model and combine it with an FS model. This strategy +leads to 12.4% relative improvement over vanilla finetuning in terms +of Hit@1. Given that mixture finetuning substantially increases the +time and cost required for finetuning (especially for higher ratios +for the pretraining task), if one wants to avoid that cost the next +best option is to use FT+FS. Alternatively, if one can afford the +training cost but needs to reduce the inference cost by using a single +model during inference and avoiding making multiple model calls +per query, the best strategy is to use a mixture finetuned model. +Finally, FS may result in better aggregate performance than FT. +6 +Conclusion +Language models (LMs), especially when finetuned, can be a great source of knowledge for constructing +(or augmenting) knowledge graphs. However, finetuning may also exhibit negative effects for knowledge +extraction that are important to understand and be aware of. In this paper, we identified Frequency Shock +and Range Shift as side-effects of finetuning, which can be helpful or harmful depending on the frequency +statistics of the dataset. We then considered two solutions, model mixing and mixture finetuning with the +pre-training task, that help remedy the negative effects and result in improved performance. 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Note that in the case of mixture finetuning (Section 5.2), +each epoch consists of more batch updates compared to single-task finetuning because a portion of the +examples in each batch come from the pre-training task. We set the batch size to 128 and the learning rate to +0.0001. We measured the Hit@1 performance on the validation set after each epoch and selected the model +parameters from the best performing epoch on the validation for evaluating on the test set. +15 + +For the few-shot experiments (FS model), for each relation type we selected 10 random examples from the +training set of the dataset. To get an estimate of the standard deviation due to the choice of different +examples, we repeated our experiment for the LowMismatch dataset 5 times each time selecting different +examples and observed a standard deviation of 0.56. +Note that while for the three datasets we use in this paper the queries have been selected in such a way +that the answer is mostly a single token according to the BERT vocabulary, those answers are already +multi-token according to the T5 vocabulary and that allows us to test the multi-token prediction ability of +the LMs. Previous work restrict the model prediction distribution to a predefined set of tokens and ignores +any predicted outputs outside that set. We disregard that pre-defined set during training and validation to +avoid unwanted artifacts introduced due to the use of that specific vocabulary set. However, we use that set +for measuring performance on the test set so that the final results are in the same footing as those of the +previously published work. +16 + diff --git a/XdFIT4oBgHgl3EQfiyst/content/tmp_files/load_file.txt b/XdFIT4oBgHgl3EQfiyst/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ba410406f6fb03088b0e07050b63f6e4e5e05b3 --- /dev/null +++ b/XdFIT4oBgHgl3EQfiyst/content/tmp_files/load_file.txt @@ -0,0 +1,881 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf,len=880 +page_content='Understanding Finetuning for Factual Knowledge Extraction from Language Models Mehran Kazemi mehrankazemi@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='com Sid Mittal sidmittal@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='com Deepak Ramachandran ramachandrand@google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='com Google Research Abstract Language models (LMs) pretrained on large corpora of text from the web have been observed to contain large amounts of various types of knowledge about the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This observation has led to a new and exciting paradigm in knowledge graph construction where, instead of manual curation or text mining, one extracts knowledge from the parameters of an LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Recently, it has been shown that finetuning LMs on a set of factual knowledge makes them produce better answers to queries from a different set, thus making finetuned LMs a good candidate for knowledge extraction and, consequently, knowledge graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In this paper, we analyze finetuned LMs for factual knowledge extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We show that along with its previously known positive effects, finetuning also leads to a (potentially harmful) phenomenon which we call Frequency Shock, where at the test time the model over-predicts rare entities that appear in the training set and under-predicts common entities that do not appear in the training set enough times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We show that Frequency Shock leads to a degradation in the predictions of the model and beyond a point, the harm from Frequency Shock can even outweigh the positive effects of finetuning, making finetuning harmful overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then consider two solutions to remedy the identified negative effect: 1- model mixing and 2- mixture finetuning with the LM’s pre-training task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The two solutions combined lead to significant improvements compared to vanilla finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 1 Introduction Recently, Language Models (LMs) pre-trained on large corpora of web documents such as CommonCrawl1 have achieved impressive results on multiple NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In their pioneering work, Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019) showed that LMs also contain a large amount of factual knowledge about the world, motivating a line of research to extract this knowledge using well-designed prompting or finetuning methods (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' It also led to probing for other types of knowledge (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Davison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' These findings motivate a new Knowledge Graph (KG) construction paradigm where instead of laboriously hand-curating or mining facts, LMs can be used as a simple and effective pipeline to translate heterogeneous data sources on the web into structured KG representations (West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Allaway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Fichtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021) show that LMs finetuned on a set of queries perform well on other factual queries and outperform other knowledge probing techniques (such as prompt tuning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Some recent work (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021) however, casts doubt on previous findings by showing that when finetuned on in-distribution data (data that follows the same distribution as the test data), there are statistical patterns in training that can be exploited by a model leading to over-estimation of the test performance of LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Moreover, Wallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021) show that finetuning may lead to forgetting the previously known facts by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Therefore, to thoroughly assess the merit of finetuned LMs for KG construction, a clear understanding of their strengths and failure modes is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' These results raise the question of whether for 1http://commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='org 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='11293v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='CL] 26 Jan 2023 constructing KGs from LMs, using a finetuned LM is a good strategy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Toward answering this question, a clear understanding of the strengths and failure modes of finetuned LMs for factual knowledge extraction is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Zero-shot Test Query: Marat Makhmutov was born in [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Correct Answer: Moscow Model Answer: Moscow Finetuned Train Data: Out of all “X was born in [MASK] .” queries: the answer to 5 of them is Moscow, the answer to 5 of them is Baku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Test Query: Marat Makhmutov was born in [MASK] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Correct Answer: Moscow Model Answer: Baku Figure 1: For the query “Marat Makhmutov was born in [MASK] .”, a pre-trained language model correctly returns “Moscow” as answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Once we finetune the language model on some data, it changes its prediction to “Baku” even though both “Moscow” and “Baku” appear as answers in the training set an equal number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In this paper, we provide a deeper understanding of fine- tuned LMs for knowledge extraction and provide an analysis that helps understand the behaviour, the advantages and disadvantages of finetuning LMs for knowledge extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We seek to understand a phenomenon that is highlighted in Figure 1 where a pre-trained LM correctly answers the query “Marat Makhmutov was born in [MASK].” with “Moscow”, whereas a finetuned LM modifies its prediction to “Baku” despite seeing “Moscow” and “Baku” an equal number of times during finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We identify three effects of finetuning (the first one already explicated, but the other two less understood): Task Learning: Finetuning makes the LM understand the semantics of the task/prompt and learn the expected output domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' the expected entity types/subtypes) for each relation type, Frequency Shock: Finetuning biases the model’s predic- tions towards the frequency statistics of the entities seen as answers during finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' When entities that are expected to be rare appear as answers in the training set, the model receives a frequency shock and tends to over-predict these entities for many queries in the test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Moreover, when entities that are expected to be common do not appear in the dataset enough times, the model receives a frequency shock and tends to under-predict these entities for the queries in the test examples, Range Shift: Finetuning makes the model mostly predict entities from those seen as answers during finetuning (this could be considered as a specific case of Frequency Shock).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' While previous work typically explains the phenomanon in Figure 1 as forgetting effect (Wallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021), our study reveals a more nuanced explanation in terms of Frequency Shock: even though both “Moscow” and “Baku” have been observed an equal number of times in the training set, since “Baku” is expected to be a less common entity2 and hence less observed during the pre-training of the language model, the finetuned model receives a frequency shock leading to an over-prediction of the entity “Baku”, hence corrupting an originally correct prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Note that Frequency Shock and Range Shift are related to the problem of out-of-distribution (OOD) generalization in machine learning, see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 for more discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We design careful experiments to better understand Frequency Shock and Range Shift and show that while Task Learning may lead to improvements, Frequency Shock and Range Shift may lead to a degradation that can even sometimes outweigh the positive effect of Task Learning such that finetuning hurts the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then propose two approaches to remedy the negative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' First, we show that mixing a finetuned model with a zero-shot or few-shot model can lead to correcting for the shock and range shift and consequently yields better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Second, we show that a version of multi-task finetuning where we mix the knowledge extraction task with the original pre-training task of the LM can also help alleviate the negative effect of Frequency Shock and Range Shift and leads to better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The two approaches combined lead to an aggregate 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4% improvement in model performance over the vanilla finetuning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Our main contributions include: 1- Identifying Frequency Shock and Range Shift as side-effects of finetuned LMs for factual knowledge extraction, 2- Creating datasets for thoroughly analyzing these effects and identifying their root causes, 3- Identifying two solutions for the side-effects leading to an aggregate 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4% improvement in model performance, and 4- Proposing a practical recipe for knowledge extraction from LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 2As an example in the LAMA probe, which is a natural subset of a large real-world knowledge graph, “Baku” appears only 4 times as answer whereas “Moscow” appears 13 times and a more common entity such as “London” appears 59 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 2 2 Related Work The works from the literature that relate to our paper can be categorized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Knowledge Probing: Pre-training makes LMs contain a large amount of factual knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' A large body of work aims at probing how much knowledge is stored in the parameters of LMs, and whether they can be used to replace KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' These works include probing for factual (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2019), commonsense (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Davison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Yin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2022), biomedical (Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021), numerical (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020a), scale (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020), and many other types of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' While we focus on factual knowledge extraction in this paper, our results can extend to other types of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Finetuning and Prompt Tuning for Better Knowledge Extraction: Most related to our paper are the works that aim at improving the knowledge extraction from LMs using prompt tuning or finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The works on prompt tuning either mine prompts from the web (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020), optimize prompts in the discrete space of words and tokens (Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020), optimize prompts in the continuous embedding space (Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021), or use adapters (Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' It has been recently shown that finetuning may result in higher performance gains compared to prompt tuning (Fichtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The merit of finetuned LMs has been also shown for common-sense knowledge extraction (Bosselut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Previous work also studies the effect of dataset size for finetuning (Wallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Fichtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Da et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021), but the negative effects finetuning (studied in this paper) remain unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For a full review of the literature on knowledge probing and extraction, we refer to (Safavi & Koutra, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' AlKhamissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' KG Construction (from LMs): Typically, KGs are either created manually (by domain experts or through crowd-sourcing) Miller (1995);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Vrandečić & Krötzsch (2014), automatically (by extracting from the web) Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Carlson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Bhakthavatsalam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020), or a combination of the two Speer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Sap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In this paper, we are mostly interested in an emerging line of work that constructs KGs directly from LMs or by leveraging LMs West et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Bosselut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Hao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Allaway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' KG Completion: A class of approaches under the umbrella of KG completion aim at predicting new facts for an incomplete KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Approaches have been developed for static Bordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Kazemi & Poole (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Trouillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2016), temporal Goel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Lacroix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020), commensense Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2016), and many other types of KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' While these works derive new facts based solely on the existing ones, the work in this paper utilizes existing facts as well as an LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Generalization in Question Answering (QA): Generalization, especially out-of-distribution (OOD), has been a hot topic of study for various QA settings including open-domain QA Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021), reading comprehension Talmor & Berant (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Fisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019), and visual QA Kervadec et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Gokhale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' These works mainly concern the statistical pattern differences of the questions or the question- answer pairs between the train and test sets and propose solutions such as multi-task learning, adversarial training, or data augmentation to reduce reliance on spurious correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Knowledge extraction can be considered as a specific case of QA where questions are based on template prompts and do not require multi-hop reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' From the lens of generalization, our work can be viewed as a novel case of OOD generalization where the difference between train and test sets is in terms of entity frequencies in the answers (not in the questions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The closest to our work is the study in Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020) where generalization is measured for novel answer entities in the test set, but our work goes beyond that and studies Frequency Shock for non-novel entities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3 Experimental Setup We start by describing the factual knowledge extraction problem and the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 Factual Knowledge Extraction Let E = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' , en} be a set of entities and R = {r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' , rm} be a set of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' A knowledge graph (KG) is a set of triples of the form (ei, rj, ek) where ei is the subject, rj is the relation, and ek is the object of the 3 Table 1: Entity coverage and Pearson correlation for the three datasets studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Dataset Entity Coverage Pearson LowMismatch 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='68 MediumMismatch 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='30 HighMismatch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='02 triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Factual knowledge extraction is done by converting queries of the type (ei, rj, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=') into natural language queries that can be answered by an LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The conversion is done by considering a prompt for each relation type containing a masked token for the object so it can be predicted by the LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' As an example, we may convert a query such as (Barack Obama, profession, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=') into: “Barack Obama is a [MASK] by profession.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The output strings generated by the LM for filling in the masked token are then ranked based on probabilities and the top output is considered the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In our experiments, we use the manual prompts from Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 Frequency Statistics Let Q be a set of factual knowledge extraction queries of the form described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 and Qr represent the subset of queries from Q that concern relation r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Let E represent a set of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We define the frequency statistics of Q as a mapping ΦQ : E → N from any entity e ∈ E to a number in N indicating how many times it appeared as answer in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For two sets Q1 and Q2, let E1,2 represent the union of the entities that appear as answers in the two sets and let τ = |E1,2| be the size of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We measure the similarity between ΦQ1 and ΦQ2 using the following two measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Pearson correlation is defined as follows: �τ i=1(ΦQ1(ei) − φQ1)(ΦQ2(ei) − φQ2) ��τ i=1 ΦQ1(ei) − φQ1 ��τ i=1 ΦQ2(ei) − φQ2 , φQ1 = �τ i=1 ΦQ1(ei) τ , φQ2 = �τ i=1 ΦQ2(ei) τ where φQ1 represents the average frequencies from the first set and φQ2 represents the average frequencies from the second set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Entity coverage of Q2 with respect to Q1 is defined as the proportion of answers for Q2 that are also the answer to at least one query in Q1: |{e | ΦQ2(e) > 0, ΦQ1(e) > 0}| |{e | ΦQ2(e) > 0} Note that if two sets are identical, their Pearson correlation is 1 and their entity coverage is also 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 Datasets We aim to create datasets that help us better understand the positive and negative effects of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We adopt the following three widely-used datasets for LM knowledge probing and modify them to suit our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LAMA (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2019) (the T-Rex subset): A natural subset of the WikiData knowledge graph (Vrandečić & Krötzsch, 2014) containing 34, 039 triples over 41 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LPAQA (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2020): another natural subset of WikiData containing 38896 triples (non-overlapping with LAMA) over the same 41 relations as LAMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LANKA (aka wiki-uni) (Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021): A subset of WikiData with 64427 triples over the same 41 relations that has been designed to have a uniform answer distribution for each relation type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' for any two entities e and e′ that appear as answers to queries for relation type r, ΦQr(e) = ΦQr(e′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 4 For our experiments, we create three datasets with development (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' train and validation) and test sets as follows: LowMismatch: uses LPAQA as development and LAMA as test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Since both LPAQA and LAMA are natural subsets of WikiData, we expect a low mismatch between the frequency statistics of the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' MediumMismatch: uses LANKA as development and LAMA as test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Since LANKA has a uniform distribution whereas LAMA is a natural subset of WikiData, we expect some amount of mismatch between the frequency statistics of the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' HighMismatch: combines all three datasets and divides the facts into two sets such that the answers in one set are mutually exclusive from the answers in the other set, then uses one set for development and the other set for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Since the entities in the train and test sets are disjoint, there is a high amount of mismatch between the frequency statistics in the train and test sets by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The entity coverage and Pearson correlations between development/test splits for the 3 datasets is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For LowMismatch both values are high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For MediumMismatch, the Pearson correlation is substantially lower so this dataset can be effectively used for studying Frequency Shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For HighMismatch, entity coverage is zero and Pearson correlation is close to zero, so this dataset can be effectively used for studying both Frequency Shock and Range Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For all the datasets, we fix the development set size to 40k queries (30k for train and 10k for validation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For LowMismatch, since LPAQA contains slightly fewer than 40K queries (38896 queries in total), we add some queries from LANKA to the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For HighMismatch, we sample 30K queries as our test to keep the number of test queries close to the other two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Since LANKA and LAMA share some facts, we remove from LANKA those triples that overlap with LAMA to avoid leakage or duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 Model Variants Used in the Experiments While the majority of previous studies have focused on encoder-only LMs such as BERT that are limited to single-token predictions (hence only applicable to a very restricted set of domains), in this paper we use an encoder-decoder LM that allows for making multi-token predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In particular, unless stated otherwise, we use the T51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 XXL Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019) (hereafter, referred to simply as T5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' T5 has been pre-trained with a span corruption task where for each sentence in the training set, multiple text spans are replaced with masked tokens and the objective of the model is to predict those tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To use T5 for factual knowledge extraction, we use the manual prompts of Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2019) to turn a query (subject, relation, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=') into a sentence with a mask token corresponding to the object entity to be predicted (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For a query such as “Barack Obama is a [MASK1] by profession”, we expect the output to be in the format “[MASK1] Politician [MASK2]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' T5 may produce extra text after [MASK2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We simply ignore any text generated after that token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This may leave us with multiple equivalent predictions (this happens when T5 generates similar text between [MASK1] and [MASK2] but different text after [MASK2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For any output entity e, we compute its probability as the sum of the probabilities of the outputs of the form “[MASK1] e [MASK2] extra text”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We experiment with the following model variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Zero-shot (ZS): simply feeding the masked query to the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Few-shot (FS): prepending to the query a few example queries and answers of the same relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Reranking (RR): using a separate discriminatively finetuned LM to rerank the outputs produced by a generative model has recently gained popularity Wallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Yadav et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021), so we also experiment with reranking for factual knowledge extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We finetune a model that learns to predict which output among the top-k outputs of a pretrained model (ZS in our experiments) is correct in a binary classification setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Entities are then ranked based on the sum of the probabilities produced by the pretrained model and the score produced by the finetuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Finetuning (FT): where we finetune a model on the knowledge extraction task on the training set before evaluating on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 5 Table 2: Performances on the three datasets (bold indicates winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' FT offers substantial gains when development and test sets have similar frequency statistics, but the gain diminishes as the gap between the frequency statistics becomes more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' eventually on HighMismatch, the negative side-effects outweighs the positive effects and finetuning becomes harmful overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LowMismatch MediumMismatch HighMismatch Strategy Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 ZS 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 FS 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 RR 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 FT 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 Metrics We report the results using the widely-used Hit@k metric computed as the percentage of queries for which the correct answer is ranked among the top k entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We compute Hit@k for each relation type separately and report the macro average, following previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 Connection to Out-of-distribution Generalization Classical machine learning settings assume train and test sets come from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Recently, there has been much effort in tackling more realistic scenarios where test distributions differ from training distributions, known as out-of-distribution (OOD) generalization (see Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021) for a survey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' While OOD generalization has been investigated in many applications, it has remained largely unexplored for factual knowledge extraction from LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This may be due to a lack of clarity on what a meaningful definition of OOD is for this task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' since test queries are written using the same template as the training queries, traditional definitions are not straightforward to apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020) for example takes the extreme approach (in the context of Open-Domain Question-Answering) of defining OOD as queries whose answers have never been seen in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Using frequency statistics to measure the distance between train and test sets could be viewed as a novel formulation of OOD for factual knowledge extraction from LMs, and the negative side-effects discussed in Section 4 and the solutions considered in Section 5 are both relevant for robust solutions to OOD generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Note, however, that the Frequency Shock phenomenon goes beyond OOD generalization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' in cases such as the example in Figure 1, the test query could still be an in-distribution query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 4 Understanding Finetuning for Factual Knowledge Extraction In this section, we design experiments that help better understand the effects of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 Finetuning Performance Depends on Frequency Statistics We first compare different model variants on the LowMismatch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to the results in Table 2, FT yields a significant boost compared to the other variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This result is consistent with what has been already observed in existing literature (Fichtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To understand where the improvement comes from, in Table 3: Performance on the three datasets when using T5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 small instead of XXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LowMismatch MediumMismatch HighMismatch Strategy Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 ZS 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 FT 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 6 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='25] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='75] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1] Entity Coverage 0 2 4 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' improvement [-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='33] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='66] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='66, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0] Pearson Correlation 0 2 4 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' improvement Figure 2: Macro average relative improvement of FT over ZS for different relation types in LowMismatch as a function of entity coverage and Pearson correlation for the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The figures show that most of the improvement comes from the relations with a high entity coverage and Pearson correlation between train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Figure 2, we plot the improvement gained by the FT model over the ZS model on the LowMismatch dataset as a function of the entity coverage and Pearson correlation between the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Specifically, for each relation type in the dataset, we measure the amount of entity coverage as well as the Pearson correlation between train and test sets, then group different relation types based on these metrics and average the relative improvements in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to Figure 2, the improvements are mostly for those relation types that have a high entity coverage and high Pearson correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Based on the above result, we hypothesize that part of the improvement obtained by the FT model on the LowMismatch dataset is due to biasing the pre-trained LM’s prediction frequencies toward that of the answer set of the training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' since the train and test sets have similar frequency statistics, the frequency bias given to the model due to finetuning matches that of the test set and that results in some improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To verify this hypothesis, we next compare FT with the other variants on the MediumMismatch dataset where entity coverage is still high but Pearson correlation is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results in Table 2 show that while FT still gives a boost in performance, the gain is much lower compared to the LowMismatch case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' As we will show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2, the gap between the performance of the FT model on LowMismatch and MediumMismatch is mainly due to the difference in the frequency statistics in the train and test sets: finetuning biases the entity frequency of the LM predictions toward that of the training data but the new frequencies do not match with that of the test set on MediumMismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Moreover, we compare FT with the other variants on the HighMismatch dataset where both entity coverage and Pearson correlation are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results in Table 2 show that the bias in prediction frequencies of the LM caused by finetuning in this case even outweigh the positive effect from Task Learning resulting in a model that actually harms the overall performance and produces inferior results compared to the ZS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To verify how the above observations are affected by the scale of the LM, we also compare the ZS and FT models on the three datasets when using the T5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 Small model (60M parameters) instead of the T5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 XXL (11B parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to the results in Table 3, the small model shows a similar behaviour where FT provides a big boost on the LowMismatch dataset, but the amount of boost diminishes on MediumMismatch and finetuning becomes harmful on HighMismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Despite the striking results obtained with finetuned LMs for factual knowledge extraction in previous work, the collective results in Table 2 show that (naively) finetuned LMs may not always be the best option for factual knowledge extraction and KG construction as the performance of these models depends heavily on the frequency statistics of the train and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 Frequency Shock and Range Shift are (Side-)Effects of Finetuning We now design experiments that explain the behaviour observed for the FT model in Table 1 in terms of two side-effects: Frequency Shock and Range Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 7 ZS FS RR FT FT+ZS FT+FS Mix+FS 1 2 3 4 5 Percentage of answers Common Percentage Rare Percentage Figure 3: Common (Rare) Percentage corresponds to the percentage of test queries for which the model predicted an entity from the Common (Rare) set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to the results, after finetuning on the MediumMis- match dataset, the LM receives a frequency shock: it under-predicts the common entities and over-predicts the rare entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Table 4: Accuracy of the models for the Common and Rare entity sets for the MediumMismatch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Due to Frequency Shock, the FT model under-predicts the Common entities and over-predicts the Rare entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' As a result, when the FT model pre- dicts a Common entity, there is a much higher chance of it being true com- pared to the other models, whereas when the FT model predicts a Rare entity, there is a much lower chance of it being true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Common Accuracy Rare Accuracy ZS 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 FS 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 FT 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 FT + FS 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 1:15 + FS 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 We selected a set of 10 cities that are expected to be commonly seen3 as well as a set of 10 cities that appear as answers in LANKA but are expected to be rarely seen in a dataset4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We named the two sets Common and Rare respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then measured the number of times the models generated an entity from Common and Rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to Figure 3, the ZS model predicts the Common entities frequently and the Rare entities infrequently (this is in part due to the frequency of the entities in the test set and in part due to the prior of the language model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For the FS and RR models, the percentages for the two sets are not substantially different from the ZS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' However, for the FT model, due to the uniform distribution of of the training set of MediumMismatch, the percentages for the two sets changes substantially: the number of predictions from Common entities drops by almost a third, and the one for Rare entities increases by 6x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This reveals that Frequency Shock is indeed a side-effect of finetuning as the difference between the frequency statistics of the training set of MediumMismatch and what the pre-trained model expects causes a shock to the model and makes it over-predict Rare entities and under-predict Common entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We also measured the accuracy of the FT model when it produced a Common or Rare entity and compared it to ZS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We observe that the accuracy for Common entities increases from 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2% to 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5% and for Rare entities decreases from 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9% to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This is because the frequency shock caused by finetuning leads the model to predict the Common entities only when it has high confidence in its prediction, but be less cautious about predicting the Rare entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This result also points out an interesting future direction for measuring the model uncertainty (or whether the model knows what it does not know) through a combination of a ZS model and a model that has been finetuned on a slightly different data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To further analyze the effect of Frequency Shock and Range Shift we compare models in terms of the overlap between their predicted entities and those in the train and test sets of the HighMismatch dataset, where 3We selected the 10 cities from Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021) (Figure 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' namely {London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Rome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Montreal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Milan} 4We do this by randomly selecting 10 cities from the LANKA answers that appear as an answer in LAMA less than 20 times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' namely {Boise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Tirana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Myanmar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Hanover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Aberdeen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Chelsea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Kentucky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Oldham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Hastings,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Parma} 8 Figure 4: The percentages of overlap between the entities predicted by the models and those of the train and test sets for the HighMismatch dataset (entities in the train and test sets are disjoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' the train and test entities are mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to the results in Figure 4, we observe that the FT model predicts the entities from the train set significantly more than the FS model (almost 62% relative increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This shows a clear case of Range Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We also manually analyzed the outputs of the ZS and FT models for the “born in” relation (as a representative relation)5 and grouped the predictions of each model into three classes: 1- the output is not a location, 2- the output is the correct location, and 3- the output is an incorrect location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then compared the number of queries in the cross-product of the categories for the ZS and FT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results are presented in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Out of the 60 queries on HighMismatch where ZS predicted the correct location and FT predicted an incorrect location, in 59 cases the top answer of the FT model was one of the entities from the training set answers, showing another clear (and perhaps more severe) case for Range Shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Moreover, on MediumMismatch, out of the 58 queries for which the answer changed from a correct location to an incorrect location after finetuning, in 19% of those cases the correct entity was “London” – a commonly occurring city (note that only for 6% of the queries the correct answer is “London”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In another 33% of those cases the correct entity is one of “Paris”, “Berlin”, “Barcelona”, “Vienna” and “Brooklyn”, whereas only for 7% of the queries the answer is one of these cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This is because the training set of MediumMismatch has a uniform distribution and finetuning on it leads to frequency shock where common entities (such as “London”) are under-predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 The positive effect of Task Learning Similar to the existing literature Fichtel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021), Table 5 provides multiple evidences showing Task Learning is a positive effect of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' First, while ZS predicts non-location outputs (mostly years) for some queries, FT correctly learns to predict a location for the queries6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Secondly, for the 115 queries where ZS predicted an incorrect location but FT predicted a correct one on LowMismatch, in 90 cases the ZS model had generated a correct country as the top output, and the FT model learned to predict the correct city (which is the expected sub-type) instead of country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We observe a similar behaviour for 39/62 queries in 5We selected this relation because it is simple to verify the model’s output types and subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 6As an interesting side note, for the queries for which the ZS model outputs a different type than a location, even though the FT model learns to predict a location, it tends to predict a wrong location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' future work can use this signal to predict when the LM does not know the answer to a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Table 5: A comparison of the ZS and FT models for the “born in” relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' NL, CL and IL stand for Not a Location, Correct Location, and Incorrect Location respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' FT LowMismatch MediumMismatch HighMismatch ZS NL CL IL NL CL IL NL CL IL NL 0 11 105 0 3 113 0 0 165 CL 0 89 26 0 57 58 0 3 60 IL 0 115 598 0 62 651 0 2 838 9 FT Model Other 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8% Train Entities Test Entities 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8%FS Model Other 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9% Train Entities 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4% Test Entities 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6%Table 6: Results for model mixing (bold indicates winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The best single model corresponds to the model that gave the best result for each dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', for T5 XXL FT is the best single model for LowMismatch and MediumMismatch, and FS for HighMismatch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' UB stands for upper-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LowMismatch MediumMismatch HighMismatch T5 Model Mixing Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 XXL Best Single Model 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 FT + ZS 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 FT + FS 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 FT + FS (UB) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 Small Best Single Model 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 FT + ZS 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 MediumMismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The other cases where the prediction changed from incorrect location to correct location can be explained by better learning the semantics of the task as a result of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 5 Improving Finetuning To avoid the side-effects identified in Section 4 and use finetuned LMs for factual knowledge extraction and KG construction, one may be tempted to create a training set that has a large coverage of various entities and that also has a high Pearson correlation with what is expected to be seen at the test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We note, however, that entity coverage and Pearson correlation are somewhat at odds with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' That is because if we add many queries to the training data whose answers are novel entities, it will cause the Pearson correlation to go down unless we also add a prohibitively large number of queries with common entities as answers to retain the proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Also, if we wish to keep the Pearson correlation high, many of the rare entities may not appear in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We aim to find solutions by changing the finetuning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Given that LMs are pre-trained on large corpora of text (typically much larger than the finetuning dataset), we may expect the original entity distribution of the LM (corresponding to its prior distribution) to be more robust to situations with different frequency statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In this section, we exploit this insight to provide two strategies to remedy the negative effect of Frequency Shock and Range Shift in finetuning while still retaining the benefits of Task Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 Model Mixing Alleviates Side-Effects As we observed in the previous sections, the FT model has the advantage of better learning the task and as a result producing better results than the alternative models in situations such as the LowMismatch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' However, it also has the disadvantage of introducing a frequency bias that may lead to low performance on situations such as the MediumMismatch and HighMismatch datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' On the other hand, the ZS and FS models have the advantage of being more robust to situations such as the MediumMismatch and HighMismatch datasets, but their performance falls short of the FT model in situations such as the LowMismatch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Here, we explore whether these models can be combined to get the best of the two worlds: the improved performance of the FT models and the robustness of the ZS and FS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We experiment with a simple mixing approach where we average the scores produced by the FT model for each output with that of the other models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' we leave more sophisticated combination strategies as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We first verify if model mixing helps alleviate Frequency Shock and is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Figure 3 indicates the percentage of queries for which the FT+ZS and FT+FS models predicted one of the Common or Rare entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' One can see from the figure that, contrary to the FT model, the distributions for these models are much closer to that of the ZS and FS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The correction effect is rather one-sided: while the Common entities are not under-predicted anymore, the Rare entities are still slightly over-predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This is also apparent from the accuracy of the FT+FS model on the Common and Rare sets in Table 4: the performance on the Common set 10 Table 7: Results for mixture finetuning with different mixture ratios (bold indicates winner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 1:0 corresponds to standard finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Mixture training consistently provides a boost in performance, especially for larger mixture ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The benefits from mixture training and model mixing can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' LowMismatch MediumMismatch HighMismatch T5 Mixture Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 Hit@1 Hit@3 Hit@5 XXL 1:0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 1:1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 27.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 1:15 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 45.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 1:15 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 becomes similar to the FS model as Common entities are not under-predicted anymore, but the performance on the Rare set is still much lower than the FS model as Rare entities are still being over-predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then look at whether model mixing provides better predictions overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results are presented in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' When using the T5 XXL model, for FT+ZS even though the difference between the two models is quite large on LowMismatch, the model results are only slightly worse than the FT model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For the other two datasets, where the difference between the two models is much smaller, model mixing leads to substantial improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' FT+FS offers higher performance than both individual models on LowMismatch and MediumMismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' For HighMismatch, all numbers improve substantially with respect to FT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' with respect to FS, however, Hit@1 goes slightly down whereas Hit@3 and Hit@5 improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The same trend holds for the T5 Small model where mixing ZS and FT performs better than both ZS and FT in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Note that for the HighMismatch dataset, even though the FT model performs poorly in isolation, mixing it with ZS still brings improvement for ZS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' While, in the past, model mixing has been shown to provide only marginal gains in different applications even when multiple models are being combined, the results in Table 6 show that a simple parameter-free combination of only two models provides large boosts of up to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6% on the MediumMismatch dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Furthermore, we have included in Table 6 an upper-bound result for FT+FS where we assume having access to an oracle that can tell if we should trust the FT model or the FS model for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The upper-bound result is significantly higher than each of the individual models, thus showing that there is a large subset of the data where one model produces the correct answer whereas the other model does not, hence indicating that the models work well on different subsets of the data and that more sophisticated combinations can potentially lead to more improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' These results confirm that besides the previously studied benefits of model mixing Naderi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Pranesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Ormerod (2022), it plays a much significant role for knowledge extraction from finetuned LMs by correcting the side-effects of finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 Mixture Training Alleviates Side-Effects We examine if the following multi-task finetuning strategy can be an effective solution to the identified side-effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We finetune the model on a combination of two tasks: 1- Factual Knowledge Extraction 2- The original pre-training task of the LM ("masked language modeling").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Let α : β represent the ratio between the number of queries from the first and the second tasks in each training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We set α = 1 and finetune models with different values for β for the three datasets, to see how mixture training with different ratios affects the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Intuitively, if we continue to finetune the model with both these tasks, the second task should help the LM observe the entities with a similar frequency as in its pre-training stage, thus avoiding frequency shock to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This approach has been previously shown to aid with the forgetting effect of finetuning on LMs (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The results in this section extend those results to the case of Frequency Shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' From the results in Table 7, we can see that mixture finetuning consistently provides improvements across the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' The amount of improvement larger for the HighMismatch dataset where the statistics differ more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 11 Finally, we combine the mixture finetuned model with the few-shot model and see the benefits from the two solutions is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We see from Figure 3 that the resulting model does not under-predict Common entities and does not over-predict Rare entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Also, from Table 4, we see that the resulting model does not produce a substantially higher accuracy on the Common set due to under-prediction, and does not produce a substantially lower accuracy on the Rare set due to over-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Finally, we see from Table 7 that the benefits from model mixing and mixture training can be combined to make yet better predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 A Knowledge Extraction Recipe So far, we have observed that 1) a finetuned model may suffer from Frequency Shock and Range Shift, 2) combining finetuned and few-shot models help improve the results, and 3) mixture finetuning helps too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Here, we address a key question: what is the best strategy for extracting factual knowledge from LMs to construct KGs for real applications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Table 8: Macro average performances of different models over the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Overall, an combination of a mixture finetuned model (with a large mixture ratio) with an FS model performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' If one wishes to use only a single model during inference, then the best option is the mixture trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' If one wants to avoid mixture training due to its higher cost, then the FS model is the winner followed by the FT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Macro average Strategy Hit@1 Hit@3 Hit@5 ZS 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 FS 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='9 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 FT 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 RR 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='6 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 FT + ZS 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 FT + FS 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 1:15 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 1:15 + ZS 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 1:15 + FS 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='8 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='1 In real applications of knowledge extraction from LMs for the purpose of knowledge graph construction, we expect to see a combi- nation of the properties in the three datasets studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' That is, we expect training data to be available mostly for some subsets of the knowledge domain that needs to be extracted, and be scarce for other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We also expect the frequency statistics of the train data and the data that needs to be extracted to be similar for parts of the knowledge domain, and differ to various degrees in other parts of the knowledge domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Therefore, we expect knowledge extraction in real applications to involve a combination of the three datasets studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To find the best strategy for factual knowledge extraction from LMs, we report the average performance of different combinations of the techniques introduced in this paper on our three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' According to the results in Table 8, the best strategy is to finetune a mixture model and combine it with an FS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' This strategy leads to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='4% relative improvement over vanilla finetuning in terms of Hit@1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Given that mixture finetuning substantially increases the time and cost required for finetuning (especially for higher ratios for the pretraining task), if one wants to avoid that cost the next best option is to use FT+FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Alternatively, if one can afford the training cost but needs to reduce the inference cost by using a single model during inference and avoiding making multiple model calls per query, the best strategy is to use a mixture finetuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Finally, FS may result in better aggregate performance than FT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 6 Conclusion Language models (LMs), especially when finetuned, can be a great source of knowledge for constructing (or augmenting) knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' However, finetuning may also exhibit negative effects for knowledge extraction that are important to understand and be aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In this paper, we identified Frequency Shock and Range Shift as side-effects of finetuning, which can be helpful or harmful depending on the frequency statistics of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We then considered two solutions, model mixing and mixture finetuning with the pre-training task, that help remedy the negative effects and result in improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Future work can look into more sophisticated ways of model mixing or other finetuning techniques that avoid the negative effects to a larger extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} 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joint candidate evidence retrieval for multi-hop question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Da Yin, Hritik Bansal, Masoud Monajatipoor, Liunian Harold Li, and Kai-Wei Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Geomlama: Geo-diverse commonsense probing on multilingual pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Factual probing is [mask]: Learning vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' learning to recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' arXiv preprint arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='05240, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Xuhui Zhou, Yue Zhang, Leyang Cui, and Dandan Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Evaluating commonsense in pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 9733–9740, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' A Implementation Details We train all models for 10 epochs on a 4x4 v3 TPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Note that in the case of mixture finetuning (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='2), each epoch consists of more batch updates compared to single-task finetuning because a portion of the examples in each batch come from the pre-training task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We set the batch size to 128 and the learning rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We measured the Hit@1 performance on the validation set after each epoch and selected the model parameters from the best performing epoch on the validation for evaluating on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 15 For the few-shot experiments (FS model), for each relation type we selected 10 random examples from the training set of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' To get an estimate of the standard deviation due to the choice of different examples, we repeated our experiment for the LowMismatch dataset 5 times each time selecting different examples and observed a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Note that while for the three datasets we use in this paper the queries have been selected in such a way that the answer is mostly a single token according to the BERT vocabulary, those answers are already multi-token according to the T5 vocabulary and that allows us to test the multi-token prediction ability of the LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' Previous work restrict the model prediction distribution to a predefined set of tokens and ignores any predicted outputs outside that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' We disregard that pre-defined set during training and validation to avoid unwanted artifacts introduced due to the use of that specific vocabulary set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' However, we use that set for measuring performance on the test set so that the final results are in the same footing as those of the previously published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdFIT4oBgHgl3EQfiyst/content/2301.11293v1.pdf'} diff --git a/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/2301.00632v1.pdf.txt b/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/2301.00632v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b95c940bdd27634638b05e16124d46d704fbbabb --- /dev/null +++ b/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/2301.00632v1.pdf.txt @@ -0,0 +1,1545 @@ +Investigating the Role of Electric Fields on Flow Harmonics in Heavy-Ion Collisions +Ankit Kumar Panda,∗ Reghukrishnan Gangadharan,† and Victor Roy‡ +School of Physical Sciences, National Institute of Science Education and Research, +An OCC of Homi Bhabha National Institute, Jatni-752050, India +(Dated: January 3, 2023) +Using the blast-wave model, we explore the effect of electric fields on spectra and flow harmonics +(especially the elliptic flow) for charged pions and protons. We incorporate the first-order correction +to the single-particle distribution function due to the electric fields and the dissipative effect while +calculating the invariant yields of hadron in the Cooper-Frey prescription at the freezeout hyper- +surface. We find a noticeable correction to the directed and elliptic flow of pions and protons for +unidirectional and azimuthal asymmetric electric fields in the transverse plane of magnitude ∼ m2 +π. +Further, we observe mass dependency of the directed flow generated due to the electric fields. The +splitting of particle and antiparticle’s elliptic flow is also discussed. +I. +INTRODUCTION +In high energy heavy-ion collisions at RHIC and LHC, +two highly Lorentz contracted nuclei colliding onto one +another produces a hot and dense novel state of matter +known as Quark Gluon Plasma (QGP). The QGP formed +in heavy-ion collisions is the same as the medium that +existed during the microsecond old universe, which con- +sisted of almost freely moving quarks and gluons that are +usually confined in colorless hadrons at low or zero tem- +peratures [1, 2]. Hydrodynamic calculations have con- +firmed that the QGP is a strongly coupled fluid with +extremely low η +s value [3–5]. Along with the QGP, an in- +tense electromagnetic field of the order of 1014 T is gener- +ated in the initial stage of heavy ion collision in Au+Au +collision at top RHIC energies [6–12]. The primary source +of intense electromagnetic (EM) fields in heavy-ion col- +lisions are the spectator nucleons (the nucleons that are +unaffected by collisions). +Relativistic +viscous +hydrodynamics +is +a +well- +established formalism for describing the space-time +evolution of the quark-gluon plasma (QGP). However, +owing to the finite electrical conductivity of the high +temperature QGP and low temperature hadronic gas +phase both phases evolving in intense EM fields, the +proper framework, we believe, is the relativistic resistive +magnetohydrodynamics (RRMHD). The straightforward +generalization of non-relativistic viscous hydrodynamics +to the relativistic regime leads to an acausal theory. +The acausality problem is usually cured by incorpo- +rating higher-order gradient terms in the entropy-four +current or in the energy-momentum tensor of the +fluid [13]. The space-time evolution of the QGP should +be studied using a causal RRMHD theory. There have +been recent developments on the formulation of causal +(second-order) magnetohydrodynamic formalisms from +the kinetic theory perspective [14–19]. +Most of these +∗ ankitkumar.panda@niser.ac.in +† reghukrishnang@niser.ac.in +‡ victor@niser.ac.in +studies have reported a few new transport coefficients +arising due to the external EM fields, some of which +were non-dissipative as they did not increase entropy. +However, a systematic investigation of many of these +transport coefficients has not yet been done. Moreover, +the effect of electric fields on bulk observables such as +transverse momentum (pT ) spectra and flow coefficients +of charged hadrons has not been extensively studied, +to our knowledge, except for a few studies [20, 21]. +Developing a RRMHD numerical code is a challenging +task [22, 23], and it is known that viscosity (and EM +fields) affects the space-time evolution of the fluid and +also the one-particle thermal distribution of particles +used in the Cooper-Frye formulation, which ultimately +gives the final experimentally observed particles from +the fluid elements. +Here, we use a simple but effective blastwave model [24] +to investigate the influence of the electric field on exper- +imental bulk observables such as first and second-order +flow harmonics (v1, v2) and transverse momentum (pT ) +spectra of charged hadrons, due to corrections only in the +Cooper-Frye prescription. These observables are known +as ‘bulk observables’ because they involve the collective +motion of many particles in the fluid.The effect of vis- +cous corrections is introduced through a relaxation time +τc, which is a free parameter in our case. We note that a +more accurate estimate of the effect of EM fields on bulk +observables requires a magnetohydrodynamic evolution +[22] of the fluid. It is important to note that in addi- +tion to second-order flow harmonics (v2) and transverse +momentum (pT ) spectra of charged hadrons, the elec- +tromagnetic field may result in a variety of other novel +phenomena such as the Chiral Magnetic Effect and Chi- +ral Separation Effect [25–27]. Other works that study the +effect of EM fields on the QGP include [28–32]. +The article is organized as follows: we discuss the blast +wave model in Section II, then we discuss the Cooper- +Frye mechanism and other relevant formulas, along with +the setup, in Sections III and IV. Results are discussed in +Section V. Finally, we conclude and summarize our study +in Section VI. Throughout the article, we use natural +units where ℏ = c = kB = 1. The metric signature used +here is gµν = diag(+, −, −, −). +arXiv:2301.00632v1 [nucl-th] 2 Jan 2023 + +2 +II. +BLAST WAVE MODEL +The blast-wave model is a simple model that considers +the collective motion of the matter produced in heavy- +ion collisions and parameterizes its four-velocity. It fur- +ther assumes that hadrons are produced from a constant +temperature freeze-out hypersurface, with the freeze-out +temperature being a free parameter. The invariant yields +of hadrons are obtained from the Cooper-Frye formalism, +which is described later. Despite its simplicity, the blast- +wave model can successfully describe experimental data +qualitatively and quantitatively in most cases. +In heavy-ion collisions, one popular parametrization of +the fluid four-velocity is inspired by the Bjorken model +of boost-invariant expansion in the longitudinal direction +of the fluid. In this work, we use the Milne coordinate +(τ, η, r, φ) with the metric gµν = diag +� +1, −τ 2, −1, −r2� +, +and the transformation between the Cartesian and Milne +coordinate is given as +� +�� +t +x +y +z +� +�� = +� +�� +coshη 0 +0 +0 +0 +0 cosφ 0 +0 +0 sinφ 0 +sinhη 0 +0 +0 +� +�� +� +�� +τ +η +r +φ +� +�� . +Where +τ = +� +t2 − z2, +η = tanh−1 z/t, +r = +� +x2 + y2, +φ = arctan2(y, x). +The parameterized form of the velocity four vector +with longitudinal boost-invariance is given as +ur = u0 +r +R +� +1 + 2 +∞ +� +n=1 +cn cos n [φ − ψn] +� +Θ (R − r) , +uφ = uη = 0, +uτ = +� +1 + (ur)2. +(1) +Where, uτ, ur, uφ, uη are the components of the fluid +four velocity; u0, and cn’s are free parameters used to +reproduce the pT spectra (invariant yield) and the flow +harmonics of charged hadrons. Θ (R − r) is the Heaviside +function which imposes the condition that if r > R, ur = +0. R is the radius of the freezeout hypersurface, φ is the +azimuthal angle in co-ordinate space and ψn is the n-th +order participant plane angle. +As we do not consider +event-by-event fluctuations in this work, we set ψn = 0 +i.e., the minor axis of the participant planes coincide with +the direction of the impact parameter. We parametrised +the temperature on the freezeout hypersurface as +T (τ, η, r, φ) = T0Θ (R − r) . +Where T0 denotes the temperature at the freezeout hy- +persurface. +III. +COOPER-FREY FORMALISM +As mentioned earlier, the invariant yield of hadrons is +obtained from the Cooper-Frye formula [33], which as- +sumes that the freeze-out hypersurface is a timelike vec- +tor dΣµ given by (τdηdrrdφ, 0, 0, 0). The invariant yield +is given by the following equation: +dN +d2pT dy = +G +(2π)3 +� +pµdΣµf(x, p). +(2) +Here, f(x, p) is the single-particle distribution function, +and x and p are the position and momentum four-vectors +of the particles, respectively. G is the degeneracy factor. +If the system is not in local thermal equilibrium, as +is the case for a rapidly expanding fireball, the single- +particle distribution function must take into account +the deviation from equilibrium when calculating hadron +yields using the Cooper-Frye prescription. +The distri- +bution function is usually decomposed into an equilib- +rium part f0 and a small non-equilibrium part δf, so +that the total distribution function becomes f = f0 +δf. +The second-order correction to f gives rise to some new +transport coefficients due to the external magnetic field. +The temperature and mass dependence of some of these +transport coefficients were discussed in [34], and we will +show the results for the remaining coefficients later in the +Results section. +However, in this work, we consider terms up to first- +order in gradients [17]. In this case, the invariant yield +(Eq. (2)) becomes +dN +d2pT dy = +G +(2π)3 +� +pµdΣµ +� +f0 + δf 1� +. +(3) +Where δf 1 ≪ f0. +In [16, 17] δf 1 is calculated using +the Boltzmann equation using the Relaxation Time Ap- +proximation (RTA). We give the expression for δf 1 in +Appendix.(B) for the sake of completeness. +IV. +SETUP +For the current study, we consider only n = 2 in the +expression for ur (Eq.(1)), resulting in +ur = u0 +r +R[1 + 2c2 cos(2φ)]Θ (R − r) . +The equilibrium distribution is f0 = +� +eβu·p−α + r +�−1, +here β is the inverse temperature, r = ±1 corresponds +to fermions and bosons respectively. We use the param- +eters given in Table (I) to obtain the invariant yield of +π+ that matches the ALICE measurement shown as red +circles in the top panel of Fig.(1)[35]. Earlier hydrody- +namic model studies [36] have shown that it is not possi- +ble to simultaneously describe π+ and p spectra for zero +baryon chemical potential, so we use a different set of val- +ues for u0, c2, and T (given in the rightmost column of + +3 +Table (I)) to explain the proton spectra. The blast-wave +model with these parameters reasonably well explains the +experimental data. The bottom panel of Fig. (1) shows +the blast-wave results for v2 of π+ and p. We do not +compare these ideal results with experimental data since +they are known to over-predict the experimental mea- +surements, but we show them for the sake of qualitative +description. +●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● +▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.1 +1 +10 +100 +pT (GeV) +d2 N / pT d pT d y (GeV)-2 +Blast wave fit +▲ +p +● +π+ +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +pT(GeV) +v2 +p +π+ +FIG. 1: (Color online)Top plot is the comparison of the +experimentally measured π+ (red filled circles) and +protons (black triangles) for Pb+ Pb collision at +√s=2.76 TeV for 20-30 % centrality with the ideal blast +wave results (lines) for the parameters in Table (I).The +lower panel shows the corresponding v2 vs pT for π+ +(blue line) and protons (red dashed line). +After setting up the parameters for the ideal case (zero +viscosity), we can now explore the effect of viscosity +(through the relaxation time τc) and electric fields on +spectra and flow harmonics by comparing the corre- +sponding results with the ideal results. +In our model, +τc and the electric field qE appear in δf 1, giving rise +to additional corrections to the invariant yields and flow +harmonics of charged hadrons. We compute the pT differ- +ential n-th order flow coefficient using the usual formula: +vn(pT , y) = +� 2π +0 +dϕ cos(nϕ) +dN +d2pT dy +� 2π +0 +dϕ +dN +d2pT dy +. +(4) +As we need the electric field distribution on the freeze- +π+ +p +u0 +1.2 +1.22 +T +130 MeV +140 MeV +R +10 fm +10 fm +τ +6 fm +6 fm +m +139.5 MeV +938 MeV +c2 +0.1 +0.15 +TABLE I: The fit parameters for π+ and p respectively +at mid-rapidity. +out hypersurface to be used in the Cooper-Frye formula +(Eq.(2)) we use a parameterized form of the EM field. +In principle, the fields generated in the initial stages of +heavy-ion collisions due to the charged protons inside the +two colliding nucleus would evolve with the QGP fluid, +but the blast wave model does not allow any such self- +consistent dynamical evolution of fields. The magnitude +of the electric fields in all the cases are kept Here we use +parameterised electric fields of four different configura- +tions in the transverse plane (XY plane) while calculating +the invariant yields. Some of these configurations does +not represent the actual scenario encountered in heavy- +ion collisions but we use them for exploratory purpose. In +the left panel of Fig.(2) we show the first setup of electric +fields which we call config-1 henceforth, this represents +isotropic fields due to a point charge of large magnitude +at the origin, the right panel shows config-2 that closely +resembles the field configuration expected in symmetric +heavy-ion collision. config-3 as shown in the left panel of +Fig.(2) is the π/2 rotated version of config-2. The right +panel of Fig.(2) represents a constant unidirectional elec- +tric fields which might be applied in a limited sense for +large-small nuclei collisions, this is config-4. To better +understand the effect of electric fields on flow harmonics +we note the following +• config-1: isotropic fields; expected to unalter the +flow harmonics, +• config-2: prolate-like fields; expected to decrease +the flow harmonics, +• config-3: oblate-like fields; expected to increase +the flow harmonics, +• config-4: directional fields; expected to alter di- +rectional flow only. +V. +RESULTS +Here we discuss our main results for different configura- +tions of electric fields (described in the previous section) + +4 +-40 +-20 +0 +20 +40 +-40 +-20 +0 +20 +40 +config-1 +0.005 +0.010 +0.015 +0.020 +0.025 +-40 +-20 +0 +20 +40 +-40 +-20 +0 +20 +40 +config-2 +0.01 +0.02 +0.03 +0.04 +-40 +-20 +0 +20 +40 +-40 +-20 +0 +20 +40 +config-3 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +-40 +-20 +0 +20 +40 +-40 +-20 +0 +20 +40 +config-4 +0.2 +0.4 +0.6 +0.8 +1.0 +FIG. 2: (Color online) Electric field configurations in +the transverse plane. Detail expressions for different +configurations are given in Appendix.(B 1). Magnitude +of electric fields (in GeV2) are shown using colour map. +on the spectra ,and v2 for pions and protons. Top panel +of Fig.(3) shows the dependence of pT spectra of π+ on +the electric fields for ideal case, for comparison we also +show the results for non-zero viscosity but with no elec- +tric fields. The magnitude of viscosity in our model is +controlled through tc which is set to 1 fm for the results +shown here. Here we see that the pT spectra hardly shows +any dependence on different transverse electric field con- +figurations (solid orange, black, red lines correspond to +config-1, config-2, and config-3 respectively). As ex- +pected the effect of finite viscosity is comparatively more +prominent on the pT spectra; we see a suppression at +higher pT region for the viscous case. Since both bulk +and shear viscosity is present in our case, the slope of +the spectra is determined by the relative contributions of +these two viscosities [37, 38].The bottom panel of Fig.(3) +shows the pT spectra for protons for ideal (with electric +fields) and viscous (without electric field) cases where we +see a small suppression in the lower pT region for the case +of config-2 and config-3 for ideal case. We also see a +suppression at higher pT for viscous case as was seen for +π+. +Before discussing the differential v2 for different cases, +it is worthwhile to explore the dependence of pT inte- +grated (0-3 GeV) dN/dφ on different field configurations +and viscosity. In the top panel of Fig.(4) we show the +dN/dφ of π+ as a function of φ for various cases. As ex- +pected we have pure cosine like dependence of dN/dφ for +ideal case (shown by the solid blue line), the viscosity re- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.001 +0.010 +0.100 +1 +10 +100 +pT (GeV) +d2N / pT dpT dy (GeV)-2 +viscosity +Ideal (config-4) +Ideal (config-3) +Ideal (config-2) +Ideal (config-1) +Ideal +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.005 +0.010 +0.050 +0.100 +0.500 +1 +pT (GeV) +d2 N / pT dpT dy (GeV)-2 +viscosity +Ideal (config-4) +Ideal (config-3) +Ideal (config-2) +Ideal (config-1) +Ideal +FIG. 3: (Color online) Upper panel represents the pT +Spectra for π+ and lower panel for proton with different +transverse field configurations as in Fig.(2) and for the +case of viscosity for tc = 1fm . +duces the multiplicity as well as the amplitude (shown by +the red dashed line). config-1 the isotropic field configu- +ration (orange solid line) almost coincides with the ideal +result, but we see noticeable change in the amplitude of +dN/dφ for config-2, config-3 as expected. Things be- +come more interesting for config-4 (black dashed line), +here we generate finite directed flow like behaviour, this is +understood from the fact that a unidirectional force shifts +the centre of mass of the distribution. Similar behaviour +was observed for protons also (shown in the bottom panel +of Fig.(4). +A more concrete way to study the angular dependence +of dN/dφ for config-4 can be achieved by using a fi- +nite Fourier series decomposition of dN/dφ as shown in +Eq.(5). +f(Φ) = N +� +1 + 2 +3 +� +n=1 +vncos(nΦ) + 2 +3 +� +n=1 +wnsin(nΦ) +� +. +(5) +A non-linear least squares fit with vn, wn and N as a free- +parameters we obtain the best fit for dN/dφ with the val- +ues of these parameters given in Table (II). Here we note +that the directional force v1 is larger than v2 for π+, and + +5 +-3 +-2 +-1 +0 +1 +2 +3 +0 +5 +10 +15 +20 +Φ (Radian) +dN / dΦ +-3 +-2 +-1 +0 +1 +2 +3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Φ (Radian) +dN / dΦ +viscosity +Ideal (config-4) +Ideal (config-3) +Ideal (config-2) +Ideal (config-1) +Ideal +FIG. 4: (Color online) Top panel: dN/dφ as a function +of φ for π+ for various field configurations. Bottom +panel: same as top panel but for proton. +they are similar in magnitude for protons. Moreover, we +notice that unlike other cases the azimuthal distribution +breaks reflection symmetry with respect to the Y axis +which gives rise to non-zero wn shown in Table (II). We +also observe a mass dependence of the directional flow as +π+ has a larger v1 compared to the protons. +π+ +p +N +11.907 ± 2.7e-05 +0.672 ± 7.1e-08 +v1 +0.218 ± 1.0e-07 +0.081 ± 7.9e-08 +v2 +0.064 ± 9.8e-08 +0.096 ± 7.9e-08 +v3 +0.017 ± 9.7e-08 +0.005 ± 7.8e-08 +w1 +0.213 ± 1.0e-07 +0.081 ± 7.9e-08 +w2 +0.000 ± 9.8e-08 +0.000 ± 7.8e-08 +w3 +0.017 ± 9.8e-08 +0.005 ± 7.8e-08 +TABLE II: Fit parameters for π+ and proton for +config-4 from Eq.(5). +To have a visual understanding of the goodness of fit, +we show a comparison of the fitted values (using Eq.(5)) +(solid blue line) and the dN/dφ from the blast wave +model (dotted-dash orange line) for π+ (top panel) and +proton (bottom panel) in Fig.(5). +3 +2 +1 +0 +1 +2 +3 + (Radian) +4 +6 +8 +10 +12 +14 +16 +18 +20 +dN/d +Pion +Fit Plot +Blastwave result +3 +2 +1 +0 +1 +2 +3 + (Radian) +0.5 +0.6 +0.7 +0.8 +0.9 +dN/d +Proton +Fit Plot +Blastwave result +FIG. 5: (Color online) Blast wave results and the fitted +curves using Eq.(5) for config-4 for π+ (top panel) and +proton (bottom panel). +More familiar and useful observables in experiments +are centrality and pT dependent flow harmonics. +In +Fig.(6) we show the dependence of the second-order flow +harmonics v2 (a.k.a elliptic flow) for different configura- +tions. Here we see that there is almost no deviation from +the ideal case for the isotropic (config-1) and directed +field case (config-4). However, the situation is different +for the other two cases, we can clearly see an increase in +v2 for config-3 and a suppression for config-2. We also +note that viscosity supresses the elliptic (red dashed line) +flow for π+ and elavates for proton. +The effect of electric fields become more interesting +when we examine the difference in v2 for particles (π+, p) +and antiparticles (π−, ¯p). This difference ∆v2 = v2(h) − +v2(¯h) is shown in Fig.(7) as a function of pT . We ob- +serve a non-monotonic variation in ∆v2 as a function of +pT for both pions and protons. Interestingly, a similar +observation was made in [29]. +Throughout this study we only consider the effect of + +6 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +pT (GeV) +v2 +viscosity +Ideal (config-4) +Ideal (config-3) +Ideal (config-2) +Ideal (config-1) +Ideal +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +pT (GeV) +v2 +viscosity +Ideal (config-4) +Ideal (config-3) +Ideal (config-2) +Ideal (config-1) +Ideal +FIG. 6: (Color online) v2 vs pT for π+ (top panel) and +proton (bottom panel) for the transverse electric field +configurations shown in Fig.(2). +electric fields and the first order correction in the δf. As +mentioned earlier, previous studies showed that there are +new transport coefficients at higher order corrections to +f which may alter the results obtained here. But these +are beyond the scope of the present exploratory study. In +Fig.(8) we show the temperature and mass dependence +of some of the transport coefficients arising due to the +magnetic fields. This may give some hints of the relative +contribution of various transport coefficients in the bulk +observables while used in the Cooper-Frye prescription. +VI. +SUMMARY AND CONCLUSION +In this work, we have studied the effect of electric +fields on the bulk observables in heavy-ion collisions such +as pT spectra, directed and elliptic flow of charged pi- +ons and protons. We use the blast-wave model and dif- +ferent configurations of electric fields in the transverse +plane to carry out this exploratory study. The pT spec- +tra of hadrons in the blast-wave model are obtained us- +ing Cooper-Frye prescription, where we incorporate non- +equilibrium correction δf due to the viscosity and the +electric fields. Since the blast-wave model does not in- +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +-0.15 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0.15 +pT (GeV) +v2(h)-v2(h) +config-3 +config-2 +FIG. 7: (Color online) v2(h) -v2(¯h) as a function of pT +for config-2 (red) and config-3 (blue) line corresponds +to π (Solid line) and p (Dashed line). +clude space-time evolution, the fluid velocity and the +electric fields are parameterized on the freezeout hyper- +surface to calculate experimental observables. +For our +case, fluid velocity fields are modulated so that it domi- +nantly generates the elliptic flow. We use four different +configurations of transverse electric fields (i) isotropic +fields, (ii) prolate-like fields, (iii) oblate-like fields, and +(iv) directed fields. The typical maximum value of the +electric field for all these configurations is ∼ m2 +π. We find +that flow harmonics for isotropic fields remain unchanged +for both pions and protons. Both prolate and oblate-like +field configuration alters the flow harmonics, and the di- +rected field gives rise to large directed flow v1 for both +pions and protons. We also observe a mass dependence +of v1 generated due to the electric fields. We also dis- +cuss the temperature and mass dependence of some of +the new transport coefficients that appear in the second- +order correction in the distribution function due to the +EM field. They can further contribute to the results we +obtained here. This we leave for a possible future study. +ACKNOWLEDGMENTS +AP acknowledges the CSIR-HRDG financial support. +RG and VR acknowledge support from the DAE, Govt. +of India. +Appendix A: Appendix +In this work we have used Milne coordinate system +(τ, η, r, φ), where τ = +√ +t2 − z2, r = +� +x2 + y2, η = +tanh−1(z/t), and φ = tan−1(y/x), due to the co-ordinate +transformation various equations changed forms com- +pared to the Cartesian coordinates. +Here we give the +details about the Jacobian and Christoffel symbols used +in this study due to the above coordinate transformation: + +7 +0.0 +0.5 +1.0 +1.5 +2.0 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +δ +πVB +Mass Dependence +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +T +emperature Dependence +m = 120 MeV +m = 100 MeV +m = 80 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +δ +ΠVB +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.10 +0.15 +0.20 +0.25 +0.30 +m = 120 MeV +m = 100 MeV +m = 80 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +0 +2 +4 +6 +8 +10 +l +VπB +, +l +VΠB +, +τ +VΠB +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +10 +20 +30 +40 +50 +60 +m = 500 MeV +m = 250 MeV +m = 100 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ρ +VVB +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +2 +4 +6 +8 +m = 500 MeV +m = 250 MeV +m = 100 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +λ +VVB +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +1 +2 +3 +4 +5 +6 +7 +m = 500 MeV +m = 250 MeV +m = 100 MeV +0.0 +0.5 +1.0 +1.5 +2.0 +m (GeV) +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +τ +VVB +T = 120 MeV +T = 100 MeV +T = 80 MeV +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +T (GeV) +0 +2 +4 +6 +8 +m = 500 MeV +m = 250 MeV +m = 100 MeV +FIG. 8: (Color online)Mass and temperature variation +of the transport coefficients arising due to external +magnetic field. +the Jacobian for volume element is √−g = τr and the +non-vanishing Christoffel symbols are Γτ +ηη = τ ,Γη +τη = 1 +τ , +Γr +φφ = −r ,Γφ +rφ = 1 +r. The space-like projection is defined +as ∆µν = gµν−uµuν ; for Milne coordinate system, differ- +ent components of ∆µν are ∆ττ = − (ur)2 , ∆ηη = − 1 +τ 2 +, ∆rr = −1 − (ur)2 , ∆φφ = − 1 +r2 , ∆τη = ∆τφ = ∆ηr = +∆ηφ = ∆rφ = 0 , ∆τr = − +� +1 + (ur)2ur. The expansion +scalar is given by θ = Dµuµ = ∂µuµ + Γµ +µαuα . +We consider real particles and the on-shell con- +dition is given by gµνpµpν += +(pτ)2 − τ 2 (pη)2 − +(pr)2 − r2 � +pφ�2 += +m2. +Following the convention +used in heavy-ion collisions we express the compo- +nents of the four momentum pµ as (E, px, py, pz) = +(mT cosh y, pT cos ϕ, pT sin ϕ, mT sinh y). +Where pT = +� +p2x + p2y , mT = +� +m2 + p2 +T , and y = tanh−1 (pz/E). +The components of the four-momentum in Milne co- +ordinates are +pτ = mT cosh (y − η), +τpη = mT sinh (y − η), +pr = pT cos (ϕ − φ), +rpφ = pT sin (ϕ − φ). +ϕ is the azimuthal angle of the particle in the momentum +space. +In Eq.(2) we have the term pµdΣµ which in our is given +by pµdΣµ = gµνpµdΣν = mT cosh (y − η)τdηrdrdφ. +Appendix B: First-order (δf) correction to the single-particle distribution +In Eq.(3) we introduced the first-order correction to the single-particle distribution while calculating invariant yield +using the Cooper-Frye formula. Here we give the detail expression of δf in terms of gradients of fluid variables and + +8 +fields: +δf = − τc +u.p +� +pµ∂µf0 + qF µνpν +∂f0 +∂pµ +� += − τc +u.p +� +−pµf0 ˜f0 [βpαDµuα + (u.p)∂µβ − ∂µα] +� +− τc +u.p +� +−f0 ˜f0qF µνpνβ ∂(u.p) +∂pµ +� += τcf0 ˜f0 +u.p +(βpµpαDµuα + (u.p)pµ∂µβ − pµ∂µα) − τcf0 ˜f0 +u.p +qβEνpν += τcf0 ˜f0 +u.p +� +βpµpα +� +uµ ˙uα + σµα + ωµα + ∆µαθ +3 +� ++ (u.p)pµ∂µβ − pµ∂µα +� +− τcf0 ˜f0 +u.p +qβEνpν += τcf0 ˜f0 +u.p +� +βpµpα +� +σµα + ∆µαθ +3 +� ++ (u.p)pµ∂µβ − pµ∂µα +� +− τcf0 ˜f0 +u.p +qβEνpν += τcf0 ˜f0 +u.p +� +βpµpα∂µuα + βpφpφrur − βpηpητuτ + (u.p)pµ∂µβ − pµ∂µα +� +− τcf0 ˜f0 +u.p +qβEνpν += τcf0 ˜f0 +u.p +� +βpφpφrur − βpηpητuτ + (u.p)pµ∂µβ − pµ∂µα +� +− τcf0 ˜f0 +u.p +qβEνpν ++τcf0 ˜f0 +u.p +� +βpτpr∂τur + βprpr∂rur + βpφpr∂φur +� ++ τcf0 ˜f0 +u.p +� +βpτpτ∂τuτ + βprpτ∂ruτ + βpφpτ∂φuτ +� += τcf0 ˜f0 +u.p +� +β +�pT sin (ϕ − φ) +r +�2 +rur − β +�mT sinh (y − η) +τ +�2 +τuτ + (u.p)pµ∂µβ − pµ∂µα +� +−τcf0 ˜f0 +u.p +qβEνpν + τcf0 ˜f0 +u.p +� +βmT cosh(y − η)pT cos(ϕ − φ)(ur)2 +ruτ +− β(pT cos(ϕ − φ))2 ur +r +� ++τcf0 ˜f0 +u.p +β pT sin(ϕ − φ) +r +pT cos(ϕ − φ)u0 +2r +R +� +nun sin n [φ − ψn] ++τcf0 ˜f0 +u.p +� +−β(mT cosh(y − η))2 (ur)3 +r(uτ)2 + βmT cosh(y − η)pT cos(ϕ − φ)(ur)2 +ruτ +� +−τcf0 ˜f0 +u.p +β pT sin(ϕ − φ) +r +mT cosh(y − η)ur +uτ u0 +2r +R +� +nun sin n [φ − ψn], +δf = τcf0 ˜f0 +u.p +β +�ur +r + sin(ϕ − φ)cos(ϕ − φ) +r +u0 +2r +R +� +unn sin n [φ − ψn] +� +p2 +T − τcf0 ˜f0 +u.p +qβEνpν ++τcf0 ˜f0 +u.p +β +� +2cosh(y − η)cos(ϕ − φ)(ur)2 +ruτ +− cosh(y − η)sin(ϕ − φ) ur +ruτ u0 +2r +R +� +unn sin n [φ − ψn] +� +mT pT +−τcf0 ˜f0 +u.p +β +� +sinh2 (y − η)uτ +τ + cosh2(y − η) (ur)3 +r(uτ)2 +� +m2 +T + τcf0 ˜f0 +u.p +((u.p)pµ∂µβ − pµ∂µα) , +where β = 1 +T and α = µ +T with Γτ +ηη = τ along with Γr +φφ = −r. The contribution due to the electric field E · p in the +above equation, when expanded, takes the following form +δf = −τcf0 ˜f0 +u.p +qβE · p, +δf = −τcf0 ˜f0 +u.p +qβ +� +Eτpτ − τ 2Eηpη − Erpr − r2Eφpφ� +. +1. +Co-ordinate transformation of Electric four vector +The electric field components in Milne-coordinates (Eτ, Eη, Er, Eφ) are connected to the Cartesian components +(Et, Ex, Ey, Ez) through the following transformation + +9 +� +�� +Eτ +Er +Eφ +Eη +� +�� = +� +��� +Cosh[η] +0 +0 +−Sinh[η] +0 +Cos[φ] Sin[φ] +0 +0 +−Sin[φ] +r +Cos[φ] +r +0 +−Sinh[η] +τ +0 +0 +Cosh[η] +τ +� +��� +� +�� +Et +Ex +Ey +Ez +� +�� . +We also note E.u = 0 and this gives rise to : +Et = EzSinhη +coshη ++ (cosφEx + sinφEy)ur +coshηuτ +, +Eη = +Ez +τCoshη − tanhηErur +τuτ +. +As mentioned in the main text we use four different configuration of transverse electric fields, they are parameterised +as : +eEx = +BZαem(x − x0)Cosh[η − η0] +((x − x0)2 + (y − y0)2 + (τSinh[η − η0])2)3/2 , +eEy = +AZαem(y − y0)Cosh[η − η0] +((x − x0)2 + (y − y0)2 + (τSinh[η − η0])2)3/2 , +eEz = 0, +where Z is the atomic number (for our case we choose Z=82), αem= 1 +137, A and B are the modulation factors which +controls the spatial configuration of the field in the transverse plane, and α is the fine structure constant. 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G 37, +094040 (2010), arXiv:1002.2394 [nucl-th]. + diff --git a/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/load_file.txt b/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98095532ebe98dbcf837b4a444f7311f3fe12650 --- /dev/null +++ b/YNAyT4oBgHgl3EQfvfmK/content/tmp_files/load_file.txt @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf,len=819 +page_content='Investigating the Role of Electric Fields on Flow Harmonics in Heavy-Ion Collisions Ankit Kumar Panda,∗ Reghukrishnan Gangadharan,† and Victor Roy‡ School of Physical Sciences, National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Jatni-752050, India (Dated: January 3, 2023) Using the blast-wave model, we explore the effect of electric fields on spectra and flow harmonics (especially the elliptic flow) for charged pions and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We incorporate the first-order correction to the single-particle distribution function due to the electric fields and the dissipative effect while calculating the invariant yields of hadron in the Cooper-Frey prescription at the freezeout hyper- surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We find a noticeable correction to the directed and elliptic flow of pions and protons for unidirectional and azimuthal asymmetric electric fields in the transverse plane of magnitude ∼ m2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Further, we observe mass dependency of the directed flow generated due to the electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The splitting of particle and antiparticle’s elliptic flow is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' INTRODUCTION In high energy heavy-ion collisions at RHIC and LHC, two highly Lorentz contracted nuclei colliding onto one another produces a hot and dense novel state of matter known as Quark Gluon Plasma (QGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The QGP formed in heavy-ion collisions is the same as the medium that existed during the microsecond old universe, which con- sisted of almost freely moving quarks and gluons that are usually confined in colorless hadrons at low or zero tem- peratures [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Hydrodynamic calculations have con- firmed that the QGP is a strongly coupled fluid with extremely low η s value [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Along with the QGP, an in- tense electromagnetic field of the order of 1014 T is gener- ated in the initial stage of heavy ion collision in Au+Au collision at top RHIC energies [6–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The primary source of intense electromagnetic (EM) fields in heavy-ion col- lisions are the spectator nucleons (the nucleons that are unaffected by collisions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Relativistic viscous hydrodynamics is a well- established formalism for describing the space-time evolution of the quark-gluon plasma (QGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' However, owing to the finite electrical conductivity of the high temperature QGP and low temperature hadronic gas phase both phases evolving in intense EM fields, the proper framework, we believe, is the relativistic resistive magnetohydrodynamics (RRMHD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The straightforward generalization of non-relativistic viscous hydrodynamics to the relativistic regime leads to an acausal theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The acausality problem is usually cured by incorpo- rating higher-order gradient terms in the entropy-four current or in the energy-momentum tensor of the fluid [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The space-time evolution of the QGP should be studied using a causal RRMHD theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' There have been recent developments on the formulation of causal (second-order) magnetohydrodynamic formalisms from the kinetic theory perspective [14–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Most of these ∗ ankitkumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='panda@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='in † reghukrishnang@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='in ‡ victor@niser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='in studies have reported a few new transport coefficients arising due to the external EM fields, some of which were non-dissipative as they did not increase entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' However, a systematic investigation of many of these transport coefficients has not yet been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Moreover, the effect of electric fields on bulk observables such as transverse momentum (pT ) spectra and flow coefficients of charged hadrons has not been extensively studied, to our knowledge, except for a few studies [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Developing a RRMHD numerical code is a challenging task [22, 23], and it is known that viscosity (and EM fields) affects the space-time evolution of the fluid and also the one-particle thermal distribution of particles used in the Cooper-Frye formulation, which ultimately gives the final experimentally observed particles from the fluid elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here, we use a simple but effective blastwave model [24] to investigate the influence of the electric field on exper- imental bulk observables such as first and second-order flow harmonics (v1, v2) and transverse momentum (pT ) spectra of charged hadrons, due to corrections only in the Cooper-Frye prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' These observables are known as ‘bulk observables’ because they involve the collective motion of many particles in the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='The effect of vis- cous corrections is introduced through a relaxation time τc, which is a free parameter in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We note that a more accurate estimate of the effect of EM fields on bulk observables requires a magnetohydrodynamic evolution [22] of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' It is important to note that in addi- tion to second-order flow harmonics (v2) and transverse momentum (pT ) spectra of charged hadrons, the elec- tromagnetic field may result in a variety of other novel phenomena such as the Chiral Magnetic Effect and Chi- ral Separation Effect [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Other works that study the effect of EM fields on the QGP include [28–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The article is organized as follows: we discuss the blast wave model in Section II, then we discuss the Cooper- Frye mechanism and other relevant formulas, along with the setup, in Sections III and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Results are discussed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Finally, we conclude and summarize our study in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Throughout the article, we use natural units where ℏ = c = kB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The metric signature used here is gµν = diag(+, −, −, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='00632v1 [nucl-th] 2 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' BLAST WAVE MODEL The blast-wave model is a simple model that considers the collective motion of the matter produced in heavy- ion collisions and parameterizes its four-velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' It fur- ther assumes that hadrons are produced from a constant temperature freeze-out hypersurface, with the freeze-out temperature being a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The invariant yields of hadrons are obtained from the Cooper-Frye formalism, which is described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Despite its simplicity, the blast- wave model can successfully describe experimental data qualitatively and quantitatively in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In heavy-ion collisions, one popular parametrization of the fluid four-velocity is inspired by the Bjorken model of boost-invariant expansion in the longitudinal direction of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In this work, we use the Milne coordinate (τ, η, r, φ) with the metric gµν = diag � 1, −τ 2, −1, −r2� , and the transformation between the Cartesian and Milne coordinate is given as � �� t x y z � �� = � �� coshη 0 0 0 0 0 cosφ 0 0 0 sinφ 0 sinhη 0 0 0 � �� � �� τ η r φ � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Where τ = � t2 − z2, η = tanh−1 z/t, r = � x2 + y2, φ = arctan2(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The parameterized form of the velocity four vector with longitudinal boost-invariance is given as ur = u0 r R � 1 + 2 ∞ � n=1 cn cos n [φ − ψn] � Θ (R − r) , uφ = uη = 0, uτ = � 1 + (ur)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (1) Where, uτ, ur, uφ, uη are the components of the fluid four velocity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' u0, and cn’s are free parameters used to reproduce the pT spectra (invariant yield) and the flow harmonics of charged hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Θ (R − r) is the Heaviside function which imposes the condition that if r > R, ur = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' R is the radius of the freezeout hypersurface, φ is the azimuthal angle in co-ordinate space and ψn is the n-th order participant plane angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' As we do not consider event-by-event fluctuations in this work, we set ψn = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=', the minor axis of the participant planes coincide with the direction of the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We parametrised the temperature on the freezeout hypersurface as T (τ, η, r, φ) = T0Θ (R − r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Where T0 denotes the temperature at the freezeout hy- persurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' COOPER-FREY FORMALISM As mentioned earlier, the invariant yield of hadrons is obtained from the Cooper-Frye formula [33], which as- sumes that the freeze-out hypersurface is a timelike vec- tor dΣµ given by (τdηdrrdφ, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The invariant yield is given by the following equation: dN d2pT dy = G (2π)3 � pµdΣµf(x, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) Here, f(x, p) is the single-particle distribution function, and x and p are the position and momentum four-vectors of the particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' G is the degeneracy factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' If the system is not in local thermal equilibrium, as is the case for a rapidly expanding fireball, the single- particle distribution function must take into account the deviation from equilibrium when calculating hadron yields using the Cooper-Frye prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The distri- bution function is usually decomposed into an equilib- rium part f0 and a small non-equilibrium part δf, so that the total distribution function becomes f = f0 +δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The second-order correction to f gives rise to some new transport coefficients due to the external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The temperature and mass dependence of some of these transport coefficients were discussed in [34], and we will show the results for the remaining coefficients later in the Results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' However, in this work, we consider terms up to first- order in gradients [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In this case, the invariant yield (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2)) becomes dN d2pT dy = G (2π)3 � pµdΣµ � f0 + δf 1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (3) Where δf 1 ≪ f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In [16, 17] δf 1 is calculated using the Boltzmann equation using the Relaxation Time Ap- proximation (RTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We give the expression for δf 1 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (B) for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' SETUP For the current study, we consider only n = 2 in the expression for ur (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (1)), resulting in ur = u0 r R[1 + 2c2 cos(2φ)]Θ (R − r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The equilibrium distribution is f0 = � eβu·p−α + r �−1, here β is the inverse temperature, r = ±1 corresponds to fermions and bosons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We use the param- eters given in Table (I) to obtain the invariant yield of π+ that matches the ALICE measurement shown as red circles in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='(1)[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Earlier hydrody- namic model studies [36] have shown that it is not possi- ble to simultaneously describe π+ and p spectra for zero baryon chemical potential, so we use a different set of val- ues for u0, c2, and T (given in the rightmost column of 3 Table (I)) to explain the proton spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The blast-wave model with these parameters reasonably well explains the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (1) shows the blast-wave results for v2 of π+ and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We do not compare these ideal results with experimental data since they are known to over-predict the experimental mea- surements, but we show them for the sake of qualitative description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' ●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 1 10 100 pT (GeV) d2 N / pT d pT d y (GeV)-2 Blast wave fit ▲ p π+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 pT(GeV) v2 p π+ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 1: (Color online)Top plot is the comparison of the experimentally measured π+ (red filled circles) and protons (black triangles) for Pb+ Pb collision at √s=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='76 TeV for 20-30 % centrality with the ideal blast wave results (lines) for the parameters in Table (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='The lower panel shows the corresponding v2 vs pT for π+ (blue line) and protons (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' After setting up the parameters for the ideal case (zero viscosity), we can now explore the effect of viscosity (through the relaxation time τc) and electric fields on spectra and flow harmonics by comparing the corre- sponding results with the ideal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In our model, τc and the electric field qE appear in δf 1, giving rise to additional corrections to the invariant yields and flow harmonics of charged hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We compute the pT differ- ential n-th order flow coefficient using the usual formula: vn(pT , y) = � 2π 0 dϕ cos(nϕ) dN d2pT dy � 2π 0 dϕ dN d2pT dy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (4) As we need the electric field distribution on the freeze- π+ p u0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='22 T 130 MeV 140 MeV R 10 fm 10 fm τ 6 fm 6 fm m 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 MeV 938 MeV c2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 TABLE I: The fit parameters for π+ and p respectively at mid-rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' out hypersurface to be used in the Cooper-Frye formula (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2)) we use a parameterized form of the EM field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In principle, the fields generated in the initial stages of heavy-ion collisions due to the charged protons inside the two colliding nucleus would evolve with the QGP fluid, but the blast wave model does not allow any such self- consistent dynamical evolution of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The magnitude of the electric fields in all the cases are kept Here we use parameterised electric fields of four different configura- tions in the transverse plane (XY plane) while calculating the invariant yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Some of these configurations does not represent the actual scenario encountered in heavy- ion collisions but we use them for exploratory purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) we show the first setup of electric fields which we call config-1 henceforth, this represents isotropic fields due to a point charge of large magnitude at the origin, the right panel shows config-2 that closely resembles the field configuration expected in symmetric heavy-ion collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' config-3 as shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) is the π/2 rotated version of config-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) represents a constant unidirectional elec- tric fields which might be applied in a limited sense for large-small nuclei collisions, this is config-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' To better understand the effect of electric fields on flow harmonics we note the following config-1: isotropic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' expected to unalter the flow harmonics, config-2: prolate-like fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' expected to decrease the flow harmonics, config-3: oblate-like fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' expected to increase the flow harmonics, config-4: directional fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' expected to alter di- rectional flow only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' RESULTS Here we discuss our main results for different configura- tions of electric fields (described in the previous section) 4 40 20 0 20 40 40 20 0 20 40 config-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='025 40 20 0 20 40 40 20 0 20 40 config-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='04 40 20 0 20 40 40 20 0 20 40 config-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='030 40 20 0 20 40 40 20 0 20 40 config-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 2: (Color online) Electric field configurations in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Detail expressions for different configurations are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (B 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Magnitude of electric fields (in GeV2) are shown using colour map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' on the spectra ,and v2 for pions and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (3) shows the dependence of pT spectra of π+ on the electric fields for ideal case, for comparison we also show the results for non-zero viscosity but with no elec- tric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The magnitude of viscosity in our model is controlled through tc which is set to 1 fm for the results shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here we see that the pT spectra hardly shows any dependence on different transverse electric field con- figurations (solid orange, black, red lines correspond to config-1, config-2, and config-3 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' As ex- pected the effect of finite viscosity is comparatively more prominent on the pT spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' we see a suppression at higher pT region for the viscous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Since both bulk and shear viscosity is present in our case, the slope of the spectra is determined by the relative contributions of these two viscosities [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (3) shows the pT spectra for protons for ideal (with electric fields) and viscous (without electric field) cases where we see a small suppression in the lower pT region for the case of config-2 and config-3 for ideal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also see a suppression at higher pT for viscous case as was seen for π+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Before discussing the differential v2 for different cases, it is worthwhile to explore the dependence of pT inte- grated (0-3 GeV) dN/dφ on different field configurations and viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (4) we show the dN/dφ of π+ as a function of φ for various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' As ex- pected we have pure cosine like dependence of dN/dφ for ideal case (shown by the solid blue line), the viscosity re- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='100 1 10 100 pT (GeV) d2N / pT dpT dy (GeV)-2 viscosity Ideal (config-4) Ideal (config-3) Ideal (config-2) Ideal (config-1) Ideal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='500 1 pT (GeV) d2 N / pT dpT dy (GeV)-2 viscosity Ideal (config-4) Ideal (config-3) Ideal (config-2) Ideal (config-1) Ideal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 3: (Color online) Upper panel represents the pT Spectra for π+ and lower panel for proton with different transverse field configurations as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) and for the case of viscosity for tc = 1fm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' duces the multiplicity as well as the amplitude (shown by the red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' config-1 the isotropic field configu- ration (orange solid line) almost coincides with the ideal result, but we see noticeable change in the amplitude of dN/dφ for config-2, config-3 as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Things be- come more interesting for config-4 (black dashed line), here we generate finite directed flow like behaviour, this is understood from the fact that a unidirectional force shifts the centre of mass of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Similar behaviour was observed for protons also (shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' A more concrete way to study the angular dependence of dN/dφ for config-4 can be achieved by using a fi- nite Fourier series decomposition of dN/dφ as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' f(Φ) = N � 1 + 2 3 � n=1 vncos(nΦ) + 2 3 � n=1 wnsin(nΦ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5) A non-linear least squares fit with vn, wn and N as a free- parameters we obtain the best fit for dN/dφ with the val- ues of these parameters given in Table (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here we note that the directional force v1 is larger than v2 for π+, and 5 3 2 1 0 1 2 3 0 5 10 15 20 Φ (Radian) dN / dΦ 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='9 Φ (Radian) dN / dΦ viscosity Ideal (config-4) Ideal (config-3) Ideal (config-2) Ideal (config-1) Ideal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 4: (Color online) Top panel: dN/dφ as a function of φ for π+ for various field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Bottom panel: same as top panel but for proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' they are similar in magnitude for protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Moreover, we notice that unlike other cases the azimuthal distribution breaks reflection symmetry with respect to the Y axis which gives rise to non-zero wn shown in Table (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also observe a mass dependence of the directional flow as π+ has a larger v1 compared to the protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' π+ p N 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='907 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='7e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='672 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1e-08 v1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='218 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0e-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='081 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='9e-08 v2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='064 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='096 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='9e-08 v3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='017 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='7e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='005 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 w1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='213 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0e-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='081 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='9e-08 w2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='000 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='000 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 w3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='017 ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='005 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8e-08 TABLE II: Fit parameters for π+ and proton for config-4 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' To have a visual understanding of the goodness of fit, we show a comparison of the fitted values (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5)) (solid blue line) and the dN/dφ from the blast wave model (dotted-dash orange line) for π+ (top panel) and proton (bottom panel) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 3 2 1 0 1 2 3 (Radian) 4 6 8 10 12 14 16 18 20 dN/d Pion Fit Plot Blastwave result 3 2 1 0 1 2 3 (Radian) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='9 dN/d Proton Fit Plot Blastwave result FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 5: (Color online) Blast wave results and the fitted curves using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (5) for config-4 for π+ (top panel) and proton (bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' More familiar and useful observables in experiments are centrality and pT dependent flow harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (6) we show the dependence of the second-order flow harmonics v2 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='a elliptic flow) for different configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here we see that there is almost no deviation from the ideal case for the isotropic (config-1) and directed field case (config-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' However, the situation is different for the other two cases, we can clearly see an increase in v2 for config-3 and a suppression for config-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also note that viscosity supresses the elliptic (red dashed line) flow for π+ and elavates for proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The effect of electric fields become more interesting when we examine the difference in v2 for particles (π+, p) and antiparticles (π−, ¯p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' This difference ∆v2 = v2(h) − v2(¯h) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (7) as a function of pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We ob- serve a non-monotonic variation in ∆v2 as a function of pT for both pions and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Interestingly, a similar observation was made in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Throughout this study we only consider the effect of 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 pT (GeV) v2 viscosity Ideal (config-4) Ideal (config-3) Ideal (config-2) Ideal (config-1) Ideal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 pT (GeV) v2 viscosity Ideal (config-4) Ideal (config-3) Ideal (config-2) Ideal (config-1) Ideal FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 6: (Color online) v2 vs pT for π+ (top panel) and proton (bottom panel) for the transverse electric field configurations shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' electric fields and the first order correction in the δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' As mentioned earlier, previous studies showed that there are new transport coefficients at higher order corrections to f which may alter the results obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' But these are beyond the scope of the present exploratory study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (8) we show the temperature and mass dependence of some of the transport coefficients arising due to the magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' This may give some hints of the relative contribution of various transport coefficients in the bulk observables while used in the Cooper-Frye prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' SUMMARY AND CONCLUSION In this work, we have studied the effect of electric fields on the bulk observables in heavy-ion collisions such as pT spectra, directed and elliptic flow of charged pi- ons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We use the blast-wave model and dif- ferent configurations of electric fields in the transverse plane to carry out this exploratory study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The pT spec- tra of hadrons in the blast-wave model are obtained us- ing Cooper-Frye prescription, where we incorporate non- equilibrium correction δf due to the viscosity and the electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Since the blast-wave model does not in- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 pT (GeV) v2(h)-v2(h) config-3 config-2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 7: (Color online) v2(h) -v2(¯h) as a function of pT for config-2 (red) and config-3 (blue) line corresponds to π (Solid line) and p (Dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' clude space-time evolution, the fluid velocity and the electric fields are parameterized on the freezeout hyper- surface to calculate experimental observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' For our case, fluid velocity fields are modulated so that it domi- nantly generates the elliptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We use four different configurations of transverse electric fields (i) isotropic fields, (ii) prolate-like fields, (iii) oblate-like fields, and (iv) directed fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The typical maximum value of the electric field for all these configurations is ∼ m2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We find that flow harmonics for isotropic fields remain unchanged for both pions and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Both prolate and oblate-like field configuration alters the flow harmonics, and the di- rected field gives rise to large directed flow v1 for both pions and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also observe a mass dependence of v1 generated due to the electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also dis- cuss the temperature and mass dependence of some of the new transport coefficients that appear in the second- order correction in the distribution function due to the EM field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' They can further contribute to the results we obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' This we leave for a possible future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' ACKNOWLEDGMENTS AP acknowledges the CSIR-HRDG financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' RG and VR acknowledge support from the DAE, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Appendix A: Appendix In this work we have used Milne coordinate system (τ, η, r, φ), where τ = √ t2 − z2, r = � x2 + y2, η = tanh−1(z/t), and φ = tan−1(y/x), due to the co-ordinate transformation various equations changed forms com- pared to the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here we give the details about the Jacobian and Christoffel symbols used in this study due to the above coordinate transformation: 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='40 δ πVB Mass Dependence T = 120 MeV T = 100 MeV T = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='40 T emperature Dependence m = 120 MeV m = 100 MeV m = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='30 δ ΠVB T = 120 MeV T = 100 MeV T = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='30 m = 120 MeV m = 100 MeV m = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0 2 4 6 8 10 l VπB , l VΠB , τ VΠB T = 120 MeV T = 100 MeV T = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 ρ VVB T = 120 MeV T = 100 MeV T = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0 2 4 6 8 m = 500 MeV m = 250 MeV m = 100 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 m = 500 MeV m = 250 MeV m = 100 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 m (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 τ VVB T = 120 MeV T = 100 MeV T = 80 MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='5 T (GeV) 0 2 4 6 8 m = 500 MeV m = 250 MeV m = 100 MeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 8: (Color online)Mass and temperature variation of the transport coefficients arising due to external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' the Jacobian for volume element is √−g = τr and the non-vanishing Christoffel symbols are Γτ ηη = τ ,Γη τη = 1 τ , Γr φφ = −r ,Γφ rφ = 1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The space-like projection is defined as ∆µν = gµν−uµuν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' for Milne coordinate system, differ- ent components of ∆µν are ∆ττ = − (ur)2 , ∆ηη = − 1 τ 2 , ∆rr = −1 − (ur)2 , ∆φφ = − 1 r2 , ∆τη = ∆τφ = ∆ηr = ∆ηφ = ∆rφ = 0 , ∆τr = − � 1 + (ur)2ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The expansion scalar is given by θ = Dµuµ = ∂µuµ + Γµ µαuα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We consider real particles and the on-shell con- dition is given by gµνpµpν = (pτ)2 − τ 2 (pη)2 − (pr)2 − r2 � pφ�2 = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Following the convention used in heavy-ion collisions we express the compo- nents of the four momentum pµ as (E, px, py, pz) = (mT cosh y, pT cos ϕ, pT sin ϕ, mT sinh y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Where pT = � p2x + p2y , mT = � m2 + p2 T , and y = tanh−1 (pz/E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The components of the four-momentum in Milne co- ordinates are pτ = mT cosh (y − η), τpη = mT sinh (y − η), pr = pT cos (ϕ − φ), rpφ = pT sin (ϕ − φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' ϕ is the azimuthal angle of the particle in the momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) we have the term pµdΣµ which in our is given by pµdΣµ = gµνpµdΣν = mT cosh (y − η)τdηrdrdφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Appendix B: First-order (δf) correction to the single-particle distribution In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (3) we introduced the first-order correction to the single-particle distribution while calculating invariant yield using the Cooper-Frye formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Here we give the detail expression of δf in terms of gradients of fluid variables and 8 fields: δf = − τc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � pµ∂µf0 + qF µνpν ∂f0 ∂pµ � = − τc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � −pµf0 ˜f0 [βpαDµuα + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)∂µβ − ∂µα] � − τc u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � −f0 ˜f0qF µνpνβ ∂(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p) ∂pµ � = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p (βpµpαDµuα + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα) − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpµpα � uµ ˙uα + σµα + ωµα + ∆µαθ 3 � + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα � − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpµpα � σµα + ∆µαθ 3 � + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα � − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpµpα∂µuα + βpφpφrur − βpηpητuτ + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα � − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpφpφrur − βpηpητuτ + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα � − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν +τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpτpr∂τur + βprpr∂rur + βpφpr∂φur � + τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βpτpτ∂τuτ + βprpτ∂ruτ + βpφpτ∂φuτ � = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � β �pT sin (ϕ − φ) r �2 rur − β �mT sinh (y − η) τ �2 τuτ + (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα � −τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν + τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � βmT cosh(y − η)pT cos(ϕ − φ)(ur)2 ruτ − β(pT cos(ϕ − φ))2 ur r � +τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p β pT sin(ϕ − φ) r pT cos(ϕ − φ)u0 2r R � nun sin n [φ − ψn] +τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p � −β(mT cosh(y − η))2 (ur)3 r(uτ)2 + βmT cosh(y − η)pT cos(ϕ − φ)(ur)2 ruτ � −τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p β pT sin(ϕ − φ) r mT cosh(y − η)ur uτ u0 2r R � nun sin n [φ − ψn], δf = τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p β �ur r + sin(ϕ − φ)cos(ϕ − φ) r u0 2r R � unn sin n [φ − ψn] � p2 T − τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβEνpν +τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p β � 2cosh(y − η)cos(ϕ − φ)(ur)2 ruτ − cosh(y − η)sin(ϕ − φ) ur ruτ u0 2r R � unn sin n [φ − ψn] � mT pT −τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p β � sinh2 (y − η)uτ τ + cosh2(y − η) (ur)3 r(uτ)2 � m2 T + τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p ((u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p)pµ∂µβ − pµ∂µα) , where β = 1 T and α = µ T with Γτ ηη = τ along with Γr φφ = −r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' The contribution due to the electric field E · p in the above equation, when expanded, takes the following form δf = −τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβE · p, δf = −τcf0 ˜f0 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='p qβ � Eτpτ − τ 2Eηpη − Erpr − r2Eφpφ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Co-ordinate transformation of Electric four vector The electric field components in Milne-coordinates (Eτ, Eη, Er, Eφ) are connected to the Cartesian components (Et, Ex, Ey, Ez) through the following transformation 9 � �� Eτ Er Eφ Eη � �� = � ��� Cosh[η] 0 0 −Sinh[η] 0 Cos[φ] Sin[φ] 0 0 −Sin[φ] r Cos[φ] r 0 −Sinh[η] τ 0 0 Cosh[η] τ � ��� � �� Et Ex Ey Ez � �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We also note E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content='u = 0 and this gives rise to : Et = EzSinhη coshη + (cosφEx + sinφEy)ur coshηuτ , Eη = Ez τCoshη − tanhηErur τuτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' As mentioned in the main text we use four different configuration of transverse electric fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' they are parameterised as : eEx = BZαem(x − x0)Cosh[η − η0] ((x − x0)2 + (y − y0)2 + (τSinh[η − η0])2)3/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' eEy = AZαem(y − y0)Cosh[η − η0] ((x − x0)2 + (y − y0)2 + (τSinh[η − η0])2)3/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' eEz = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' where Z is the atomic number (for our case we choose Z=82),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' αem= 1 137,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' A and B are the modulation factors which controls the spatial configuration of the field in the transverse plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' and α is the fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' We get the first 3 configurations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' (2) by choosing (i) A=B=10 for config-1, (ii) A=20 , B=1 for config-2 (iii) A=1 , B=20 for config-3, and qEx = qEy = m2 π for config-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Wilson, Confinement of quarks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' D 10, 2445 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YNAyT4oBgHgl3EQfvfmK/content/2301.00632v1.pdf'} 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b/aNFST4oBgHgl3EQfBTj3/content/tmp_files/2301.13702v1.pdf.txt @@ -0,0 +1,1755 @@ +Tachyons and Misaligned Supersymmetry in +Closed String Vacua +Carlo Angelantonj1, Ioannis Florakis2 and Giorgio Leone1 +1 Dipartimento di Fisica, Università di Torino and INFN Sezione di Torino +Via Pietro Giuria 1, I-10125 Torino +2 Department of Physics, University of Ioannina, GR-45110, Ioannina +ABSTRACT +In a remarkable paper, Dienes discovered that the absence of physical tachyons in closed string +theory is intimately related to oscillations in the net number of bosonic minus fermionic degrees +of freedom, a pattern known as misaligned supersymmetry. The average of these oscillations was +linked to an exponential growth controlled by an effective central charge Ceff smaller than the +expected inverse Hagedorn temperature. Dienes also conjectured that Ceff should vanish when +tachyons are absent. +In this paper, we revisit this problem and show that misaligned supersymmetry is actually re- +alised even when tachyons are present in the physical spectrum. In fact, we prove that the average +growth rate Ceff is set by the mass of the lightest state, be it massless or tachyonic, and coincides +with the inverse Hagedorn temperature only when fermions are absent from the spectrum. We also +provide a general proof that the necessary and sufficient condition for classical stability is Ceff = 0, +in agreement with Dienes’ conjecture. +E-mail: carlo.angelantonj@unito.it +iflorakis@uoi.gr +giorgio.leone@unito.it +arXiv:2301.13702v1 [hep-th] 31 Jan 2023 + +Contents +1 +Introduction +1 +2 +Asymptotic Mass Degeneracies +5 +3 +A Proof of Misaligned Supersymmetry +7 +4 +Non-Supersymmetric Heterotic Vacua in D = 10 +10 +5 +Scherk-Schwarz Reductions at Rational Points +13 +5.1 +A Comment on Phase Transitions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 +6 +Conclusions +18 +A The Scherk-Schwarz Mechanism in RCFT’s +19 +1 +Introduction +Superstring vacua are typically unstable when space-time supersymmetry is absent. The instabil- +ity can be ascribed to the emergence of tadpoles for scalar fields or, more explicitly, to the pres- +ence of tachyons in the classical spectrum. In the first case, much progress has been made in the +last years in the determination of the corrected geometry which can describe, for instance, spon- +taneous compactification or cosmological evolution [1–15]. The second case, instead, is much +less under control. In fact, although the condensation of open-string tachyons is well under- +stood [16, 17], very little is known about the rolling of closed-string ones. Moreover, in critical +strings tadpoles typically emerge at higher genus and, as such, induce quantum corrections to the +classical moduli space. In this case, perturbation theory is relatively under control at least up to +the order associated to the tadpole itself. On the contrary, tree-level tachyons imply that the string +vacuum cannot be trusted even classically, and it is impossible to extract any meaningful informa- +tion from it. Therefore, if one insists that a non-supersymmetric vacuum ought to be predictive, +at least classically, one is bound to consider only string configurations where tachyonic excitations +are absent. This turns out to be rather non-trivial to achieve and requires a precise correlation in +the growth rates of massive bosonic and fermionic string states, since these two features are strictly +related by modular invariance. +A first study of the interplay between massive states and tachyons was done in [18]. There it +was shown that the infra-red finiteness of the one-loop vacuum energy of closed oriented strings +implies the overall cancellation +lim +Λ→∞ +� +n +(dB(n)−dF(n)) e−4πn/Λ2 ≡ lim +Λ→∞ +� +n +d(n)e−4πn/Λ2 = 0, +(1.1) +between the total bosonic dB(n) and fermionic dF(n) degrees of freedom, at mass-squared n. This +property goes under the name asymptotic supersymmetry, since it is reminiscent of the Bose-Fermi +degeneracy one typically encounters when supersymmetry is present. +1 + +A more refined analysis based on the Rankin-Selberg-Zagier transform [19–21] done in [22], +actually showed that, when physical tachyons are absent and thus the one-loop vacuum energy +Ω is finite, the distribution of bosonic and fermionic degrees of freedom follows a precise pattern +dictated by +� +n +d(n)e−4πn/Λ2 ≃ Λ2−D 3Ω +π + +� +ζ∗(ρ)=0 +Cρ Λρ1−D cos +� +ρ2 logΛ+ϕρ +� +, +(1.2) +where the sum in the RHS runs over the zeroes ρ = ρ1 +iρ2 of the dressed Riemann zeta-function, +ζ∗(s) = π−s/2 Γ(s/2)ζ(s), and Cρ and ϕρ are model-dependent real constants. Indeed, this expres- +sion implies that not only must the full string spectrum in D > 2 non-compact space-time dimen- +sions enjoy asymptotic supersymmetry but, for finite large Λ the excess of bosonic and fermionic +degrees of freedom must alternate with a frequency dictated by the non-trivial zeroes of the Rie- +mann zeta-function. This oscillatory behaviour of a classically stable string spectrum was actually +discovered already in [23], where it was named misaligned supersymmetry. Notice that the analysis +of [18] and [22] crucially rely on the finiteness of the one-loop modular integral, and thus the results +(1.1) and (1.2) are necessary conditions for the tree-level stability of a string vacuum. Whether these +conditions are also sufficient is an open problem which cannot be addressed using the techniques +in [18,22]. +5 +10 +15 +20 +25 +-40 +-20 +20 +40 +Figure 1.1: The signed logarithm of the net number of degrees of freedom at each mass level for the +non-tachyonic SO(16)×SO(16) heterotic string in D = 10 highlighting the presence of misaligned +supersymmetry. Positive (negative) contributions are ascribed to the excess of bosonic (fermionic) +states. +A somehow different route to relate the distribution of degrees of freedom to the classical sta- +bility of closed-string vacua was followed by Dienes in the remarkable paper [23]. There, he ob- +served that the degrees of freedom of non-tachyonic, non-supersymmetric vacua follow a pre- +cise pattern as in Figure 1.1, where the net excess between bosons and fermions alternates as the +mass-level increases1. He then observed that the presence of oscillations can be traced back to the +growth property of a special quantity, termed the sector averaged sum, to be introduced shortly. +1Clearly, in the case of unbroken supersymmetry there is an exact degeneracy between bosonic and fermionic states +and, therefore, no oscillations are observed in the full spectrum. +2 + +In a given string background, the partition function admits the double power-series expansion +Z = +� +n,m +� +a,b +Nab dab(n,m)qn ¯qm , +(1.3) +with dab(n,m) counting the net number of bosonic minus fermionic states with left mass-squared +n and right mass-squared m in the sector ab. The matrix Nab enforces the GSO projection and +builds the string vacuum. Propagating degrees of freedom in the ab sector are selected by the +level matching condition which identifies the left-moving and right-moving masses, and are then +counted by the integral functions dab(n) ≡ dab(n,n), whose domain is a countable set associated +to the string oscillators, Kaluza-Klein momenta and/or windings. Two-dimensional conformal in- +variance and modularity imply a universal exponential growth of dab(n), +dab(n) ∼ eCtot +�n , +(1.4) +controlled by the total central charge Ctot = 4π(�cL/24 + �cR/24), which also defines the inverse +Hagedorn temperature2. This growth is then inherited by the total net number of physical states +d(n) = � +a,b Nab dab(n), obtained by summing over all sectors. +The connection between the asymptotic growth of states and the absence of tachyons cannot +be formulated in terms of the physical d(n), but requires the introduction of the continuous en- +veloping functions Φab(n), which reproduce the dab(n) for the special values of n associated to the +actual masses [23]. Assuming that such functions exist, Dienes introduced the sector averaged sum +〈d(n)〉 = +� +a,b +Nab Φab(n), +(1.5) +which discriminates between tachyonic and non-tachyonic vacua. In fact, 〈d(n)〉 is no-longer +bound to exhibit the universal exponential growth (1.4), controlled by the total central charge, but +Dienes showed [23] that in a classically stable vacuum +〈d(n)〉 ∼ eCeff +�n , +(1.6) +with Ceff < Ctot. The cancellation underlying this slower growth clearly implies the oscillatory be- +haviour of fig. 1.1 observed in the string spectrum and it was claimed in [23] to be a necessary and +sufficient condition for the IR finiteness of String Theory. There it was also conjectured that the +absence of physical tachyons actually implies Ceff = 0, a conjecture which was later shown to be +true [24] in the case of the ten-dimensional SO(16)×SO(16) heterotic string. +Simple numerical experiments on non-supersymmetric string vacua, actually reveal a slightly +more complicated story. In fact, we find that the oscillatory behaviour of the net number of physi- +cal states is not an exclusive prerogative of classically stable vacua, but is also present in all tachy- +onic string theories, in ten and lower dimensions, which have both bosonic and fermionic states. +2Of course, the total central charge of the full CFT is cL +cR. However, by abuse of language, we shall refer to Ctot as +the total central charge, as in [23]. +3 + +5 +10 +15 +20 +25 +-40 +-20 +20 +40 +Figure 1.2: The signed logarithm of the net number of degrees of freedom at each mass level for +the tachyonic SO(16)×E8 heterotic string in D = 10. The plots exhibits misaligned supersymmetry +even in the presence of physical tachyons. Positive (negative) contributions are ascribed to the +excess of bosonic (fermionic) states. +For instance, in the case of the ten-dimensional non-supersymmetric heterotic string with gauge +group SO(16)×E8 the signed multiplicities follow the oscillatory pattern shown in Figure 1.2. This +is so despite the fact that the low-lying spectrum contains a physical tachyon of mass-squared +m2 +T = −2/α′ in the representation (16,1). Moreover, the exponential growth of the sector averaged +sum is controlled by Ceff < Ctot, thus providing a notable counter-example to the argument given +in [23]. In fact, in the present paper we will prove, independently of the details of the string con- +struction, that Ctot only sets the growth rate in purely bosonic theories, like the bosonic string and +the type 0A/0B superstrings while, whenever fermions are present in the spectrum, the growth +rate is slower than Ctot, and is actually set by the mass of the lightest states, whether tachyonic or +massless, +Ceff = 4π +� +|α′m2 +lightest| < Ctot . +(1.7) +Importantly, our result implies that the necessary and sufficient condition for classical stability is +an at most polynomial growth of the sector averaged sum, i.e. Ceff = 0. This proves the conjecture +of Dienes for any critical closed-string vacuum, regardless of its CFT realisation. +The paper is organised as follows. In Section 2, we review the asymptotic properties of the de- +generacies of states in Rational CFT’s (RCFT’s). Section 3 deals with the analysis of the sector aver- +aged sum and contains the main result of our work. In Sections 4 and 5, we illustrate our results by +studying concrete examples in ten and lower dimensions, and we conjecture the presence of first- +order phase transitions on the world-sheet of an averaged CFT associated to the Scherk-Schwarz +reduction. Our conclusions and perspectives are gathered in Section 6. Finally, the Appendix con- +tains the relevant character decomposition of shift orbifolds of one-dimensional Narain lattices at +rational points. +4 + +2 +Asymptotic Mass Degeneracies +As anticipated, in order to uncover the origin of misaligned supersymmetry and of the absence +of physical tachyons, one needs to study the large-mass behaviour of the string spectrum. For +simplicity, we assume that the world-sheet fermions and the compact bosons contribute with a +finite number M of characters, both in the holomorphic and anti-holomorphic sectors. In princi- +ple, these sets of characters need not be the same, as for instance, in the heterotic string. Still, for +convenience, we shall use the same symbol χ to label the contributions χa from the holomorphic +sector and ¯χa from the anti-holomorphic one. The D non-compact bosons contribute instead with +the Dedekind η-function, so that the one-loop vacuum energy reads +� +F +dµZ = +� +F +dµ +1 +(�τ2 ¯ηη)D−2 +� +a,b +¯χaNabχb , +(2.1) +where τ = τ1+iτ2 is the complex structure modulus of the world-sheet torus, F is the fundamental +domain of SL(2;Z) and dµ = τ−2 +2 dτ1 dτ2 is the modular invariant measure. Again, Nab is a matrix +enforcing the GSO projection and is subject to the conditions +N = T † +R N TL , +N = S† +RN SL , +(2.2) +where TL,R and SL,R represent the action of the generators of SL(2;Z) on the characters χa and ¯χa, +respectively. Notice that, since any element of the modular group can be decomposed in terms of +T and S, the previous conditions also imply +N = M† +RN ML , +∀ML,R ∈ SL(2;Z). +(2.3) +In string theory, the characters χa of the RCFT always come together with the contribution of +the non-compact bosons, and it is natural to define the dressed pseudo-characters ˆχa = η2−D χa. +Strictly speaking, these are no longer characters of a RCFT, since they carry the non-trivial modular +weight 1−D/2. As a result, the generators of the modular group involve extra phases, and act as +T : +ˆχa → ˆTab ˆχb = e−2πi(D−2)/24 Tabχb +ηD−2 , +S : +ˆχa → τ1−D/2 ˆSab ˆχb = τ1−D/2 i D/2−1 Sabχb +ηD−2 . +(2.4) +The pseudo-characters ˆχa, from now on simply referred to as the characters χa with the hat +omitted, admit a power series expansion in the nome q = e2πiτ of the form +χa(q) = +∞ +� +n=0 +da(n)qHa+n , +(2.5) +with Ha = ha − c/24 expressed in terms of the conformal weight ha and the central charge c of +the full theory, including the D − 2 non-compact bosons. As a result, the total net number of the +5 + +physical degrees of freedom at mass level n reads +d(n) = +� +a,b +Nab ¯da(n + Hb − ¯Ha)db(n), +(2.6) +where we have imposed the level-matching condition. +The coefficients da(n), counting the degeneracies of states of χa, can be computed using Cauchy +theorem, +da(n) = +1 +2πi +� +Γ +dq χa(q) +qna+1 , +(2.7) +where na = n + Ha, and Γ is any closed contour lying entirely inside the unit disk in the complex +q-plane and containing the origin q = 0. The large-n behaviour of this integral can be extracted +using the circle method of Hardy and Ramanujan [25] and takes the form [26] +da(n) = +M−1 +� +b=0 +O([�n]) +� +ℓ=1 +Q(ℓ,n) +ab +fb(ℓ,na). +(2.8) +The matrix +Q(ℓ,n) +ab += i 1−D/2 +ℓ−1 +� +p=0 +gcd(ℓ,p)=1 +� +M−1 +ℓ,p +� +ab e− 2πi +ℓ (pna−p′Hb) +(2.9) +is expressed in terms of the representation Mℓ,p of the modular transformation +γℓ,p = +� +−p′ +1+pp′ +ℓ +−ℓ +p +� +γℓ,p ∈ SL(2,Z) +(2.10) +on the space of characters, +χa → (−ℓτ+ p)1−D/2 (Mℓ,p)abχb , +(2.11) +while +fb(ℓ,na) = 2π +ℓ db(0) +� Hb +na +�D/4 +J−D/2 +�4π +ℓ +� +Hb na +� +, +(2.12) +with Jk(x) being the Bessel function of the first kind. Notice that the Q(ℓ,n) +ab +are real functions, as +follows from the conjugation property of the modular transformations [26], in accordance with the +fact that da(n) count physical states and are thus real. +The large-n behaviour of the degeneracies da(n) is clearly dictated by the fb(ℓ,na) functions, +and receive contributions from all the characters χb. These are power-like if Hb ≥ 0 and expo- +nential whenever Hb < 0, as can be seen from the asymptotic expansions of the Bessel functions +6 + +Jα(x) ∼ +� +2 +πx +� +cos +� +x − πα +2 − π +4 +� +− 4α2 −1 +8x +sin +� +x − πα +2 − π +4 +� ++... +� +, +(2.13) +and +Jα(ix) ≡ i αIα(x) ∼ i α +ex +� +2πx +� +1− 4α2 −1 +8x ++... +� +, +(2.14) +valid for positive x. It is therefore clear that the asymptotic growth of the degeneracies is domi- +nated by the tachyonic characters, which are always present in the CFT of any string construction, +although they may be eliminated from the physical spectrum by the GSO projection. Schemati- +cally, we have +da(n) ∼ +� +b |Hb<0 +db(0) +� +2 +|Hb|(D−1)/4 +n(D+1)/4 +� +Q(1,n) +ab +e4π�|Hb|n + +Q(2,n) +ab +� +2 +e2π�|Hb|n + +Q(3,n) +ab +� +3 +e +4π +3 +�|Hb|n +... +� +, (2.15) +with Q(1,n) +ab += i 1−D/2 Sab simply determined by the S matrix. The tachyon with the smallest confor- +mal weight, included in χ0, is associated to the ubiquitous NS vacuum and plays the role of the +identity in the RCFT. As a result, it dictates the leading growth +da(n) ∼ e4π +� +cn/24 +... , +(2.16) +in agreement with the Cardy formula [27], and this contribution is universal since Sa0 = i D/2−1/ +� +M. +The sub-leading exponentials depend instead on the data of the RCFT. +3 +A Proof of Misaligned Supersymmetry +We have now all the ingredients to connect the large-n behaviour of d(n) with the classical stability +of a string vacuum. To this end, following [23], we continue the integer n to the reals, introduce the +enveloping functions da(n) → Φa(n), and construct the sector averaged sum +〈d(n)〉 = +� +a,b +Nab ¯Φa(n + Hb − ¯Ha)Φb(n), +(3.1) +whose asymptotic behaviour dictates the presence/absence of physical tachyons. +From eqs (2.6) and (2.16) it follows that the SL(2;C) invariant vacuum in principle determines +the leading large-n behaviour, +〈d(n)〉 ∼ e4π�n( +�cL/24+�cR/24) � +a,b +Nab = eCtot +�n � +a,b +Nab . +(3.2) +However, whether the asymptotic growth is controlled by Ctot or not depends entirely on the prop- +erties of N . Recall that the GSO matrix must yield a modular invariant partition function, and is +7 + +thus subject to the conditions (2.2). As a result, +N00 = 1 +M +� +a,b +Nab +(3.3) +since, for a RCFT, where all M characters have been resolved, Sa0 = i D−2−1/ +� +M. Moreover, the +entries Nab can only be ±1 or zero, which yield to the following two scenarios: +1. If the spectrum does contain the leading tachyon, i.e. N00 = 1, one has +� +a,b +Nab ̸= 0 +and +〈d(n)〉 ∼ eCtot +�n . +(3.4) +2. If the spectrum does not contain the leading tachyon, i.e. N00 = 0, then +� +a,b +Nab = 0 +and +〈d(n)〉 ∼ eCeff +�n +(3.5) +with an effective central charge Ceff < Ctot. +The condition N00 = 0 clearly requires that fermions are present in the spectrum and, in particular +the number of bosonic sectors must equal the number of fermionic ones. On the contrary, in +the absence of extended symmetries, namely when each character appears only once in (2.1), the +condition N00 = 1 implies that the spectrum only contains bosonic excitations. These observations +were already contained in [23] and, in fact, represent the main result of Dienes’ paper. However, it +is important to stress that, contrary to the result of [23], Ceff < Ctot does not imply classical stability +since, although it is true that the leading tachyon must be absent, the condition N00 = 0 does not +automatically exclude the possibility that other tachyons be present. Indeed, as we shall see in the +following Sections, this is precisely what happens in most non-supersymmetric theories. +The analysis of the sub-dominant contributions to 〈d(n)〉 is a bit more involved. To start, we +notice that the Q’s are periodic functions of n, with period ℓ, +Q(ℓ,n+ℓm) +ab += Q(ℓ,n) +ab +, +(3.6) +for any integer m. Moreover, since the number of physical states (2.6) depends on the prod- +uct of the Q functions from the holomorphic and anti-holomorphic characters, it is convenient +to decompose the degrees of freedom into classes organised by the common periodicity vℓ, ¯ℓ = +ℓ ¯ℓ/gcd(ℓ, ¯ℓ) of the two Q’s and associate to each class its own enveloping function Φa(n,{w}), as +in [24]. As a result, +〈d(n)〉 = +� +a,b +� +{w} +Nab ¯Φa(n,{w + Hb − ¯Ha})Φb(n,{w}) +≡ +� +a,b,c,d +[�n] +� +ℓ, ¯ℓ=1 +vℓ, ¯ℓ−1 +� +w=0 +Nab Q(ℓ,w) +bc +¯Q( ¯ℓ,w+Hb− ¯Ha) +ad +fc(ℓ,n) ¯fd( ¯ℓ,n). +(3.7) +8 + +Moreover, since one can always write w = kℓ +ℓr, and +vℓ, ¯ℓ−1 +� +w=0 += +ℓ−1 +� +kℓ=0 +¯ℓ +gcd(ℓ, ¯ℓ) −1 +� +r=0 +, +(3.8) +the periodicity of the Q’s can be exploited to cast the sector averaged sum as +〈d(n)〉 = +� +a,b,c,d +[�n] +� +ℓ, ¯ℓ=1 +ℓ−1 +� +kℓ=0 +¯ℓ +gcd(ℓ, ¯ℓ) −1 +� +r=0 +Nab Q(ℓ,kℓ) +bc +¯Q( ¯ℓ,kℓ+rℓ+Hb− ¯Ha) +ad +fc(ℓ,n) ¯fd( ¯ℓ,n). +(3.9) +This is the expression from which we can extract our main results. We have to distinguish the two +cases ℓ = ¯ℓ and ℓ ̸= ¯ℓ. In the latter case, if ℓ does not divide ¯ℓ, there is no contribution to the sector +averaged sum, since +¯ℓ +gcd(ℓ, ¯ℓ) −1 +� +r=0 +¯Q( ¯ℓ,kℓ+rℓ+Hb− ¯Ha) +ad += 0, +(3.10) +which follows from +¯ℓ +gcd(ℓ, ¯ℓ) −1 +� +r=0 +¯Q( ¯ℓ,k+rℓ) +ad += +¯ℓ +gcd(ℓ, ¯ℓ) −1 +� +r=0 +¯ℓ−1 +� +¯p=0 +gcd( ¯ℓ, ¯p)=1 +e− 2πi +¯ℓ ( ¯p(k+rℓ)+ ¯p′ ¯Hd) � +M−1 +¯ℓ, ¯p +� +ad += +¯ℓ−1 +� +¯p=0 +gcd( ¯ℓ, ¯p)=1 +e− 2πi +¯ℓ ( ¯pk+ ¯p′ ¯Hd) � +M−1 +¯ℓ, ¯p +� +ad +1−e +−2πi ¯p +¯ℓ +gcd(ℓ, ¯ℓ) +1−e−2πi ¯pℓ +¯ℓ += 0, +(3.11) +since ¯ℓ is an integer multiple of gcd(ℓ, ¯ℓ). Similarly, if ¯ℓ = mℓ, for some integer m, eq. (3.10) still +holds since ¯p and ¯ℓ must be co-prime, and thus +m−1 +� +r=0 +e−2πi ¯pr/m = 0. +(3.12) +The case ℓ = ¯ℓ is a bit more involved and, as we shall see, discriminates between tachyonic and +non-tachyonic string vacua. The Q functions have now the same periodicity, which implies that in +eq. (3.9) the sum over r is trivial, since nothing depends on r, and +〈d(n)〉 = +� +a,b,c,d +[�n] +� +ℓ=1 +ℓ−1 +� +p=0 +gcd(ℓ,p)=1 +ℓ−1 +� +¯p=0 +gcd(ℓ, ¯p)=1 +Nab +� +M−1 +ℓ,p +� +bc +� +M−1 +ℓ, ¯p +�∗ +ad e− 2πi +ℓ [(p− ¯p)Hb−(p′Hc− ¯p′ ¯Hd)] fc(ℓ,n) ¯fd(ℓ,n) +× +ℓ−1 +� +kℓ=0 +e− 2πi +ℓ (p− ¯p)kℓ . +(3.13) +9 + +The sum over kℓ imposes the condition p = ¯p which identifies the holomorphic and anti-holomor- +phic modular transformations. Using the condition (2.3) of modular invariance on the GSO matrix +Nab, one arrives at the final result +〈d(n)〉 = +[�n] +� +ℓ=1 +ℓ−1 +� +p=0 +gcd(ℓ,p)=1 +� +ab +Nab e +2πi +ℓ p′(Hb− ¯Ha)ℓ fb(ℓ,n) ¯fa(ℓ,n). +(3.14) +The exponential growth of the sector averaged sum is thus directly linked to the presence of tachyons +in the physical string spectrum. In fact, if tachyons are absent, the f functions are expressed in +terms of the J Bessel functions which only admit the asymptotic power-law growth (2.13) and +therefore Ceff = 0. On the contrary, if physical tachyons, with Hb = ¯Ha < 0, are present in the string +spectrum, then +〈d(n)〉 ∼ +� +a,b +Hb= ¯Ha<0 +Nab db(0) ¯da(0) |Hb|(d−1)/2 +2n(d+1)/2 +[�n] +� +ℓ=1 +ϕ(ℓ)e +8π +ℓ +�|Hb|n , +(3.15) +where ϕ(ℓ) is the Euler totient function. The exponential growth is then dictated by the conformal +weight Hb of the lightest physical tachyon, Ceff = 8π +� +|Hb|. +To reiterate, we have proven that the asymptotic growth rate of the sector averaged sum is +dictated by the mass of the lightest states, whether tachyonic or massless, +Ceff = 4π +� +|α′m2 +lightest| ≤ Ctot , +(3.16) +and thus the necessary and sufficient condition for classical stability is the vanishing of the effective +central charge, as conjectured by Dienes in [23]. +Our analysis is fully general and applies to any vacuum of oriented closed strings. It extends the +discussion of [24] which heavily relies on the representation of the characters in terms of eta quo- +tients of special type, which is a rather restrictive requirement, not met by most non-supersymme- +tric string vacua. +4 +Non-Supersymmetric Heterotic Vacua in D = 10 +A simple arena where to test our result is ten-dimensional closed-string vacua with no space- +time supersymmetry. Indeed, one-loop modular invariance allows for many consistent construc- +tions in ten dimensions, most of which do not enjoy space-time supersymmetry. These non- +supersymmetric vacua can be divided into three different classes: tachyonic theories with only +bosonic excitations, tachyonic theories with both fermionic and bosonic fields, and a single the- +ory with no tachyons. The unique representative of the last class is the SO(16) × SO(16) heterotic +theory [28,29] with partition function +Z16 = O8 ( ¯V16 ¯C16 + ¯C16 ¯V16)+V8( ¯O16 ¯O16 + ¯S16 ¯S16) +−S8 ( ¯O16 ¯S16 + ¯S16 ¯O16)−C8 ( ¯V16 ¯V16 + ¯C16 ¯C16), +(4.1) +10 + +while the first class comprises the type 0A and 0B strings [28], with partition functions +Z0A = |O8|2 +|V8|2 +S8 ¯C8 +C8 ¯S8 , +Z0B = |O8|2 +|V8|2 +|S8|2 +|C8|2 . +(4.2) +The second class is richer and contains five heterotic vacua with gauge groups SO(32), SO(16)×E8, +SO(8)×SO(24), (E7 ×SU(2))2 and SU(16) [28]. These theories present a similar behaviour and, for +simplicity, we shall concentrate on the SO(32) theory with partition function +Z32 = O8 ¯V32 +V8 ¯O32 −S8 ¯S32 −C8 ¯C32 . +(4.3) +Notice that in writing eqs. (4.1), (4.2) and (4.3) we have used the SO(2n) characters, as defined +in [30]. +As anticipated in the previous Section, strictly speaking these string theories do not correspond +to RCFTs because of the presence of non-compact bosons. Still, following [26], we can overcome +this problem by defining the pseudo-characters +(O2n,V2n,S2n,C2n) → +�O2n +η8 , V2n +η8 , S2n +η8 , C2n +η8 +� +, +(4.4) +and including suitable phases in the modular transformations +T : +(O2n,V2n,S2n,C2n) → e−iπ(n+8)/12 (O2n,−V2n,eiπn/4 S2n,eiπn/4C2n), +(4.5) +and +S : +� +����� +O2n +V2n +S2n +C2n +� +����� +→ τ−4 1 +2 +� +����� +1 +1 +1 +1 +1 +1 +−1 +−1 +1 +−1 +i −n +−i −n +1 +−1 +−i −n +i −n +� +����� +� +����� +O2n +V2n +S2n +C2n +� +����� +. +(4.6) +We can discuss all these theories at once by noticing that their partition functions can be com- +pactly written as +ZA = +3� +a,b=0 +¯R A +a Nab Lb , +(4.7) +where L = (O8,V8,S8,C8) denotes the left-moving characters, which are common to all ten-dimen- +sional non-supersymmetric theories, and ¯R A +a denotes the right-moving characters, which depend +on the specific model A = 16,0,32, and can be extracted from eqs. (4.1), (4.2) and (4.3), respectively. +The corresponding GSO matrices read +N0A = +� +����� +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +1 +0 +� +����� +, +N0B = +� +����� +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +� +����� +, +(4.8) +11 + +for the type 0A and type 0B theories, while +Nhet = +� +����� +0 +1 +0 +0 +1 +0 +0 +0 +0 +0 +−1 +0 +0 +0 +0 +−1 +� +����� +, +(4.9) +for the heterotic theories. +The universal holomorphic characters La have Ha = (− 1 +2,0,0,0), and using the explicit form +(4.5) and (4.6) of the T and S modular matrices, one finds +Φa(n;{wℓ}) = +1 +2 +15 +4 n +11 +4 +� +e4π +� +n/2 − +� +2δa0 (−1)w2 e2π +� +n/2 ++ 2 +� +3 cos +� +2 +3π(w3 +(−1)δa0) +� +e +4π +3 +� +n/2 +... +� +, +(4.10) +for the asymptotic behaviour of the degeneracies, where the wℓ take integer values in the interval +0,...,ℓ − 1. Notice that L0 = O8 is the unique tachyonic character, associated to the NS vacuum, +which reflects the presence of a single leading exponential growth e4π +� +n/2. +As for the anti-holomorphic characters ¯R A +a , they carry weights +¯Ha = (−1, 1 +2,0,0) +for the SO(16)×SO(16) heterotic theory +¯Ha = (− 1 +2,0,0,0) +for the 0A and 0B theories +¯Ha = (−1,− 1 +2,1,1) +for the SO(32) heterotic theory +(4.11) +and using the explicit expression for the T and S modular matrices, which can be read/derived +from eqs. (4.5) and (4.6), one finds +¯Φ(16) +a +(n;{ ¯wl}) = +1 +2 +3 +2 n +11 +4 +� +e4π�n + +� +2δa1 (−1) ¯w2 e2π�n ++ 2 +� +3 cos +� 2 +3π( ¯w3 +δa0) +� +e +4π +3 +�n +... +� +, +(4.12) +for the SO(16)×SO(16) theory, +¯Φ(32) +a +(n,{ ¯wℓ}) = +1 +2 +3 +2 n +11 +4 +� +e4π�n +2 +11 +4 (−1)δa2+δa3 e4π +� +n/2 ++ +� +2δa1 (−1) ¯w2 e2π�n −2 +13 +4 δa0 (−1) ¯w2 e2π +� +n/2 ++ 2 +� +3 cos +� 2 +3π( ¯w3 −δa1) +� +e +4π +3 +�n +− 2 +15 +4 +� +3 (−1)δa1 cos +� 2 +3π( ¯w3 +1−δa1) +� +e +4π +3 +� +n/2 +... +� +, +(4.13) +for the SO(32) theory, while ¯Φ(0) +a (n,{ ¯wℓ}) = Φa(n,{ ¯wℓ}) for the 0A and 0B theories. Notice, that in +¯Φ(32) +a +(n,{ ¯wℓ}) there are two leading exponential growths e4π�n and e4π +� +n/2 associated, respectively, +to the two tachyonic characters ¯R0 and ¯R1, with ¯H0 = −1 and ¯H1 = − 1 +2. +12 + +The sector averaged sum is then given by +〈d(n)〉A = +3� +a,b=0 +� +{wℓ} +Nab ¯Φ(A) +a (n,{wℓ + Hb − ¯Ha})Φb(n,{wℓ}), +(4.14) +where the choice ¯wℓ = wℓ + Hb − ¯Ha follows from the level matching condition, and the sum is +extended over all integers wℓ = 0,...,ℓ−1, for each ℓ = 2,3,.... It is straightforward to see that the +growth of 〈d(n)〉0 ∼ e4π +� +2n for the type 0A and 0B theories is dictated by the total central charge +Ctot = 4π +� +2, while for the heterotic models the leading term e4π(1+1/ +� +2)�n cancels upon summing +over the sectors a,b = 0,...,3. This reflects the fact that the type 0 theories only have bosonic excita- +tions while the heterotic ones have both bosons and fermions in their spectrum, and � +ab Nab = 0. +Moreover, while for the tachyonic SO(32) theory +〈d(n)〉32 = +1 +� +2n +11 +2 +� +e4π +� +2n +e2π +� +2n +2e +2π +3 +� +2n +... +� +, +(4.15) +and Ceff = 4π +� +2 < Ctot, for the non-tachyonic SO(16)×SO(16) model Ceff = 0 since all exponential +growths cancel leaving, at most, a power-law behaviour. This is in agreement with our general +result given in the previous Section and with the result of [24] for the SO(16)×SO(16) theory. +5 +Scherk-Schwarz Reductions at Rational Points +A more interesting class of non-supersymmetric vacua can be constructed in lower dimensions by +employing the Scherk-Schwarz mechanism [31, 32]. This can be conveniently realised as a freely +acting orbifold [33], where the action of a supersymmetry breaking generator g is combined with a +suitable shift δ along compact directions. In its simplest nine-dimensional incarnation g = (−1)F, +with F the space-time fermion number, while δ acts as y → y + πR on the compact coordinate y +parametrising the circle S1(R) of radius R. In the heterotic case, one can consider more general +orbifolds where the space-time fermion number is combined with an action on the gauge degrees +of freedom, in accordance with modular invariance. In this Section, however, we shall consider +the Scherk-Schwarz reduction of the type IIB superstring since it shares the main features with any +generic construction, with the advantage of being quite simple. The torus partition function +Z = 1 +2|V8 −S8|2 � +m,n +Λm,n(R)+ 1 +2|V8 +S8|2 � +m,n +(−1)m Λm,n(R) ++ 1 +2|O8 −C8|2 � +m,n +Λm,n+ 1 +2 (R)+ 1 +2|O8 +C8|2 � +m,n +(−1)m Λm,n+ 1 +2 (R), +(5.1) +clearly exhibits in the first line the action of the (−1)F δ generator on the original spectrum of the +IIB superstring, while the second line, involving the flipped GSO projection, is required by mod- +ular invariance. The Kaluza-Klein momenta and windings associated to the compact direction +contribute with the standard Narain lattice +Λm,n = q +α′ +4 +� +m +R + nR +α′ +�2 +η +¯q +α′ +4 +� +m +R − nR +α′ +�2 +¯η +. +(5.2) +13 + +In the decompactification limit, R → ∞, the orbifold action is trivialised and one recovers the su- +persymmetric IIB theory, while for generic values of the radius R supersymmetry is spontaneously +broken and the gravitini acquire a mass m ≃ 1/R. The excitations of the NS-NS vacuum in |O8|2 +now survive the GSO projection in the twisted sector, and the lightest state has mass +m2 +|O8|2 = − 1 +2α′ + 1 +4 +� R +2α′ +�2 +. +(5.3) +This scalar is then massive for large values of R, but turns tachyonic below the critical radius Rc = +2 +� +2α′. As we decrease the radius, more and more states become tachyonic, and it is then clear +that these models represent an ideal ground to study the realisation of misaligned supersymmetry +in string theory. +To illustrate the analysis of Section 3 on the degeneracies of states, we need to select rational +values for R2/α′ = s/t ∈ Q since, in this case, the Narain lattice reduces to an RCFT +� +m,n +Λm,n(R) → +2st−1 +� +α=0 +λα ¯λαl , +(5.4) +with the 2st characters defined as +λα(q) = +� +m +qst +� +m+ α +2st +�2 +η(q) +. +(5.5) +Notice, that λ0 and λst are real, while λα and λ2st−α, α = 1,...,st − 1, form conjugate pairs. The +λ’s have conformal weight hα = α2/4st, and thus Hα = hα − 1 +24, for α = 0,...,st, with conjugate +pairs carrying the same weight. In eq. (5.4) the anti-holomorphic characters have index αl, where +l = r t + sv, with the integers r and v satisfying the relation r t − vs = 1, and the label αl is defined +modulo 2st. As shown in the Appendix, these characters are eigenstates of the shift operator δ only +for even s, but must be broken into sub-characters for odd s. For simplicity, here we shall restrict +the discussion to the even-s case, where +δ : +λα → (−1)α/2t λα , +(5.6) +and we also take t = 1, so that the condition r t − vs = 1 can be easily solved by r = 1 and v = 0, for +any s. Other choices for s and t yield equivalent results. The action of the modular group on these +characters is encoded in the T and S matrices +Tαβ = eiπ +� +α2 +2s − 1 +12 +� +δαβ , +Sαβ = e2πi αβ +2s +� +2s +. +(5.7) +14 + +Taking all this into account, the torus partition function becomes +Z = +s−1 +� +a=0 +� +|χ2a+2s|2 +|χ2a+4s|2� +− +s−1 +� +a=0 +� +χ2a+1+2s ¯χ2a+1+4s +χ2a+1+4s ¯χ2a+1+2s +� ++ +s +2 −1 +� +a=0 +� +χ2a+σ ¯χ2a+σ+s +χ2a+σ+s ¯χ2a+σ ++χ2a+σ+6s ¯χ2a+σ+7s +χ2a+σ+7s ¯χ2a+σ+6s +� +− +s +2 −1 +� +a=0 +� +χ2a+1−σ ¯χ2a+1−σ+7s +χ2a+1−σ+7s ¯χ2a+1−σ ++χ2a+1−σ+s ¯χ2a+1−σ+6s +χ2a+1−σ+6s ¯χ2a+1−σ+s +� +, +(5.8) +where in the third and fourth sums one has to distinguish the two cases s = 2(2m +σ) with σ = 0,1 +while, as dictated by spin-statistics, the minus signs reflect the presence of space-time fermions. +The new characters are +{χa}8s−1 +a=0 = (O8,V8,S8,C8)⊗{λα}2s−1 +α=0 . +(5.9) +The characters χa and χ2s−a have shifted conformal weight Ha = a2 +4s − 1 +2, and therefore are +tachyonic for a < +� +2s. However, since they appear in the partition function in the combination +χ2a+σ ¯χ2a+σ+s +χ2a+σ+s ¯χ2a+σ +(5.10) +the only states which are level-matched are +χs/2 ¯χ3s/2 +χ3s/2 ¯χs/2 , +(5.11) +which are tachyonic for s < 8. This agrees with the result of eq. (5.3) valid at irrational values of R, +since now R2 = s α′. +The effective degrees of freedom at a given mass level can be straightforwardly extracted from +the partition function (5.8) by Taylor expanding the various characters. We find that misaligned +supersymmetry is present for any choice of the compactification radius, even in the tachyonic +regime, as shown in Figure 5.1. This is in accordance with our general discussion of Section 3 and +is corroborated by the large-n behaviour of the sector averaged sum 〈d(n)〉. +Indeed, the degeneracy of each character χa has in principle many exponential growth rates +associated to all tachyonic characters of the RCFT. However, the leading exponential, associated +to χ0, is universal and cancels in the sector averaged sum since the partition function involves 4s +bosonic and fermionic sectors. Therefore, we can conclude that, for any s, Ceff < Ctot. Moreover, +for s > 8 no physical tachyons are present in the spectrum and thus Ceff = 0. Finally, in the tachy- +onic region s < 8, the state which dictates the exponential growth of 〈d(n)〉 is the physical tachyon +15 + +5 +10 +15 +-20 +-10 +10 +20 +5 +10 +15 +-20 +-10 +10 +20 +Figure 5.1: The signed logarithm of the net number of degrees of freedom at each mass level for the +type IIB Scherk-Schwarz reduction. Positive (negative) contributions are ascribed to the excess of +bosonic (fermionic) states. The left figure refers to R2 = 2α′, and despite the presence of tachyons, +the spectrum exhibits misaligned supersymmetry. The right figure corresponds to R2 = 8α′, within +the non-tachyonic region. The step-like shape of the enveloping functions reflects the fact that the +first few Kaluza-Klein excitations have masses smaller than the string scale. +χs/2 ¯χ3s/2 and its conjugate, so that +Ceff = 2π +� +8− s . +(5.12) +Indeed, explicit calculations yield +〈d(n)〉 = +81 +4096n5 +� +e2π +� +6n +eπ +� +6n +2e2π +� +2n/3 +... +� +, +(5.13) +for s = 2, +〈d(n)〉 = +1 +256n5 +� +e4π�n +e2π�n +2e +4 +3 π�n +... +� +, +(5.14) +for s = 4, while 〈d(n)〉 = 0 for s > 8. +It is tempting to continue the behaviour (5.12) to arbitrary irrational values of the compactifi- +cation radius, so that the asymptotic growth of the irrational sector averaged sum reads +Ceff = +� +� +� +2π +� +8−R2/α′ +for +R2 < 8α′ , +0 +for +R2 ≥ 8α′ , +(5.15) +as shown in Figure 5.2, and Ceff → Ctot as R → 0, in accordance with the fact that, in this limit, one +recovers the purely bosonic type 0B theory. +5.1 +A Comment on Phase Transitions +It is tempting to consider the partition function Z of the world-sheet CFT as a function of q = +e−β+iµ, where we now interpret β as the world-sheet inverse temperature and µ as the spin po- +tential, conjugate to the worldsheet momentum operator. Clearly, the sum over states in Z is ab- +solutely convergent, and the corresponding world-sheet free energy F = −(1/β)logZ is analytic, +signifying the absence of phase transitions, as in any theory with a finite number of local degrees +16 + +4 +6 +8 +10 +12 +14 +5 +10 +15 +Figure 5.2: The figure displays the dependence of the effective central charge on R and shows that +below the critical radius the Scherk-Schwarz reduction is not a deformation of the original theory. +of freedom. For instance, in bosonic string theory the density of states grows universally according +to the Cardy formula +d(n) ∼ e4π +� +cn/24 , +(5.16) +so that the level matched partition function +�1 +0 +dτ1Z (τ1,β/2π) = +� +n +d(n)e−2βn , +(5.17) +converges for all values of β. +In light of our discussion on Ceff, it is interesting to construct a deformed version of the above +partition function, obtained by analytically continuing the integer n to the reals and by averaging +the net degrees of freedom in terms of the enveloping functions Φ(n). This amounts to replacing +the physical net degeneracies d(n) by their sector averaged versions 〈d(n)〉, and defining a sector +averaged partition function for the world-sheet CFT as +〈Z 〉 = +� +n +〈d(n)〉e−2βn . +(5.18) +This deformation effectively introduces an infinite number of degrees of freedom by averaging +the interpolation of the mass levels in terms of the enveloping functions Φ. In doing so, one can +estimate +〈Z 〉 ∼ +� +dn eCeff +�n−2βn ∼ e +C2 +eff +8β , +(5.19) +so that the sector averaged free energy reads +〈F〉 ∼ − +C 2 +eff +8β2 , +(5.20) +and is controlled by the square of the effective central charge. In the case of Scherk-Schwarz su- +persymmetry breaking, Ceff is a function of the compactification radius R, as shown in eq. (5.15), +17 + +which, from the point of view of the worldsheet CFT, should be treated as an external background +field. In this sense, the sector averaged free energy 〈F〉 displays a first order phase transition as +the radius crosses the critical value Rc, since its first derivative is discontinuous. +This would suggest a possible interpretation in terms of phase transitions in a suitable holo- +graphic dual. In fact, it is known that two-dimensional CFT’s may admit an AdS3 gravitational +description, and in certain deformations of symmetric orbifold CFTs, in the large central charge +limit, Hagedorn-like phase transitions of the CFT are mapped to Hawking-Page transitions of the +gravity theory, dominated by the entropy of BTZ black holes [34]. It is thus tempting to interpret +the phase transition displayed by our system in terms of a holographic dual where a similar kind of +averaging procedure is introduced. However, this investigation lies beyond the scope of this work. +6 +Conclusions +In this paper we have revisited the problem of classical stability of non-supersymmetric string +vacua in various dimensions in relation to the growth rate of the net number of degrees of freedom. +Contrary to common belief, we find that misaligned supersymmetry is not an exclusive property of +non-tachyonic strings, but manifests itself in all theories, tachyonic or not, containing space-time +fermions. In fact, we show that the growth rate of the sector averaged sum 〈d(n)〉 is set by the mass +of the lightest state, whether it be tachyonic or massless, and is strictly smaller than Ctot in the +presence of fermions. We also prove that the necessary and sufficient condition for the tree-level +stability of the vacuum is an at most power-like growth of 〈d(n)〉 corresponding to vanishing Ceff, +as conjectured in [23]. Our result is model independent and applies to any closed-string vacuum +in any dimension, and agrees with the recent analysis of [24] conducted for the SO(16) × SO(16) +heterotic string. +Following [23], our analysis is based on the sector average of the string degrees of freedom +whose discrete masses are analytically continued to real values. The cancellations required by mis- +aligned supersymmetry highly depend on this analytic continuation and not just on the properties +of the discrete spectrum. This is to be contrasted to the works of [18] and [22] which, using num- +ber theoretic methods, relate misaligned supersymmetry to the real physical degrees of freedom +of the vacuum, without having to resort to an analytic continuation of the mass levels. It would be +interesting to obtain a more direct connection between these two a priori different approaches. +Any quantitative analysis on misaligned supersymmetry heavily relies on modular invariance +of the torus partition function, and can thus only be formulated for closed oriented strings. Still +orientifold vacua [35–38,40] provide an appealing phenomenology and afford more general ways +to break supersymmetry [41–51] which cannot be realised in closed strings. Although some at- +tempts have been made to understand the role played by misaligned supersymmetry in the classi- +cal stability of non-supersymmetric orientifolds [24,52–54], a thorough quantitative analysis is still +lacking. Modular transformations relate in them the direct and transverse channels, so that there +is no obvious link between the IR and the UV properties of the spectrum, and therefore different +tools need to be employed to uncover this connection. We hope to return to this problem in the +near future. +18 + +Acknowledgements +It is a pleasure to thank Ivano Basile, Flavio Tonioni and in particular Niccolò Cribiori for enlight- +ening discussions on misaligned supersymmetry and non-supersymmetric string vacua. We are +grateful to Augusto Sagnotti for constructive feedback on the manuscript. I.F. would like to thank +the Physics Department of the University of Torino for hospitality during the final stages of this +project. The work of C.A. is partially supported by the MIUR-PRIN contract 2017CC72MK-003. +A +The Scherk-Schwarz Mechanism in RCFT’s +Although the Narain partition function for a real boson Y compactified on a circle S1(R) of ra- +dius R does not fully factorise into the product of holomorphic and anti-holomorphic contribu- +tions, things simplify considerably whenever R2/α′ = s/t ∈ Q takes rational values. In this case, the +CFT of the compact boson becomes rational, the Kaluza-Klein momenta and windings admit the +parametrisation +m = s(k + ¯k)+ 1+l +2t α, +n = t(k − ¯k)+ 1−l +2s α, +(A.1) +with l defined after eq. (5.5) and k, ¯k ∈ Z, and only a finite number N = 2st of representations are +unitary, and are associated to the characters +λα = 1 +η +� +k∈Z +q +N +2 (k+ α +N )2. +(A.2) +Although the λ’s provide a natural decomposition of the Narain partition function (5.2) as the +sesquilinear combination (5.4), in general they do not provide a suitable basis when shift orbifolds +act on S1(R). In fact, already for the simple order-two shift δ : Y → Y +πR, the Narain lattice picks +up a phase +� +m,n +Λm,n → +� +m,n +(−1)m Λm,n +(A.3) +and in view of (A.1), it is clear that the characters λα are eigenstates of δ only for even s so that +(−1)m → (−1) +1+l +2t α. For odd s, instead, the k-th excitation in (A.2) acquires an additional sign de- +pending on the parity of k, that would suggest the decomposition +λα → ξi +α = 1 +η +� +k∈Z +q +N +2 +� +2k+i+ α +N +�2 +, +i = 0,1. +(A.4) +The ξ’s however, are not closed under the action of SL(2,Z), since they fail to capture the twisted +sector with half-integer windings. Therefore, the correct choice of the δ eigenstates is +ζα = 1 +η +� +k +q2N +� +k+ α +4N +�2 +(A.5) +with now α = 0,...,4N −1. +19 + +One can thus decompose the various orbifold blocks as +� +m,n +Λm,n = +� +� +� +�N−1 +α=0 λα ¯λlα +for even s , +�1 +a,b=0 +�N−1 +α=0 ζ2(α+aN) ¯ζ2(lα+bN) +for odd s , +� +m,n +(−1)m Λm,n = +� +� +� +�N−1 +α=0 (−1) +1+l +2t α λα ¯λlα +for even s , +�1 +a,b=0 +�N−1 +α=0 (−1)a+b+ 1+l +2t α ζ2(α+aN) ¯ζ2(lα+bN) +for odd s , +� +m,n +Λm,n+ 1 +2 = +� +� +� +�N−1 +α=0 λlα−(1+l)s/2 ¯λα +for even s , +�1 +c=0 +�2N−1 +α=0 ζl(2α+1)−(1+l)s+2Nc ¯ζ2α+1 +for odd s , +� +m,n +(−1)m Λm,n+ 1 +2 = +� +� +� +�N−1 +α=0 (−1) +(lα− 1+l +2 +s)2−α2 +N +λlα−(1+l)s/2 ¯λα +for even s , +�1 +c=0 +�2N−1 +α=0 (−1) +(l(2α+1)−(1+l)s+2Nc)2−(2α+1)2 +4N +ζl(2α+1)−(1+l)s+2Nc ¯ζ2α+1 +for odd s . +(A.6) +Using the representation of the SL(2,Z) generators +Tαβ = eiπ +� +α2 +M − 1 +12 +� +, +Sαβ = e +2πiαβ +M +� +M +, +(A.7) +on the space of characters, with M = N for the λ’s, and M = 4N for the ζ’s, it is straightforward to +show that the relations +� +m,n +(−1)m Λm,n +S +−→ +� +m,n +Λm,n+ 1 +2 +T +−→ +� +m,n +(−1)m Λm,n+ 1 +2 +(A.8) +hold both for odd and even s. +20 + +References +[1] E. 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Wrase, Modular invariance, misalignment and +finiteness in non-supersymmetric strings, JHEP 01 (2022), 127 [arXiv:2110.11973 [hep-th]]. +24 + diff --git a/aNFST4oBgHgl3EQfBTj3/content/tmp_files/load_file.txt b/aNFST4oBgHgl3EQfBTj3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e58353a718e0ec5d56f94db67b0d5fe2ec789c30 --- /dev/null +++ b/aNFST4oBgHgl3EQfBTj3/content/tmp_files/load_file.txt @@ -0,0 +1,749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf,len=748 +page_content='Tachyons and Misaligned Supersymmetry in Closed String Vacua Carlo Angelantonj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Ioannis Florakis2 and Giorgio Leone1 1 Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Università di Torino and INFN Sezione di Torino Via Pietro Giuria 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' I-10125 Torino 2 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' University of Ioannina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' GR-45110,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Ioannina ABSTRACT In a remarkable paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Dienes discovered that the absence of physical tachyons in closed string theory is intimately related to oscillations in the net number of bosonic minus fermionic degrees of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' a pattern known as misaligned supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The average of these oscillations was linked to an exponential growth controlled by an effective central charge Ceff smaller than the expected inverse Hagedorn temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Dienes also conjectured that Ceff should vanish when tachyons are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In this paper, we revisit this problem and show that misaligned supersymmetry is actually re- alised even when tachyons are present in the physical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, we prove that the average growth rate Ceff is set by the mass of the lightest state, be it massless or tachyonic, and coincides with the inverse Hagedorn temperature only when fermions are absent from the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We also provide a general proof that the necessary and sufficient condition for classical stability is Ceff = 0, in agreement with Dienes’ conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' E-mail: carlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='angelantonj@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='it iflorakis@uoi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='gr giorgio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='leone@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='it arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13702v1 [hep-th] 31 Jan 2023 Contents 1 Introduction 1 2 Asymptotic Mass Degeneracies 5 3 A Proof of Misaligned Supersymmetry 7 4 Non-Supersymmetric Heterotic Vacua in D = 10 10 5 Scherk-Schwarz Reductions at Rational Points 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1 A Comment on Phase Transitions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 16 6 Conclusions 18 A The Scherk-Schwarz Mechanism in RCFT’s 19 1 Introduction Superstring vacua are typically unstable when space-time supersymmetry is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The instabil- ity can be ascribed to the emergence of tadpoles for scalar fields or, more explicitly, to the pres- ence of tachyons in the classical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In the first case, much progress has been made in the last years in the determination of the corrected geometry which can describe, for instance, spon- taneous compactification or cosmological evolution [1–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The second case, instead, is much less under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, although the condensation of open-string tachyons is well under- stood [16, 17], very little is known about the rolling of closed-string ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, in critical strings tadpoles typically emerge at higher genus and, as such, induce quantum corrections to the classical moduli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In this case, perturbation theory is relatively under control at least up to the order associated to the tadpole itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' On the contrary, tree-level tachyons imply that the string vacuum cannot be trusted even classically, and it is impossible to extract any meaningful informa- tion from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Therefore, if one insists that a non-supersymmetric vacuum ought to be predictive, at least classically, one is bound to consider only string configurations where tachyonic excitations are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This turns out to be rather non-trivial to achieve and requires a precise correlation in the growth rates of massive bosonic and fermionic string states, since these two features are strictly related by modular invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' A first study of the interplay between massive states and tachyons was done in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' There it was shown that the infra-red finiteness of the one-loop vacuum energy of closed oriented strings implies the overall cancellation lim Λ→∞ � n (dB(n)−dF(n)) e−4πn/Λ2 ≡ lim Λ→∞ � n d(n)e−4πn/Λ2 = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) between the total bosonic dB(n) and fermionic dF(n) degrees of freedom, at mass-squared n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This property goes under the name asymptotic supersymmetry, since it is reminiscent of the Bose-Fermi degeneracy one typically encounters when supersymmetry is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 1 A more refined analysis based on the Rankin-Selberg-Zagier transform [19–21] done in [22], actually showed that, when physical tachyons are absent and thus the one-loop vacuum energy Ω is finite, the distribution of bosonic and fermionic degrees of freedom follows a precise pattern dictated by � n d(n)e−4πn/Λ2 ≃ Λ2−D 3Ω π + � ζ∗(ρ)=0 Cρ Λρ1−D cos � ρ2 logΛ+ϕρ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) where the sum in the RHS runs over the zeroes ρ = ρ1 +iρ2 of the dressed Riemann zeta-function, ζ∗(s) = π−s/2 Γ(s/2)ζ(s), and Cρ and ϕρ are model-dependent real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Indeed, this expres- sion implies that not only must the full string spectrum in D > 2 non-compact space-time dimen- sions enjoy asymptotic supersymmetry but, for finite large Λ the excess of bosonic and fermionic degrees of freedom must alternate with a frequency dictated by the non-trivial zeroes of the Rie- mann zeta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This oscillatory behaviour of a classically stable string spectrum was actually discovered already in [23], where it was named misaligned supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Notice that the analysis of [18] and [22] crucially rely on the finiteness of the one-loop modular integral, and thus the results (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) are necessary conditions for the tree-level stability of a string vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Whether these conditions are also sufficient is an open problem which cannot be addressed using the techniques in [18,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 5 10 15 20 25 40 20 20 40 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1: The signed logarithm of the net number of degrees of freedom at each mass level for the non-tachyonic SO(16)×SO(16) heterotic string in D = 10 highlighting the presence of misaligned supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Positive (negative) contributions are ascribed to the excess of bosonic (fermionic) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' A somehow different route to relate the distribution of degrees of freedom to the classical sta- bility of closed-string vacua was followed by Dienes in the remarkable paper [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' There, he ob- served that the degrees of freedom of non-tachyonic, non-supersymmetric vacua follow a pre- cise pattern as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1, where the net excess between bosons and fermions alternates as the mass-level increases1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' He then observed that the presence of oscillations can be traced back to the growth property of a special quantity, termed the sector averaged sum, to be introduced shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 1Clearly, in the case of unbroken supersymmetry there is an exact degeneracy between bosonic and fermionic states and, therefore, no oscillations are observed in the full spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 2 In a given string background, the partition function admits the double power-series expansion Z = � n,m � a,b Nab dab(n,m)qn ¯qm , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) with dab(n,m) counting the net number of bosonic minus fermionic states with left mass-squared n and right mass-squared m in the sector ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The matrix Nab enforces the GSO projection and builds the string vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Propagating degrees of freedom in the ab sector are selected by the level matching condition which identifies the left-moving and right-moving masses, and are then counted by the integral functions dab(n) ≡ dab(n,n), whose domain is a countable set associated to the string oscillators, Kaluza-Klein momenta and/or windings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Two-dimensional conformal in- variance and modularity imply a universal exponential growth of dab(n), dab(n) ∼ eCtot �n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) controlled by the total central charge Ctot = 4π(�cL/24 + �cR/24), which also defines the inverse Hagedorn temperature2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This growth is then inherited by the total net number of physical states d(n) = � a,b Nab dab(n), obtained by summing over all sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The connection between the asymptotic growth of states and the absence of tachyons cannot be formulated in terms of the physical d(n), but requires the introduction of the continuous en- veloping functions Φab(n), which reproduce the dab(n) for the special values of n associated to the actual masses [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Assuming that such functions exist, Dienes introduced the sector averaged sum 〈d(n)〉 = � a,b Nab Φab(n), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) which discriminates between tachyonic and non-tachyonic vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, 〈d(n)〉 is no-longer bound to exhibit the universal exponential growth (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4), controlled by the total central charge, but Dienes showed [23] that in a classically stable vacuum 〈d(n)〉 ∼ eCeff �n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) with Ceff < Ctot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The cancellation underlying this slower growth clearly implies the oscillatory be- haviour of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1 observed in the string spectrum and it was claimed in [23] to be a necessary and sufficient condition for the IR finiteness of String Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' There it was also conjectured that the absence of physical tachyons actually implies Ceff = 0, a conjecture which was later shown to be true [24] in the case of the ten-dimensional SO(16)×SO(16) heterotic string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Simple numerical experiments on non-supersymmetric string vacua, actually reveal a slightly more complicated story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, we find that the oscillatory behaviour of the net number of physi- cal states is not an exclusive prerogative of classically stable vacua, but is also present in all tachy- onic string theories, in ten and lower dimensions, which have both bosonic and fermionic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 2Of course, the total central charge of the full CFT is cL +cR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' However, by abuse of language, we shall refer to Ctot as the total central charge, as in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 3 5 10 15 20 25 40 20 20 40 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2: The signed logarithm of the net number of degrees of freedom at each mass level for the tachyonic SO(16)×E8 heterotic string in D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The plots exhibits misaligned supersymmetry even in the presence of physical tachyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Positive (negative) contributions are ascribed to the excess of bosonic (fermionic) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' For instance, in the case of the ten-dimensional non-supersymmetric heterotic string with gauge group SO(16)×E8 the signed multiplicities follow the oscillatory pattern shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This is so despite the fact that the low-lying spectrum contains a physical tachyon of mass-squared m2 T = −2/α′ in the representation (16,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, the exponential growth of the sector averaged sum is controlled by Ceff < Ctot, thus providing a notable counter-example to the argument given in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, in the present paper we will prove, independently of the details of the string con- struction, that Ctot only sets the growth rate in purely bosonic theories, like the bosonic string and the type 0A/0B superstrings while, whenever fermions are present in the spectrum, the growth rate is slower than Ctot, and is actually set by the mass of the lightest states, whether tachyonic or massless, Ceff = 4π � |α′m2 lightest| < Ctot .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) Importantly, our result implies that the necessary and sufficient condition for classical stability is an at most polynomial growth of the sector averaged sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Ceff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This proves the conjecture of Dienes for any critical closed-string vacuum, regardless of its CFT realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In Section 2, we review the asymptotic properties of the de- generacies of states in Rational CFT’s (RCFT’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Section 3 deals with the analysis of the sector aver- aged sum and contains the main result of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In Sections 4 and 5, we illustrate our results by studying concrete examples in ten and lower dimensions, and we conjecture the presence of first- order phase transitions on the world-sheet of an averaged CFT associated to the Scherk-Schwarz reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Our conclusions and perspectives are gathered in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Finally, the Appendix con- tains the relevant character decomposition of shift orbifolds of one-dimensional Narain lattices at rational points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 4 2 Asymptotic Mass Degeneracies As anticipated, in order to uncover the origin of misaligned supersymmetry and of the absence of physical tachyons, one needs to study the large-mass behaviour of the string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' For simplicity, we assume that the world-sheet fermions and the compact bosons contribute with a finite number M of characters, both in the holomorphic and anti-holomorphic sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In princi- ple, these sets of characters need not be the same, as for instance, in the heterotic string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Still, for convenience, we shall use the same symbol χ to label the contributions χa from the holomorphic sector and ¯χa from the anti-holomorphic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The D non-compact bosons contribute instead with the Dedekind η-function, so that the one-loop vacuum energy reads � F dµZ = � F dµ 1 (�τ2 ¯ηη)D−2 � a,b ¯χaNabχb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) where τ = τ1+iτ2 is the complex structure modulus of the world-sheet torus, F is the fundamental domain of SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='Z) and dµ = τ−2 2 dτ1 dτ2 is the modular invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Again, Nab is a matrix enforcing the GSO projection and is subject to the conditions N = T † R N TL , N = S† RN SL , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) where TL,R and SL,R represent the action of the generators of SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='Z) on the characters χa and ¯χa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Notice that, since any element of the modular group can be decomposed in terms of T and S, the previous conditions also imply N = M† RN ML , ∀ML,R ∈ SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) In string theory, the characters χa of the RCFT always come together with the contribution of the non-compact bosons, and it is natural to define the dressed pseudo-characters ˆχa = η2−D χa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Strictly speaking, these are no longer characters of a RCFT, since they carry the non-trivial modular weight 1−D/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As a result, the generators of the modular group involve extra phases, and act as T : ˆχa → ˆTab ˆχb = e−2πi(D−2)/24 Tabχb ηD−2 , S : ˆχa → τ1−D/2 ˆSab ˆχb = τ1−D/2 i D/2−1 Sabχb ηD−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) The pseudo-characters ˆχa, from now on simply referred to as the characters χa with the hat omitted, admit a power series expansion in the nome q = e2πiτ of the form χa(q) = ∞ � n=0 da(n)qHa+n , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) with Ha = ha − c/24 expressed in terms of the conformal weight ha and the central charge c of the full theory, including the D − 2 non-compact bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As a result, the total net number of the 5 physical degrees of freedom at mass level n reads d(n) = � a,b Nab ¯da(n + Hb − ¯Ha)db(n), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) where we have imposed the level-matching condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The coefficients da(n), counting the degeneracies of states of χa, can be computed using Cauchy theorem, da(n) = 1 2πi � Γ dq χa(q) qna+1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) where na = n + Ha, and Γ is any closed contour lying entirely inside the unit disk in the complex q-plane and containing the origin q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The large-n behaviour of this integral can be extracted using the circle method of Hardy and Ramanujan [25] and takes the form [26] da(n) = M−1 � b=0 O([�n]) � ℓ=1 Q(ℓ,n) ab fb(ℓ,na).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) The matrix Q(ℓ,n) ab = i 1−D/2 ℓ−1 � p=0 gcd(ℓ,p)=1 � M−1 ℓ,p � ab e− 2πi ℓ (pna−p′Hb) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='9) is expressed in terms of the representation Mℓ,p of the modular transformation γℓ,p = � −p′ 1+pp′ ℓ −ℓ p � γℓ,p ∈ SL(2,Z) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='10) on the space of characters, χa → (−ℓτ+ p)1−D/2 (Mℓ,p)abχb , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='11) while fb(ℓ,na) = 2π ℓ db(0) � Hb na �D/4 J−D/2 �4π ℓ � Hb na � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='12) with Jk(x) being the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Notice that the Q(ℓ,n) ab are real functions, as follows from the conjugation property of the modular transformations [26], in accordance with the fact that da(n) count physical states and are thus real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The large-n behaviour of the degeneracies da(n) is clearly dictated by the fb(ℓ,na) functions, and receive contributions from all the characters χb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' These are power-like if Hb ≥ 0 and expo- nential whenever Hb < 0, as can be seen from the asymptotic expansions of the Bessel functions 6 Jα(x) ∼ � 2 πx � cos � x − πα 2 − π 4 � − 4α2 −1 8x sin � x − πα 2 − π 4 � +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13) and Jα(ix) ≡ i αIα(x) ∼ i α ex � 2πx � 1− 4α2 −1 8x +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='14) valid for positive x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' It is therefore clear that the asymptotic growth of the degeneracies is domi- nated by the tachyonic characters, which are always present in the CFT of any string construction, although they may be eliminated from the physical spectrum by the GSO projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Schemati- cally, we have da(n) ∼ � b |Hb<0 db(0) � 2 |Hb|(D−1)/4 n(D+1)/4 � Q(1,n) ab e4π�|Hb|n + Q(2,n) ab � 2 e2π�|Hb|n + Q(3,n) ab � 3 e 4π 3 �|Hb|n +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='15) with Q(1,n) ab = i 1−D/2 Sab simply determined by the S matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The tachyon with the smallest confor- mal weight, included in χ0, is associated to the ubiquitous NS vacuum and plays the role of the identity in the RCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As a result, it dictates the leading growth da(n) ∼ e4π � cn/24 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='16) in agreement with the Cardy formula [27], and this contribution is universal since Sa0 = i D/2−1/ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The sub-leading exponentials depend instead on the data of the RCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 3 A Proof of Misaligned Supersymmetry We have now all the ingredients to connect the large-n behaviour of d(n) with the classical stability of a string vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' To this end, following [23], we continue the integer n to the reals, introduce the enveloping functions da(n) → Φa(n), and construct the sector averaged sum 〈d(n)〉 = � a,b Nab ¯Φa(n + Hb − ¯Ha)Φb(n), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) whose asymptotic behaviour dictates the presence/absence of physical tachyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' From eqs (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='16) it follows that the SL(2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='C) invariant vacuum in principle determines the leading large-n behaviour, 〈d(n)〉 ∼ e4π�n( �cL/24+�cR/24) � a,b Nab = eCtot �n � a,b Nab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) However, whether the asymptotic growth is controlled by Ctot or not depends entirely on the prop- erties of N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Recall that the GSO matrix must yield a modular invariant partition function, and is 7 thus subject to the conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As a result, N00 = 1 M � a,b Nab (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) since, for a RCFT, where all M characters have been resolved, Sa0 = i D−2−1/ � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, the entries Nab can only be ±1 or zero, which yield to the following two scenarios: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' If the spectrum does contain the leading tachyon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' N00 = 1, one has � a,b Nab ̸= 0 and 〈d(n)〉 ∼ eCtot �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' If the spectrum does not contain the leading tachyon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' N00 = 0, then � a,b Nab = 0 and 〈d(n)〉 ∼ eCeff �n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) with an effective central charge Ceff < Ctot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The condition N00 = 0 clearly requires that fermions are present in the spectrum and, in particular the number of bosonic sectors must equal the number of fermionic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' On the contrary, in the absence of extended symmetries, namely when each character appears only once in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1), the condition N00 = 1 implies that the spectrum only contains bosonic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' These observations were already contained in [23] and, in fact, represent the main result of Dienes’ paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' However, it is important to stress that, contrary to the result of [23], Ceff < Ctot does not imply classical stability since, although it is true that the leading tachyon must be absent, the condition N00 = 0 does not automatically exclude the possibility that other tachyons be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Indeed, as we shall see in the following Sections, this is precisely what happens in most non-supersymmetric theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The analysis of the sub-dominant contributions to 〈d(n)〉 is a bit more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' To start, we notice that the Q’s are periodic functions of n, with period ℓ, Q(ℓ,n+ℓm) ab = Q(ℓ,n) ab , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) for any integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, since the number of physical states (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) depends on the prod- uct of the Q functions from the holomorphic and anti-holomorphic characters, it is convenient to decompose the degrees of freedom into classes organised by the common periodicity vℓ, ¯ℓ = ℓ ¯ℓ/gcd(ℓ, ¯ℓ) of the two Q’s and associate to each class its own enveloping function Φa(n,{w}), as in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As a result, 〈d(n)〉 = � a,b � {w} Nab ¯Φa(n,{w + Hb − ¯Ha})Φb(n,{w}) ≡ � a,b,c,d [�n] � ℓ, ¯ℓ=1 vℓ, ¯ℓ−1 � w=0 Nab Q(ℓ,w) bc ¯Q( ¯ℓ,w+Hb− ¯Ha) ad fc(ℓ,n) ¯fd( ¯ℓ,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) 8 Moreover, since one can always write w = kℓ +ℓr, and vℓ, ¯ℓ−1 � w=0 = ℓ−1 � kℓ=0 ¯ℓ gcd(ℓ, ¯ℓ) −1 � r=0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) the periodicity of the Q’s can be exploited to cast the sector averaged sum as 〈d(n)〉 = � a,b,c,d [�n] � ℓ, ¯ℓ=1 ℓ−1 � kℓ=0 ¯ℓ gcd(ℓ, ¯ℓ) −1 � r=0 Nab Q(ℓ,kℓ) bc ¯Q( ¯ℓ,kℓ+rℓ+Hb− ¯Ha) ad fc(ℓ,n) ¯fd( ¯ℓ,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='9) This is the expression from which we can extract our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We have to distinguish the two cases ℓ = ¯ℓ and ℓ ̸= ¯ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In the latter case, if ℓ does not divide ¯ℓ, there is no contribution to the sector averaged sum, since ¯ℓ gcd(ℓ, ¯ℓ) −1 � r=0 ¯Q( ¯ℓ,kℓ+rℓ+Hb− ¯Ha) ad = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='10) which follows from ¯ℓ gcd(ℓ, ¯ℓ) −1 � r=0 ¯Q( ¯ℓ,k+rℓ) ad = ¯ℓ gcd(ℓ, ¯ℓ) −1 � r=0 ¯ℓ−1 � ¯p=0 gcd( ¯ℓ, ¯p)=1 e− 2πi ¯ℓ ( ¯p(k+rℓ)+ ¯p′ ¯Hd) � M−1 ¯ℓ, ¯p � ad = ¯ℓ−1 � ¯p=0 gcd( ¯ℓ, ¯p)=1 e− 2πi ¯ℓ ( ¯pk+ ¯p′ ¯Hd) � M−1 ¯ℓ, ¯p � ad 1−e −2πi ¯p ¯ℓ gcd(ℓ, ¯ℓ) 1−e−2πi ¯pℓ ¯ℓ = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='11) since ¯ℓ is an integer multiple of gcd(ℓ, ¯ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Similarly, if ¯ℓ = mℓ, for some integer m, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='10) still holds since ¯p and ¯ℓ must be co-prime, and thus m−1 � r=0 e−2πi ¯pr/m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='12) The case ℓ = ¯ℓ is a bit more involved and, as we shall see, discriminates between tachyonic and non-tachyonic string vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The Q functions have now the same periodicity, which implies that in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='9) the sum over r is trivial, since nothing depends on r, and 〈d(n)〉 = � a,b,c,d [�n] � ℓ=1 ℓ−1 � p=0 gcd(ℓ,p)=1 ℓ−1 � ¯p=0 gcd(ℓ, ¯p)=1 Nab � M−1 ℓ,p � bc � M−1 ℓ, ¯p �∗ ad e− 2πi ℓ [(p− ¯p)Hb−(p′Hc− ¯p′ ¯Hd)] fc(ℓ,n) ¯fd(ℓ,n) × ℓ−1 � kℓ=0 e− 2πi ℓ (p− ¯p)kℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13) 9 The sum over kℓ imposes the condition p = ¯p which identifies the holomorphic and anti-holomor- phic modular transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Using the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) of modular invariance on the GSO matrix Nab, one arrives at the final result 〈d(n)〉 = [�n] � ℓ=1 ℓ−1 � p=0 gcd(ℓ,p)=1 � ab Nab e 2πi ℓ p′(Hb− ¯Ha)ℓ fb(ℓ,n) ¯fa(ℓ,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='14) The exponential growth of the sector averaged sum is thus directly linked to the presence of tachyons in the physical string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, if tachyons are absent, the f functions are expressed in terms of the J Bessel functions which only admit the asymptotic power-law growth (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13) and therefore Ceff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' On the contrary, if physical tachyons, with Hb = ¯Ha < 0, are present in the string spectrum, then 〈d(n)〉 ∼ � a,b Hb= ¯Ha<0 Nab db(0) ¯da(0) |Hb|(d−1)/2 2n(d+1)/2 [�n] � ℓ=1 ϕ(ℓ)e 8π ℓ �|Hb|n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='15) where ϕ(ℓ) is the Euler totient function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The exponential growth is then dictated by the conformal weight Hb of the lightest physical tachyon, Ceff = 8π � |Hb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' To reiterate, we have proven that the asymptotic growth rate of the sector averaged sum is dictated by the mass of the lightest states, whether tachyonic or massless, Ceff = 4π � |α′m2 lightest| ≤ Ctot , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='16) and thus the necessary and sufficient condition for classical stability is the vanishing of the effective central charge, as conjectured by Dienes in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Our analysis is fully general and applies to any vacuum of oriented closed strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' It extends the discussion of [24] which heavily relies on the representation of the characters in terms of eta quo- tients of special type, which is a rather restrictive requirement, not met by most non-supersymme- tric string vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 4 Non-Supersymmetric Heterotic Vacua in D = 10 A simple arena where to test our result is ten-dimensional closed-string vacua with no space- time supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Indeed, one-loop modular invariance allows for many consistent construc- tions in ten dimensions, most of which do not enjoy space-time supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' These non- supersymmetric vacua can be divided into three different classes: tachyonic theories with only bosonic excitations, tachyonic theories with both fermionic and bosonic fields, and a single the- ory with no tachyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The unique representative of the last class is the SO(16) × SO(16) heterotic theory [28,29] with partition function Z16 = O8 ( ¯V16 ¯C16 + ¯C16 ¯V16)+V8( ¯O16 ¯O16 + ¯S16 ¯S16) −S8 ( ¯O16 ¯S16 + ¯S16 ¯O16)−C8 ( ¯V16 ¯V16 + ¯C16 ¯C16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) 10 while the first class comprises the type 0A and 0B strings [28], with partition functions Z0A = |O8|2 +|V8|2 +S8 ¯C8 +C8 ¯S8 , Z0B = |O8|2 +|V8|2 +|S8|2 +|C8|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) The second class is richer and contains five heterotic vacua with gauge groups SO(32), SO(16)×E8, SO(8)×SO(24), (E7 ×SU(2))2 and SU(16) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' These theories present a similar behaviour and, for simplicity, we shall concentrate on the SO(32) theory with partition function Z32 = O8 ¯V32 +V8 ¯O32 −S8 ¯S32 −C8 ¯C32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) Notice that in writing eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) we have used the SO(2n) characters, as defined in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As anticipated in the previous Section, strictly speaking these string theories do not correspond to RCFTs because of the presence of non-compact bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Still, following [26], we can overcome this problem by defining the pseudo-characters (O2n,V2n,S2n,C2n) → �O2n η8 , V2n η8 , S2n η8 , C2n η8 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) and including suitable phases in the modular transformations T : (O2n,V2n,S2n,C2n) → e−iπ(n+8)/12 (O2n,−V2n,eiπn/4 S2n,eiπn/4C2n), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) and S : � ����� O2n V2n S2n C2n � ����� → τ−4 1 2 � ����� 1 1 1 1 1 1 −1 −1 1 −1 i −n −i −n 1 −1 −i −n i −n � ����� � ����� O2n V2n S2n C2n � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) We can discuss all these theories at once by noticing that their partition functions can be com- pactly written as ZA = 3� a,b=0 ¯R A a Nab Lb , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) where L = (O8,V8,S8,C8) denotes the left-moving characters, which are common to all ten-dimen- sional non-supersymmetric theories, and ¯R A a denotes the right-moving characters, which depend on the specific model A = 16,0,32, and can be extracted from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The corresponding GSO matrices read N0A = � ����� 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 � ����� , N0B = � ����� 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 � ����� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) 11 for the type 0A and type 0B theories, while Nhet = � ����� 0 1 0 0 1 0 0 0 0 0 −1 0 0 0 0 −1 � ����� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='9) for the heterotic theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The universal holomorphic characters La have Ha = (− 1 2,0,0,0), and using the explicit form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) of the T and S modular matrices, one finds Φa(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='{wℓ}) = 1 2 15 4 n 11 4 � e4π � n/2 − � 2δa0 (−1)w2 e2π � n/2 + 2 � 3 cos � 2 3π(w3 +(−1)δa0) � e 4π 3 � n/2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='10) for the asymptotic behaviour of the degeneracies, where the wℓ take integer values in the interval 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',ℓ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Notice that L0 = O8 is the unique tachyonic character, associated to the NS vacuum, which reflects the presence of a single leading exponential growth e4π � n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As for the anti-holomorphic characters ¯R A a , they carry weights ¯Ha = (−1, 1 2,0,0) for the SO(16)×SO(16) heterotic theory ¯Ha = (− 1 2,0,0,0) for the 0A and 0B theories ¯Ha = (−1,− 1 2,1,1) for the SO(32) heterotic theory (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='11) and using the explicit expression for the T and S modular matrices, which can be read/derived from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6), one finds ¯Φ(16) a (n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='{ ¯wl}) = 1 2 3 2 n 11 4 � e4π�n + � 2δa1 (−1) ¯w2 e2π�n + 2 � 3 cos � 2 3π( ¯w3 +δa0) � e 4π 3 �n +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='12) for the SO(16)×SO(16) theory, ¯Φ(32) a (n,{ ¯wℓ}) = 1 2 3 2 n 11 4 � e4π�n +2 11 4 (−1)δa2+δa3 e4π � n/2 + � 2δa1 (−1) ¯w2 e2π�n −2 13 4 δa0 (−1) ¯w2 e2π � n/2 + 2 � 3 cos � 2 3π( ¯w3 −δa1) � e 4π 3 �n − 2 15 4 � 3 (−1)δa1 cos � 2 3π( ¯w3 +1−δa1) � e 4π 3 � n/2 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13) for the SO(32) theory, while ¯Φ(0) a (n,{ ¯wℓ}) = Φa(n,{ ¯wℓ}) for the 0A and 0B theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Notice, that in ¯Φ(32) a (n,{ ¯wℓ}) there are two leading exponential growths e4π�n and e4π � n/2 associated, respectively, to the two tachyonic characters ¯R0 and ¯R1, with ¯H0 = −1 and ¯H1 = − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 12 The sector averaged sum is then given by 〈d(n)〉A = 3� a,b=0 � {wℓ} Nab ¯Φ(A) a (n,{wℓ + Hb − ¯Ha})Φb(n,{wℓ}), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='14) where the choice ¯wℓ = wℓ + Hb − ¯Ha follows from the level matching condition, and the sum is extended over all integers wℓ = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',ℓ−1, for each ℓ = 2,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='. It is straightforward to see that the growth of 〈d(n)〉0 ∼ e4π � 2n for the type 0A and 0B theories is dictated by the total central charge Ctot = 4π � 2, while for the heterotic models the leading term e4π(1+1/ � 2)�n cancels upon summing over the sectors a,b = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This reflects the fact that the type 0 theories only have bosonic excita- tions while the heterotic ones have both bosons and fermions in their spectrum, and � ab Nab = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, while for the tachyonic SO(32) theory 〈d(n)〉32 = 1 � 2n 11 2 � e4π � 2n +e2π � 2n +2e 2π 3 � 2n +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='15) and Ceff = 4π � 2 < Ctot, for the non-tachyonic SO(16)×SO(16) model Ceff = 0 since all exponential growths cancel leaving, at most, a power-law behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This is in agreement with our general result given in the previous Section and with the result of [24] for the SO(16)×SO(16) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 5 Scherk-Schwarz Reductions at Rational Points A more interesting class of non-supersymmetric vacua can be constructed in lower dimensions by employing the Scherk-Schwarz mechanism [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This can be conveniently realised as a freely acting orbifold [33], where the action of a supersymmetry breaking generator g is combined with a suitable shift δ along compact directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In its simplest nine-dimensional incarnation g = (−1)F, with F the space-time fermion number, while δ acts as y → y + πR on the compact coordinate y parametrising the circle S1(R) of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In the heterotic case, one can consider more general orbifolds where the space-time fermion number is combined with an action on the gauge degrees of freedom, in accordance with modular invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In this Section, however, we shall consider the Scherk-Schwarz reduction of the type IIB superstring since it shares the main features with any generic construction, with the advantage of being quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The torus partition function Z = 1 2|V8 −S8|2 � m,n Λm,n(R)+ 1 2|V8 +S8|2 � m,n (−1)m Λm,n(R) + 1 2|O8 −C8|2 � m,n Λm,n+ 1 2 (R)+ 1 2|O8 +C8|2 � m,n (−1)m Λm,n+ 1 2 (R), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) clearly exhibits in the first line the action of the (−1)F δ generator on the original spectrum of the IIB superstring, while the second line, involving the flipped GSO projection, is required by mod- ular invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The Kaluza-Klein momenta and windings associated to the compact direction contribute with the standard Narain lattice Λm,n = q α′ 4 � m R + nR α′ �2 η ¯q α′ 4 � m R − nR α′ �2 ¯η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) 13 In the decompactification limit, R → ∞, the orbifold action is trivialised and one recovers the su- persymmetric IIB theory, while for generic values of the radius R supersymmetry is spontaneously broken and the gravitini acquire a mass m ≃ 1/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The excitations of the NS-NS vacuum in |O8|2 now survive the GSO projection in the twisted sector, and the lightest state has mass m2 |O8|2 = − 1 2α′ + 1 4 � R 2α′ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) This scalar is then massive for large values of R, but turns tachyonic below the critical radius Rc = 2 � 2α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As we decrease the radius, more and more states become tachyonic, and it is then clear that these models represent an ideal ground to study the realisation of misaligned supersymmetry in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' To illustrate the analysis of Section 3 on the degeneracies of states, we need to select rational values for R2/α′ = s/t ∈ Q since, in this case, the Narain lattice reduces to an RCFT � m,n Λm,n(R) → 2st−1 � α=0 λα ¯λαl , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) with the 2st characters defined as λα(q) = � m qst � m+ α 2st �2 η(q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) Notice, that λ0 and λst are real, while λα and λ2st−α, α = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',st − 1, form conjugate pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The λ’s have conformal weight hα = α2/4st, and thus Hα = hα − 1 24, for α = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',st, with conjugate pairs carrying the same weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) the anti-holomorphic characters have index αl, where l = r t + sv, with the integers r and v satisfying the relation r t − vs = 1, and the label αl is defined modulo 2st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' As shown in the Appendix, these characters are eigenstates of the shift operator δ only for even s, but must be broken into sub-characters for odd s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' For simplicity, here we shall restrict the discussion to the even-s case, where δ : λα → (−1)α/2t λα , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) and we also take t = 1, so that the condition r t − vs = 1 can be easily solved by r = 1 and v = 0, for any s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Other choices for s and t yield equivalent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The action of the modular group on these characters is encoded in the T and S matrices Tαβ = eiπ � α2 2s − 1 12 � δαβ , Sαβ = e2πi αβ 2s � 2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) 14 Taking all this into account, the torus partition function becomes Z = s−1 � a=0 � |χ2a+2s|2 +|χ2a+4s|2� − s−1 � a=0 � χ2a+1+2s ¯χ2a+1+4s +χ2a+1+4s ¯χ2a+1+2s � + s 2 −1 � a=0 � χ2a+σ ¯χ2a+σ+s +χ2a+σ+s ¯χ2a+σ +χ2a+σ+6s ¯χ2a+σ+7s +χ2a+σ+7s ¯χ2a+σ+6s � − s 2 −1 � a=0 � χ2a+1−σ ¯χ2a+1−σ+7s +χ2a+1−σ+7s ¯χ2a+1−σ +χ2a+1−σ+s ¯χ2a+1−σ+6s +χ2a+1−σ+6s ¯χ2a+1−σ+s � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) where in the third and fourth sums one has to distinguish the two cases s = 2(2m +σ) with σ = 0,1 while, as dictated by spin-statistics, the minus signs reflect the presence of space-time fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The new characters are {χa}8s−1 a=0 = (O8,V8,S8,C8)⊗{λα}2s−1 α=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='9) The characters χa and χ2s−a have shifted conformal weight Ha = a2 4s − 1 2, and therefore are tachyonic for a < � 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' However, since they appear in the partition function in the combination χ2a+σ ¯χ2a+σ+s +χ2a+σ+s ¯χ2a+σ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='10) the only states which are level-matched are χs/2 ¯χ3s/2 +χ3s/2 ¯χs/2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='11) which are tachyonic for s < 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This agrees with the result of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) valid at irrational values of R, since now R2 = s α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The effective degrees of freedom at a given mass level can be straightforwardly extracted from the partition function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) by Taylor expanding the various characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We find that misaligned supersymmetry is present for any choice of the compactification radius, even in the tachyonic regime, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This is in accordance with our general discussion of Section 3 and is corroborated by the large-n behaviour of the sector averaged sum 〈d(n)〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Indeed, the degeneracy of each character χa has in principle many exponential growth rates associated to all tachyonic characters of the RCFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' However, the leading exponential, associated to χ0, is universal and cancels in the sector averaged sum since the partition function involves 4s bosonic and fermionic sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Therefore, we can conclude that, for any s, Ceff < Ctot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Moreover, for s > 8 no physical tachyons are present in the spectrum and thus Ceff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Finally, in the tachy- onic region s < 8, the state which dictates the exponential growth of 〈d(n)〉 is the physical tachyon 15 5 10 15 20 10 10 20 5 10 15 20 10 10 20 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1: The signed logarithm of the net number of degrees of freedom at each mass level for the type IIB Scherk-Schwarz reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Positive (negative) contributions are ascribed to the excess of bosonic (fermionic) states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The left figure refers to R2 = 2α′, and despite the presence of tachyons, the spectrum exhibits misaligned supersymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The right figure corresponds to R2 = 8α′, within the non-tachyonic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The step-like shape of the enveloping functions reflects the fact that the first few Kaluza-Klein excitations have masses smaller than the string scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' χs/2 ¯χ3s/2 and its conjugate, so that Ceff = 2π � 8− s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='12) Indeed, explicit calculations yield 〈d(n)〉 = 81 4096n5 � e2π � 6n +eπ � 6n +2e2π � 2n/3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='13) for s = 2, 〈d(n)〉 = 1 256n5 � e4π�n +e2π�n +2e 4 3 π�n +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='14) for s = 4, while 〈d(n)〉 = 0 for s > 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' It is tempting to continue the behaviour (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='12) to arbitrary irrational values of the compactifi- cation radius, so that the asymptotic growth of the irrational sector averaged sum reads Ceff = � � � 2π � 8−R2/α′ for R2 < 8α′ , 0 for R2 ≥ 8α′ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='15) as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2, and Ceff → Ctot as R → 0, in accordance with the fact that, in this limit, one recovers the purely bosonic type 0B theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1 A Comment on Phase Transitions It is tempting to consider the partition function Z of the world-sheet CFT as a function of q = e−β+iµ, where we now interpret β as the world-sheet inverse temperature and µ as the spin po- tential, conjugate to the worldsheet momentum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Clearly, the sum over states in Z is ab- solutely convergent, and the corresponding world-sheet free energy F = −(1/β)logZ is analytic, signifying the absence of phase transitions, as in any theory with a finite number of local degrees 16 4 6 8 10 12 14 5 10 15 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2: The figure displays the dependence of the effective central charge on R and shows that below the critical radius the Scherk-Schwarz reduction is not a deformation of the original theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' For instance, in bosonic string theory the density of states grows universally according to the Cardy formula d(n) ∼ e4π � cn/24 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='16) so that the level matched partition function �1 0 dτ1Z (τ1,β/2π) = � n d(n)e−2βn , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='17) converges for all values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In light of our discussion on Ceff, it is interesting to construct a deformed version of the above partition function, obtained by analytically continuing the integer n to the reals and by averaging the net degrees of freedom in terms of the enveloping functions Φ(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This amounts to replacing the physical net degeneracies d(n) by their sector averaged versions 〈d(n)〉, and defining a sector averaged partition function for the world-sheet CFT as 〈Z 〉 = � n 〈d(n)〉e−2βn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='18) This deformation effectively introduces an infinite number of degrees of freedom by averaging the interpolation of the mass levels in terms of the enveloping functions Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In doing so, one can estimate 〈Z 〉 ∼ � dn eCeff �n−2βn ∼ e C2 eff 8β , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='19) so that the sector averaged free energy reads 〈F〉 ∼ − C 2 eff 8β2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='20) and is controlled by the square of the effective central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In the case of Scherk-Schwarz su- persymmetry breaking, Ceff is a function of the compactification radius R, as shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='15), 17 which, from the point of view of the worldsheet CFT, should be treated as an external background field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In this sense, the sector averaged free energy 〈F〉 displays a first order phase transition as the radius crosses the critical value Rc, since its first derivative is discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This would suggest a possible interpretation in terms of phase transitions in a suitable holo- graphic dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, it is known that two-dimensional CFT’s may admit an AdS3 gravitational description, and in certain deformations of symmetric orbifold CFTs, in the large central charge limit, Hagedorn-like phase transitions of the CFT are mapped to Hawking-Page transitions of the gravity theory, dominated by the entropy of BTZ black holes [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' It is thus tempting to interpret the phase transition displayed by our system in terms of a holographic dual where a similar kind of averaging procedure is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' However, this investigation lies beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 6 Conclusions In this paper we have revisited the problem of classical stability of non-supersymmetric string vacua in various dimensions in relation to the growth rate of the net number of degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Contrary to common belief, we find that misaligned supersymmetry is not an exclusive property of non-tachyonic strings, but manifests itself in all theories, tachyonic or not, containing space-time fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, we show that the growth rate of the sector averaged sum 〈d(n)〉 is set by the mass of the lightest state, whether it be tachyonic or massless, and is strictly smaller than Ctot in the presence of fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We also prove that the necessary and sufficient condition for the tree-level stability of the vacuum is an at most power-like growth of 〈d(n)〉 corresponding to vanishing Ceff, as conjectured in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Our result is model independent and applies to any closed-string vacuum in any dimension, and agrees with the recent analysis of [24] conducted for the SO(16) × SO(16) heterotic string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Following [23], our analysis is based on the sector average of the string degrees of freedom whose discrete masses are analytically continued to real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The cancellations required by mis- aligned supersymmetry highly depend on this analytic continuation and not just on the properties of the discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' This is to be contrasted to the works of [18] and [22] which, using num- ber theoretic methods, relate misaligned supersymmetry to the real physical degrees of freedom of the vacuum, without having to resort to an analytic continuation of the mass levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' It would be interesting to obtain a more direct connection between these two a priori different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Any quantitative analysis on misaligned supersymmetry heavily relies on modular invariance of the torus partition function, and can thus only be formulated for closed oriented strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Still orientifold vacua [35–38,40] provide an appealing phenomenology and afford more general ways to break supersymmetry [41–51] which cannot be realised in closed strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Although some at- tempts have been made to understand the role played by misaligned supersymmetry in the classi- cal stability of non-supersymmetric orientifolds [24,52–54], a thorough quantitative analysis is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Modular transformations relate in them the direct and transverse channels, so that there is no obvious link between the IR and the UV properties of the spectrum, and therefore different tools need to be employed to uncover this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We hope to return to this problem in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 18 Acknowledgements It is a pleasure to thank Ivano Basile, Flavio Tonioni and in particular Niccolò Cribiori for enlight- ening discussions on misaligned supersymmetry and non-supersymmetric string vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' We are grateful to Augusto Sagnotti for constructive feedback on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' would like to thank the Physics Department of the University of Torino for hospitality during the final stages of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' The work of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' is partially supported by the MIUR-PRIN contract 2017CC72MK-003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' A The Scherk-Schwarz Mechanism in RCFT’s Although the Narain partition function for a real boson Y compactified on a circle S1(R) of ra- dius R does not fully factorise into the product of holomorphic and anti-holomorphic contribu- tions, things simplify considerably whenever R2/α′ = s/t ∈ Q takes rational values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In this case, the CFT of the compact boson becomes rational, the Kaluza-Klein momenta and windings admit the parametrisation m = s(k + ¯k)+ 1+l 2t α, n = t(k − ¯k)+ 1−l 2s α, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1) with l defined after eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) and k, ¯k ∈ Z, and only a finite number N = 2st of representations are unitary, and are associated to the characters λα = 1 η � k∈Z q N 2 (k+ α N )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) Although the λ’s provide a natural decomposition of the Narain partition function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) as the sesquilinear combination (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4), in general they do not provide a suitable basis when shift orbifolds act on S1(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' In fact, already for the simple order-two shift δ : Y → Y +πR, the Narain lattice picks up a phase � m,n Λm,n → � m,n (−1)m Λm,n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='3) and in view of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='1), it is clear that the characters λα are eigenstates of δ only for even s so that (−1)m → (−1) 1+l 2t α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' For odd s, instead, the k-th excitation in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='2) acquires an additional sign de- pending on the parity of k, that would suggest the decomposition λα → ξi α = 1 η � k∈Z q N 2 � 2k+i+ α N �2 , i = 0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='4) The ξ’s however, are not closed under the action of SL(2,Z), since they fail to capture the twisted sector with half-integer windings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Therefore, the correct choice of the δ eigenstates is ζα = 1 η � k q2N � k+ α 4N �2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='5) with now α = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=',4N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 19 One can thus decompose the various orbifold blocks as � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n Λm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n = � � � �N−1 α=0 λα ¯λlα for even s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' �1 a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='b=0 �N−1 α=0 ζ2(α+aN) ¯ζ2(lα+bN) for odd s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n (−1)m Λm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n = � � � �N−1 α=0 (−1) 1+l 2t α λα ¯λlα for even s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' �1 a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='b=0 �N−1 α=0 (−1)a+b+ 1+l 2t α ζ2(α+aN) ¯ζ2(lα+bN) for odd s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n Λm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n+ 1 2 = � � � �N−1 α=0 λlα−(1+l)s/2 ¯λα for even s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' �1 c=0 �2N−1 α=0 ζl(2α+1)−(1+l)s+2Nc ¯ζ2α+1 for odd s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' � m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n (−1)m Λm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='n+ 1 2 = � � � �N−1 α=0 (−1) (lα− 1+l 2 s)2−α2 N λlα−(1+l)s/2 ¯λα for even s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' �1 c=0 �2N−1 α=0 (−1) (l(2α+1)−(1+l)s+2Nc)2−(2α+1)2 4N ζl(2α+1)−(1+l)s+2Nc ¯ζ2α+1 for odd s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='6) Using the representation of the SL(2,Z) generators Tαβ = eiπ � α2 M − 1 12 � , Sαβ = e 2πiαβ M � M , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='7) on the space of characters, with M = N for the λ’s, and M = 4N for the ζ’s, it is straightforward to show that the relations � m,n (−1)m Λm,n S −→ � m,n Λm,n+ 1 2 T −→ � m,n (−1)m Λm,n+ 1 2 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content='8) hold both for odd and even s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' 20 References [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNFST4oBgHgl3EQfBTj3/content/2301.13702v1.pdf'} +page_content=' Dudas and J.' metadata={'source': 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Continuous-Discrete State Space Models +for Irregularly-Sampled Time Series +Abdul Fatir Ansari 1 † Alvin Heng 2 Andre Lim 2 Harold Soh 2 +Abstract +Learning accurate predictive models of real-world +dynamic phenomena (e.g., climate, biological) +remains a challenging task. One key issue is +that the data generated by both natural and ar- +tificial processes often comprise time series that +are irregularly sampled and/or contain missing +observations. +In this work, we propose the +Neural Continuous-Discrete State Space Model +(NCDSSM) for continuous-time modeling of +time series through discrete-time observations. +NCDSSM employs auxiliary variables to disen- +tangle recognition from dynamics, thus requiring +amortized inference only for the auxiliary vari- +ables. Leveraging techniques from continuous- +discrete filtering theory, we demonstrate how to +perform accurate Bayesian inference for the dy- +namic states. We propose three flexible parame- +terizations of the latent dynamics and an efficient +training objective that marginalizes the dynamic +states during inference. Empirical results on mul- +tiple benchmark datasets across various domains +show improved imputation and forecasting perfor- +mance of NCDSSM over existing models. +1. Introduction +State space models (SSMs) provide an elegant framework +for modeling time series data. Combinations of SSMs with +neural networks have proven effective for various time series +tasks such as segmentation, imputation, and forecasting (Kr- +ishnan et al., 2015; Fraccaro et al., 2017; Rangapuram et al., +2018; Kurle et al., 2020; Ansari et al., 2021). However, +most existing models are limited to the discrete time (i.e., +uniformly sampled) setting, whereas data from various phys- +ical and industrial systems in the real world are sometimes +only available at irregular (often sparse) intervals. Such sys- +†Work done while at National University of Singapore, prior to +joining Amazon. 1AWS AI Labs 2School of Computing, National +University of Singapore. Correspondence to: Abdul Fatir Ansari +. +Preprint. +Figure 1. (Top) Generative model of Neural Continuous-Discrete +State Space Model. The bold red arrows indicate that the state, +zt, evolves continuously in time. The auxiliary variables, ak, +and observations, yk, are emitted at arbitrary discrete timesteps +tk ∈ {t0, t1, . . . , tT }. (Bottom) Amortized inference for auxil- +iary variables and continuous-discrete Bayesian inference for states. +Samples from the amortized variational distribution over auxiliary +variables are used as pseudo-observations to condition and perform +inference in the continuous-discrete SSM at the bottom. +tems are best modeled as continuous-time latent processes +with irregularly-sampled discrete-time observations. Desir- +able features of such a time series model include modeling +of stochasticity (uncertainty) in the system, and efficient +and accurate inference of the system state from potentially +high-dimensional observations (e.g., video frames). +Recently, latent variable models based on neural differen- +tial equations have gained popularity for continuous-time +modeling of time series (Chen et al., 2018; Rubanova et al., +2019; Yildiz et al., 2019; Li et al., 2020; Liu et al., 2020; +Solin et al., 2021). However, these models suffer from lim- +itations. The ordinary differential equation (ODE)-based +models employ deterministic latent dynamics and/or encode +arXiv:2301.11308v1 [cs.LG] 26 Jan 2023 + +Zto +Zt2 +ZtT +Zt1 +ao +a1 +a2 +aT +yo +y1 +y2 +YT +yo +Y1 +Y2 +yT +ao +a1 +a2 +aT +ao +a1 +a2 +aT +Zto +Zt2 +ZtT +Zt1Neural Continuous-Discrete State Space Models +the entire context window into an initial state, creating a +restrictive bottleneck. On the other hand, stochastic differ- +ential equation (SDE)-based models use stochastic latent +dynamics, but typically perform a variational approximation +of the latent trajectories via posterior SDEs. The posterior +SDEs incorporate new observations in an ad-hoc manner, +potentially resulting in a disparity between the posterior and +generative transition dynamics, and a non-Markovian state +space. +To address these issues, we propose the Neural Continuous- +Discrete State Space Model (NCDSSM) that uses discrete- +time observations to model continuous-time stochastic +Markovian dynamics (Fig. 1). By using auxiliary variables, +NCDSSM disentangles recognition of high-dimensional ob- +servations from dynamics (encoded by the state) (Fraccaro +et al., 2017; Kurle et al., 2020). We leverage the rich lit- +erature on continuous-discrete filtering theory (Jazwinski, +1970), which has remained relatively unexplored in the mod- +ern deep learning context. Our proposed inference algorithm +only performs amortized variational inference for the auxil- +iary variables since they enable classic continuous-discrete +Bayesian inference (Jazwinski, 1970) for the states, using +only the generative model. This obviates the need for pos- +terior SDEs and allows incorporation of new observations +via a principled Bayesian update, resulting in accurate state +estimation. As a result, NCDSSM enables online prediction +and naturally provides state uncertainty estimates. We pro- +pose three dynamics parameterizations for NCDSSM (linear +time-invariant, non-linear and locally-linear) and a training +objective that can be easily computed during inference. +We evaluated NCDSSM on imputation and forecasting tasks +on multiple benchmark datasets. Our experiments demon- +strate that NCDSSM accurately captures the underlying +dynamics of the time series and extrapolates it consistently +beyond the training context, significantly outperforming +baseline models. From a practical perspective, we found +that NCDSSM is less sensitive to random initializations and +requires fewer parameters than the baselines. +In summary, the key contributions of this work are: +• NCDSSM, a continuous-discrete SSM with auxiliary +variables for continuous-time modeling of irregularly- +sampled (high dimensional) time series; +• An accurate inference algorithm that performs amor- +tized inference for auxiliary variables and classic +Bayesian inference for the dynamic states; +• An efficient learning algorithm and its stable imple- +mentation using square root factors; +• Experiments on multiple benchmark datasets demon- +strating that NCDSSM learns accurate models of the +underlying dynamics and extrapolates it consistently +into the future. +2. Approximate Continuous-Discrete +Inference +We begin with a review of approximate continuous-discrete +Bayesian filtering and smoothing, inference techniques em- +ployed by our proposed model. Consider the following Itˆo +SDE, +dzt = f(zt, t)dt + G(zt, t)dBt, +(1) +where zt ∈ Rm is the state, Bt ∈ Rm denotes a Brownian +motion with diffusion matrix Q, f(·, t) : Rm → Rm is the +drift function and G(·, t) : Rm → Rm×m is the diffusion +function. The initial density of the state, p(z0), is assumed +to be known and independent of the Brownian motion, Bt. +The evolution of the marginal density of the state, pt(zt), +is governed by the Fokker-Plank-Kolmogorov (FPK) equa- +tion (Jazwinski, 1970, Ch. 4), +∂pt(zt) +∂t += L ∗pt, +(2) +where L ∗ is the forward diffusion operator given by +L ∗ϕ = − +d +� +i=1 +∂ +∂xi +[ϕfi] + 1 +2 +d +� +i=1 +d +� +j=1 +� +ϕ(GQG⊤)ij +� +. +In practice, we only have access to noisy transformations +(called measurements or observations), yk ∈ Rd, of the +state, zt, at discrete timesteps tk ∈ {t0, . . . , tT }. The +continuous-discrete state space model (Jazwinski, 1970, +Ch. 6) is an elegant framework for modeling such time +series. +Definition 2.1 (Continuous-Discrete State Space Model). A +continuous-discrete state space model is one where the latent +state, zt, follows the continuous-time dynamics governed +by Eq. (1) and the measurement, yk, at time tk is obtained +from the measurement model p(yk|ztk). +In this work, we consider linear Gaussian measurement +models, yk ∼ N(yk; Hztk, R), where H ∈ Rd×m is the +measurement matrix and R ⪰ 0 ∈ Rd×d is the covariance +matrix. Given observations Yτ = {yk : tk ≤ τ}, we are +interested in answering two types of inference queries: the +posterior distribution of the state, zt, conditioned on obser- +vations up to time t, pt(zt|Yt), and the posterior distribution +of the state, zt, conditioned on all available observations, +pt(zt|YT ). These are known as the filtering and smoothing +problems, respectively. +The filtering density, pt(zt|Yt), satisfies the FPK equation +(Eq. 2) for t ∈ [tk, tk+1) between observations, with the +initial condition pt(zt|Ytk) at time tk. Observations can be +incorporated via a Bayesian update, +pt(ztk|Ytk) = p(ytk|ztk)p(ztk|Ytk−1) +p(yk|Ytk−1) +. +(3) + +Neural Continuous-Discrete State Space Models +The smoothing density satisfies a backward partial differen- +tial equation related to the FPK equation. We refer the reader +to Anderson (1972) and S¨arkk¨a & Solin (2019, Ch. 10) for +details and discuss a practical approximate filtering proce- +dure in the following (cf. Appendix B.1 for smoothing). +2.1. Continuous-Discrete Bayesian Filtering +Solving Eq. (2) for arbitrary f and G is intractable; hence, +several approximations have been considered in the litera- +ture (S¨arkk¨a & Solin, 2019, Ch. 9). The Gaussian assumed +density approximation uses a Gaussian approximation, +pt(zt) ≈ N(zt; mt, Pt), +(4) +for the solution to the FPK equation, characterized by the +time-varying mean, mt, and covariance matrix, Pt. Further, +linearization of the drift f via Taylor expansion results in +the following ODEs that govern the evolution of the mean +and covariance matrix, +dmt +dt += f(mt, t), +(5a) +dPt +dt = Fz(mt, t)Pt + PtF⊤ +z (mt, t) + D(mt, t), (5b) +where Fz(mt, t) is the Jacobian of f(z, t) with respect +to z at mt and D(·, t) = G(·, t)QG⊤(·, t). Thus, for +t ∈ [tk, tk+1) between observations, the filter distribution +pt(zt|Yt) can be approximated as a Gaussian with mean +and covariance matrix given by solving Eq. (5), with initial +conditions mtk and Ptk at time tk. This is known as the +prediction step. +The +Gaussian +assumed +density +approximation +of +p(ztk|Ytk−1) described above makes the Bayesian update +in Eq. (3) analytically tractable as p(ytk|ztk) is also a +Gaussian distribution with mean Hzk and covariance +matrix R. The parameters, mk and Pk, of the Gaussian +approximation of pt(ztk|Ytk) are then given by, +Sk = HP− +k H⊤ + R, +(6a) +Kk = P− +k H⊤S−1 +k , +(6b) +mk = m− +k + Kk +� +yk − Hm− +k +� +, +(6c) +Pk = P− +k − KkSkK⊤ +k , +(6d) +where m− +k and P− +k are the parameters of pt(ztk|Ytk−1) +given by the prediction step. Eq. (6) constitutes the up- +date step which is exactly the same as the update step +in the Kalman filter for discrete-time linear Gaussian +SSMs. The continuous-time prediction step together with +the discrete-time update step is sometimes also referred +to as the hybrid Kalman filter. As a byproduct, the up- +date step also provides the conditional likelihood terms, +p(yk|Ytk−1) = N(yk; Hm− +k , Sk), which can be com- +bined to give the likelihood of the observed sequence, +p(YtT ) = p(y0) �T +k=1 p(yk|Ytk−1). +3. Neural Continuous-Discrete State Space +Models +In this section, we describe our proposed model: Neural +Continuous-Discrete State Space Model (NCDSSM). We +begin by formulating NCDSSM as a continuous-discrete +SSM with auxiliary variables that serve as succinct represen- +tations of high-dimensional observations. We then discuss +how to perform efficient inference along with parameter +learning and a stable implementation for NCDSSM. +3.1. Model Formulation +NCDSSM is a continuous-discrete SSM in which the la- +tent state, zt ∈ Rm, evolves in continuous time, emitting +linear-Gaussian auxiliary variables, at ∈ Rh, which in turn +emit observations, yt ∈ Rd. Thus, NCDSSM possesses +two types of latent variables: (a) the states that encode the +hidden dynamics, and (b) the auxiliary variables that can +be viewed as succinct representations of the observations. +The inclusion of auxiliary variables offers two benefits; (i) it +allows disentangling representation learning (or recognition) +from dynamics (encoded by zt) and (ii) it enables the use +of arbitrary decoders to model the conditional distribution +p(yt|at). We discuss this further in Section 3.2. +Consider the case when we have observations available +at discrete timesteps t0, . . . , tT . Following the graphical +model in Fig. 1, the joint distribution over the states z0:T , +the auxiliary variables a0:T , and the observations y0:T fac- +torises as +pθ(z0:T ,a0:T , y0:T ) = +T +� +k=0 +p(yk|ak)p(ak|zk)p(zk|zk−1), +where x0:T denotes the set {xt0, . . . , xtT } and p(z0|z−1) = +p(z0). We model the initial (prior) distribution of the states +as a multivariate Gaussian distribution, +p(z0) = N(z0; µ0, Σ0), +(7) +where µ0 ∈ Rm and Σ0 ⪰ 0 ∈ Rm×m are the mean and +covariance matrix, respectively. The transition distribution +of the states, p(zk|zk−1), follows the dynamics governed by +the SDE in Eq. (1). The conditional emission distributions +of the auxiliary variables and observations are modeled as +multivariate Gaussian distributions given by, +p(ak|zk) = N(ak; Hzk, R), +(8) +p(yk|ak) = N(yk; f µ(ak), f Σ(ak)), +(9) +where H +∈ +Rh×m, R +⪰ +0 +∈ +Rh×h, and f µ +and f Σ are functions parameterized by neural networks +that output the mean and the covariance matrix of the dis- +tribution, respectively. We use θ to denote the parame- +ters of the generative model, including SSM parameters + +Neural Continuous-Discrete State Space Models +{µ0, Σ0, f, Q, G, H, R} and observation emission distri- +bution parameters {f µ, f Σ}. +We propose three variants of NCDSSM, depending on the +parameterization of f and G functions in Eq. (1) that govern +the dynamics of the state: +Linear time-invariant dynamics is obtained by parameter- +izing f and G as +f(zt, t) = Fzt +and +G(z, t) = I, +(10) +respectively, where F ∈ Rm×m is a Markov transition ma- +trix and I is the m-dimensional identity matrix. In this case, +Eqs. (4) and (5) become exact and the ODEs in Eq. (5) can +be solved analytically using matrix exponentials (cf. Ap- +pendix B.2). Unfortunately, the restriction of linear dynam- +ics is limiting for practical applications. We denote this +linear time-invariant variant as NCDSSM-LTI. +Non-linear dynamics is obtained by parameterizing f and +G using neural networks. With sufficiently powerful neural +networks, this parameterization is flexible enough to model +arbitrary non-linear dynamics. However, the neural net- +works need to be carefully regularized (cf. Appendix B.3) +to ensure optimization and inference stability. Inference +in this variant also requires computation of the Jacobian +of a neural network for solving Eq. (5). We denote this +non-linear variant as NCDSSM-NL. +Locally-linear dynamics is obtained by parameterizing f +and G as +f(zt, t) = F(zt)zt +and +G(z, t) = I, +(11) +respectively, where the matrix F(zt) ∈ Rm×m is given by +a convex combination of K base matrices {F(j)}K +j=1, +F(zt) = +K +� +j=1 +α(j)(zt)F(j), +(12) +and the combination weights, α(zt), are given by +α(zt) = softmax(g(zt)), +(13) +where g is a neural network. +Such parameterizations +smoothly interpolate between linear SSMs and can be +viewed as “soft” switching SSMs. Locally-linear dynam- +ics has previously been used for discrete-time SSMs (Karl +et al., 2016; Klushyn et al., 2021); we extend it to the con- +tinuous time setting by evaluating Eq. (12) continuously in +time. Unlike non-linear dynamics, this parameterization +does not require careful regularization and its flexibility can +be controlled by choosing the number of base matrices, K. +Furthermore, the Jacobian of f in Eq. (5) can be approxi- +mated as F(mt), avoiding the expensive computation of the +Jacobian of a neural network (Klushyn et al., 2021). We +denote this locally-linear variant as NCDSSM-LL. +3.2. Inference +Exact inference in the model described above is intractable +when the dynamics is non-linear and/or the observation +emission distribution, p(yk|ak), is modeled by arbitrary +non-linear functions. In the modern deep learning con- +text, a straightforward approach would be to approximate +the posterior distribution over the states and auxiliary vari- +ables, q(z0:T , a0:T |y0:T ), using recurrent neural networks +(e.g., using ODE-RNNs when modeling in continuous time). +However, such parameterizations have been shown to lead +to poor optimization of the transition model in discrete-time +SSMs, leading to inaccurate learning of system dynam- +ics (Klushyn et al., 2021). Alternatively, directly applying +continuous-discrete inference techniques to non-linear emis- +sion models requires computation of Jacobian matrices and +inverses of d × d matrices (cf. Eq. 6) which scales poorly +with the data dimensionality. +The introduction of linear-Gaussian auxiliary variables of- +fers a middle ground between the two options above. It +allows efficient use of continuous-discrete Bayesian infer- +ence techniques for the inference of states, avoiding fully +amortized inference for auxiliary variables and states. Con- +cretely, we split our inference procedure into two inference +steps: (i) for auxiliary variables and (ii) for states. +Inference for auxiliary variables. +We perform amor- +tized inference for the auxiliary variables, factorizing the +variational distribution as, +qφ(a0:T |y0:T ) = +T +� +k=0 +q(ak|yk), +(14) +where q(ak|yk) = N(ak; f µ +φ (yk), f Σ +φ (yk)) and f µ +φ , f Σ +φ are +neural networks. This can be viewed as the recognition +network in a variational autoencoder, per timestep. This +flexible factorization permits use of arbitrary recognition +networks, thereby allowing arbitrary non-linear emission +distributions, p(yk|ak). +Inference for states. +Given the variational distribu- +tion qφ(a0:T |y0:T ) in Eq. (14), we can draw samples, +˜a0:T ∼ qφ(a0:T |y0:T ), from it. Viewing ˜a0:T as pseudo- +observations, we treat the remaining SSM (i.e., the states +and auxiliary variables) separately. Specifically, conditioned +on the auxiliary variables, ˜ +Aτ = {˜ak : tk ≤ τ}, we can +answer inference queries over the states zt in continuous +time. This does not require additional inference networks +and can be performed only using the generative model via +classic continuous-discrete Bayesian inference techniques +in Section 2. To infer the filtered density, pt(zt| ˜ +At), we can +use Eq. (5) for the prediction step and Eq. (6) for the update +step, replacing yk by ˜ak. Similarly, we can use Eq. (23) +(Appendix) to infer the smoothed density, pt(zt| ˜ +AT ). + +Neural Continuous-Discrete State Space Models +As the inference of states is now conditioned on auxiliary +variables, only the inversion of h × h matrices is required +which is computationally feasible as ak generally has lower +dimensionality than yk. Notably, this inference scheme +does not require posterior SDEs for inference (as in other +SDE-based models; cf. Section 4) and does not suffer from +poor optimization of the transition model as we employ the +(generative) transition model for the inference of states. +3.3. Learning +The parameters of the generative model {θ} and the infer- +ence network {φ} can be jointly optimized by maximizing +the following evidence lower bound (ELBO) of the log- +likelihood, log pθ(y0:T ), +log pθ(y0:T ) +≥ Eqφ(a0:T |y0:T ) +� +log +�T +k=0 pθ(yk|ak)pθ(a0:T ) +�T +k=0 qφ(ak|yk) +� +=: LELBO(θ, φ). +(15) +The distributions pθ(yk|ak) and qφ(ak|yk) in LELBO are +immediately available via the emission and recognition +networks, respectively. What remains is the computation +of pθ(a0:T ). Fortunately, pθ(a0:T ) can be computed as a +byproduct of the inference (filtering) procedure described +in Section 3.2. The distribution factorizes as +p(a0:T ) = p(a0) +T +� +k=1 +p(ak|Atk−1), +where p(ak|Atk−1) = N(ak; Hm− +k , Sk), and m− +k and Sk +are computed during the prediction and update steps, respec- +tively. The pθ(a0:T ) term can be viewed as a “prior” over +the auxiliary variables. However, unlike the fixed standard +Gaussian prior in a vanilla variational autoencoder, pθ(a0:T ) +is a learned prior given by the marginalization of the states, +zt, from the underlying SSM. Algorithm 1 summarizes the +learning algorithm for a single time series; in practice, mini- +batches of time series are sampled from the dataset. +3.4. Stable Implementation +A naive implementation of the numerical integration of +ODEs (Eqs. 5 and 23) and other operations (Eq. 6) re- +sults in unstable training and crashing due to violation +of the positive definite constraint for the covariance ma- +trices. Commonly employed tricks such as symmetrization, +P = (P + P⊤)/2, and addition of a small positive number +(ϵ) to the diagonal elements, P = P + ϵI, did not solve +these training issues. Therefore, we implemented our al- +gorithms in terms of square root (Cholesky) factors, which +proved critical to the stable training of NCDSSM. Several +square root factors’ based inference algorithms have been +Algorithm 1 Learning in Neural Continuous-Discrete State +Space Models +Require: Observations {(yk, tk)}T +k=0 and model parame- +ters {θ, φ}. +1: repeat +2: +Compute qφ(a0:T |y0:T ) using Eq. (14). +3: +Sample ˜a0:T ∼ qφ(a0:T |y0:T ). +4: +, log pθ(˜a0:T ) ← FILTER(˜a0:T , t0:T ; θ) +▷ cf. Algorithm 3 (Appendix) for FILTER. +5: +Compute �T +k=0 pθ(yk|˜ak) using Eq. (9). +6: +Optimize LELBO(θ, φ). +7: until end of training. +previously proposed (Zonov, 2019; Jorgensen et al., 2007; +Kailath et al., 2000, Ch. 12). In the following, we discuss our +implementation which is based on Zonov (2019). Further +discussion on implementation stability, particularly in the +case of non-linear dynamics, can be found in Appendix B.3. +We begin with a lemma that shows that the square root factor +of the sum of two matrices with square root factors can be +computed using QR decomposition. +Lemma 3.1. Let A and B be two n × n matrices with +square root factors A1/2 and B1/2, respectively. The matrix +C = A + B also has a square root factor, C1/2, given by +Θ, +� +C1/2 +0n×n +�⊤ = QR +�� +A1/2 +B1/2�⊤� +, +where Θ is the orthogonal Q matrix given by QR decompo- +sition and 0n×n is an n × n matrix of zeros. +Prediction step. +The solution of matrix differential equa- +tions of the form in Eq. (5b) — called Lyapunov differential +equations — over [t0, t1] is given by (Abou-Kandil et al., +2012, Corollary 1.1.6) +Pt1 = Φt1Pt0Φ⊤ +t1 + +� t1 +t0 +ΦtDtΦ⊤ +t dt, +(16) +where Φt, called the fundamental matrix, is defined by +dΦt +dt += Fz(mt, t)Φt and Φt0 = I. +(17) +This initial value problem can be solved using an off-the- +shelf ODE solver. Let { ˜Φ1 = I, ˜Φ2, . . . , ˜Φn} be intermedi- +ate solutions of Eq. (17) given by an ODE solver with step +size η, Eq. (16) can be approximated as +Pt1 ≈ ˜ΦnPt0 ˜Φ⊤ +n ++ η +2 +� +˜Φ1D1 ˜Φ⊤ +1 + 2 ˜Φ2D2 ˜Φ⊤ +2 + · · · + ˜ΦnDn ˜Φ⊤ +n +� +. +(18) +The additions in Eq. (18) are performed using Lemma 3.1 +with square root factors ˜ΦnP1/2 +t0 +and { ˜ΦjD1/2 +j +}n +j=1. + +Neural Continuous-Discrete State Space Models +Update step. +Using similar arguments as in the proof of +Lemma 3.1 (cf. Appendix B.3 for details), the update step +(Eq. 6) can be performed by the QR decomposition of the +square root factor +�R1/2 +H(P− +k )1/2 +0m×d +(P− +k )1/2 +�⊤ +. +(19) +Let +�X +0 +Y +Z +�⊤ +be the upper triangular R matrix obtained +from the QR decomposition of (19). The square root factor +of the updated covariance matrix, P1/2 +k +, and the Kalman +gain matrix, Kk, are then given by P1/2 +k += Z and Kk = +YX−1, respectively. +4. Related Work +Several previous works (Chung et al., 2015; Krishnan et al., +2015; Karl et al., 2016; Krishnan et al., 2017; Doerr et al., +2018) have proposed SSM-like models for discrete-time +sequential data, trained via amortized variational inference. +Unlike NCDSSM, these models approximate sequential +Bayesian inference (i.e., filtering and smoothing) via deter- +ministic RNNs and are limited to the discrete time setting. +Bayesian inference for a subset of latent variables com- +bined with amortized inference for others has previously +been studied for SSMs. SNLDS (Dong et al., 2020) and +REDSDS (Ansari et al., 2021) perform amortized infer- +ence for the states and exact inference for discrete random +variables (switches and duration counts) in switching SSMs. +KVAE (Fraccaro et al., 2017), EKVAE (Klushyn et al., 2021) +and ARSGLS (Kurle et al., 2020) introduce auxiliary vari- +ables and perform classic Bayesian filtering and smoothing +for the state variables, similar to NCDSSM. However, these +models use specific parameterizations of state dynamics and +operate on discrete-time sequential data. In contrast, we +propose a general framework for continuous-time modeling +of irregularly-sampled time series with multiple possible +parameterizations of the dynamics. +Since the introduction of NeuralODE (Chen et al., 2018), +various models based on neural differential equations have +been proposed for continuous-time modeling of time se- +ries (Rubanova et al., 2019; De Brouwer et al., 2019; Yildiz +et al., 2019; Li et al., 2020; Liu et al., 2020; Kidger et al., +2020; Solin et al., 2021). Amongst these, we focus on the +latent variable models as they are closely related to SSMs. +LatentODE (Rubanova et al., 2019) encodes the entire con- +text window into an initial state using an encoder (e.g., +ODE-RNN) and uses a NeuralODE to model latent dynam- +ics. ODE2VAE (Yildiz et al., 2019) decomposes the latent +state into position and velocity components to explicitly +model the acceleration and parameterize the ODE dynam- +ics by Bayesian neural networks to account for uncertainty. +LatentSDE (Li et al., 2020) uses a posterior SDE in the la- +tent space to infer the latent dynamics together with a prior +(generative) SDE in a variational setup. Solin et al. (2021) +proposed a variant of LatentSDE trained by exploiting the +Gaussian assumed density approximation of the non-linear +SDE. VSDN (Liu et al., 2020) uses ODE-RNNs to provide +historical information about the time series to the SDE drift +and diffusion functions. These existing ODE-based mod- +els either use deterministic latent dynamics and/or create +a restrictive bottleneck by encoding the entire time series +into an initial state. The SDE-based models require pos- +terior SDEs to infer the dynamics; new observations are +incorporated in an ad-hoc fashion, potentially resulting in +a disparity between posterior and generative dynamics and +a non-Markovian state space. Contrary to previous models, +NCDSSM uses (i) stochastic Markovian dynamics, (ii) in- +corporates observations via a principled Bayesian update, +(iii) disentangles recognition from dynamics using auxiliary +variables and (iv) performs continuous-discrete Bayesian +inference for the state variables (dynamics), obviating the +need for posterior SDEs. +5. Experiments +In this section, we present empirical results on time series +imputation and forecasting tasks. Our primary focus was +to investigate the models’ ability to capture the underly- +ing dynamics of the time series, gauged by the accuracy of +long-term forecasts beyond the training context. We exper- +imented with the three variants of our model described in +Section 3.1: NCDSSM-LTI, NCDSSM-NL, and NCDSSM- +LL. Our main baselines were LatentODE and LatentSDE, +two popular continuous-time latent variable models with +deterministic and stochastic dynamics, respectively. We +also compared NCDSSM against several other baselines +for individual experiments. We first discuss experiment +results on the low-dimensional bouncing ball and damped +pendulum datasets, then move to higher dimensional set- +tings: walking sequences from the CMU Motion Capture +(MoCap) dataset, the USHCN daily climate dataset, and two +32x32 dimensional video datasets (Box and Pong). +5.1. Bouncing Ball and Damped Pendulum +The bouncing ball and damped pendulum datasets have +known ground truth dynamics, which facilitates quality as- +sessment of the dynamics learned by a given model. For +details on these datasets, please refer to Appendix C.1. +In brief, the univariate bouncing ball dataset exhibits +piecewise-linear dynamics, whilst bivariate damped pen- +dulum dataset (Karl et al., 2016; Kurle et al., 2020) exhibits +non-linear latent dynamics. +We trained all the models on 10s/5s sequences (with a dis- +cretization of 0.1s) for bouncing ball/damped pendulum +with 0%, 30%, 50% and 80% timesteps missing at ran- + +Neural Continuous-Discrete State Space Models +Table 1. Imputation and forecasting results for bouncing ball and damped pendulum datasets averaged over 50 sample trajectories. Mean +± standard deviation are computed over 5 independent runs. +Dataset +Model +Imputation MSE (↓) (% Missing) +Forecast MSE (↓) (% Missing) +30% +50% +80% +0% +30% +50% +80% +Bouncing Ball +LatentODE (Rubanova et al., 2019) +0.007 ± 0.000 +0.008 ± 0.001 +0.011 ± 0.000 +0.386 ± 0.025 +0.489 ± 0.133 +0.422 ± 0.053 +0.412 ± 0.048 +LatentSDE (Li et al., 2020) +0.006 ± 0.000 +0.007 ± 0.000 +0.011 ± 0.001 +0.408 ± 0.043 +1.209 ± 1.115 +1.567 ± 2.263 +0.352 ± 0.077 +NCDSSM-LTI +0.020 ± 0.001 +0.026 ± 0.001 +0.067 ± 0.002 +0.592 ± 0.106 +0.557 ± 0.014 +0.556 ± 0.025 +0.555 ± 0.022 +NCDSSM-NL +0.006 ± 0.000 +0.006 ± 0.000 +0.007 ± 0.000 +0.037 ± 0.018 +0.036 ± 0.007 +0.041 ± 0.007 +0.115 ± 0.029 +NCDSSM-LL +0.006 ± 0.000 +0.006 ± 0.000 +0.008 ± 0.001 +0.037 ± 0.028 +0.034 ± 0.016 +0.049 ± 0.034 +0.076 ± 0.017 +Damped +Pendulum +LatentODE (Rubanova et al., 2019) +0.151 ± 0.002 +0.155 ± 0.002 +0.206 ± 0.013 +0.097 ± 0.042 +0.117 ± 0.001 +0.119 ± 0.001 +0.148 ± 0.007 +LatentSDE (Li et al., 2020) +0.092 ± 0.076 +0.148 ± 0.001 +0.229 ± 0.001 +0.046 ± 0.046 +0.084 ± 0.058 +0.147 ± 0.020 +0.357 ± 0.096 +NCDSSM-LTI +0.036 ± 0.001 +0.057 ± 0.001 +0.120 ± 0.002 +0.282 ± 0.084 +1.017 ± 1.363 +1.527 ± 1.440 +0.231 ± 0.050 +NCDSSM-NL +0.008 ± 0.000 +0.011 ± 0.000 +0.033 ± 0.002 +0.011 ± 0.004 +0.011 ± 0.003 +0.012 ± 0.003 +0.034 ± 0.019 +NCDSSM-LL +0.008 ± 0.000 +0.011 ± 0.000 +0.037 ± 0.003 +0.025 ± 0.030 +0.010 ± 0.001 +0.020 ± 0.008 +0.055 ± 0.007 +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +2 +4 +6 +8 +10 +12 +14 +Time +NCDSSM-LL +Figure 2. Predictions from different models on the damped pen- +dulum dataset in the 80% missing data setting. The ground truth +is shown using dashed lines with observed points in the context +window (gray shaded region) shown as filled circles. The verti- +cal dashed gray line marks the beginning of the forecast horizon. +Solid lines indicate median predictions with 90% prediction inter- +vals shaded around them. The purple and orange colors indicate +observation dimensions. NCDSSM-NL and NCDSSM-LL are +significantly better at forecasting compared to the baselines. +dom to simulate irregularly-sampled data. The models were +evaluated on imputation of the missing timesteps and fore- +casts of 20s/10s beyond the training regime for bouncing +ball/damped pendulum. +Table 1 reports the imputation and forecast mean squared +error (MSE) for different missing data settings. In summary, +the NCDSSM models with non-linear and locally-linear +dynamics (NCDSSM-NL and NCDSSM-LL) perform well +across datasets, settings, and random initializations, signifi- +cantly outperforming the baselines. Furthermore, for these +low-dimensional datasets, learning latent representations in +the form of auxiliary variables is not required and we can +set the recognition and emission functions in Eq. (14) and +Eq. (9) to identity functions. This results in NCDSSM mod- +els requiring 2-5 times fewer parameters than LatentODE +and LatentSDE (cf. Table 5 in the Appendix). +Fig. 2 shows example predictions from the best performing +run of every model for 80% missing data for the pendu- +lum (cf. Appendix D for other settings). NCDSSM-NL +and NCDSSM-LL generates far better predictions both in- +side and outside the context window compared to the base- +lines. Ordinary least squares (OLS) goodness-of-fit results +Table 2. Forecasting results for the CMU MoCap walking dataset +averaged over 50 sample trajectories with 95% prediction interval +based on the t-statistic in parentheses. †Baseline results from Solin +et al. (2021). +Model +MSE (↓) +†Setup 1 +Setup 2 +npODE (Heinonen et al., 2018) +22.96 +– +NeuralODE (Chen et al., 2018) +22.49 (0.88) +– +ODE2VAE-KL (Yildiz et al., 2019) +8.09 (1.95) +– +LatentODE (Rubanova et al., 2019) +5.98 (0.28) +31.62 (0.05) +LatentSDE (Li et al., 2020) +4.03 (0.20) +9.52 (0.21) +LatentApproxSDE (Solin et al., 2021) +7.55 (0.05) +– +NCDSSM-LTI +13.90 (0.02) +5.22 (0.02) +NCDSSM-NL +5.69 (0.01) +6.73 (0.02) +NCDSSM-LL +9.96 (0.01) +4.74 (0.01) +in Table 6 (Appendix) suggest that this performance can +be attributed to our models having learnt the correct dy- +namics; latent states from NCDSSM-NL and NCDSSM-LL +are highly correlated with the ground truth angle and angu- +lar velocity for all missingness scenarios. In other words, +the models have learnt a Markovian state space which is +informative about the dynamics at a specific time. +5.2. CMU Motion Capture (Walking) +This dataset comprises walking sequences of subject 35 +from the CMU MoCap database containing joint angles of +subjects performing everyday activities. We used a prepro- +cessed version of the dataset from Yildiz et al. (2019) that +has 23 50-dimensional sequences of length 300. We tested +the models under two setups. Setup 1 (Yildiz et al., 2019; Li +et al., 2020; Solin et al., 2021) involves training on complete +300 timestep sequences from the training set and using only +the first 3 timesteps as context to predict the remaining 297 +timesteps during test time. Although challenging, this setup +does not evaluate the model’s performance beyond the train- +ing context. Thus, we propose Setup 2 in which we train the +model only using the first 200 timesteps. During test time, +we give the first 100 timesteps as context and predict the +remaining 200 timesteps. +The forecast MSE results for both setups are reported in + +Neural Continuous-Discrete State Space Models +Figure 3. Sample predictions from NCDSSM-NL on the Pong dataset. The top row is the ground truth with some missing observations in +the context window. The next two rows show trajectories sampled from NCDSSM-NL upto 20 forecast steps. NCDSSM-NL is able to +both impute and forecast accurately. Best viewed zoomed-in on a computer. More examples in Appendix D. +Table 3. Forecasting results for the USHCN climate dataset. Mean +± standard deviation are computed over 5 folds as described in +De Brouwer et al. (2019). †Results from De Brouwer et al. (2019). +‡Results from Liu et al. (2020). +Model +MSE (↓) +†NeuralODE-VAE (Chen et al., 2018) +0.83 ± 0.10 +†SequentialVAE (Krishnan et al., 2015) +0.83 ± 0.07 +†GRU-D (Che et al., 2018) +0.53 ± 0.06 +†T-LSTM (Baytas et al., 2017) +0.59 ± 0.11 +†GRUODE-B (De Brouwer et al., 2019) +0.43 ± 0.07 +‡ODE-RNN (Rubanova et al., 2019) +0.39 ± 0.06 +‡LatentODE (Rubanova et al., 2019) +0.77 ± 0.09 +‡LatentSDE (Li et al., 2020) +0.74 ± 0.11 +‡VSDN-F (IWAE) (Liu et al., 2020) +0.37 ± 0.06 +NCDSSM-LTI +0.38 ± 0.07 +NCDSSM-NL +0.34 ± 0.06 +NCDSSM-LL +0.37 ± 0.06 +Table 2. NCDSSM-NL performs better than all baselines +except LatentSDE on Setup 1 while NCDSSM models per- +form significantly better than baselines on Setup 2. This +showcases NCDSSM’s ability to correctly model the latent +dynamics, aiding accurate long-term predictions beyond the +training context. +5.3. USHCN Climate Indicators +We evaluated the models on the United States Historical Cli- +matology Network (USHCN) dataset that comprises mea- +surements of five climate indicators across the United States. +The preprocessed version of this dataset from De Brouwer +et al. (2019) contains sporadic time series (i.e., with mea- +surements missing both over the time and feature axes) +from 1,114 meteorological stations over 4 years. Follow- +ing De Brouwer et al. (2019), we trained the models on +sequences from the training stations and evaluated them on +the task of predicting the next 3 measurements given the +first 3 years as context from the held-out test stations. The +results in Table 3 show that NCDSSM-NL outperforms all +the baselines with NCDSSM-LTI and NCDSSM-LL per- +forming better than most of the baselines. +5.4. Pymunk Physical Environments +Finally, we evaluated the models on two high-dimensional +(video) datasets of physical environments used in Frac- +caro et al. (2017), simulated using the Pymunk Physics +engine (Blomqvist, 2022): Box and Pong. The box dataset +consists of videos of a ball moving in a 2-dimensional box +and the pong dataset consists of videos of a Pong-like envi- +ronment where two paddles move to keep a ball in the frame +at all times. Each frame is a 32x32 binary image. +Table 4. Forecasting results for the Box and Pong datasets averaged +over 16 sample trajectories. +Model +EMD (↓) +Box +Pong +LatentODE (Rubanova et al., 2019) +1.792 +4.543 +LatentSDE (Li et al., 2020) +1.925 +3.505 +NCDSSM-LTI +1.685 +3.265 +NCDSSM-NL +0.692 +1.714 +NCDSSM-LL +0.632 +1.891 +We trained the models on sequences of 20 frames with 20% +of these frames randomly dropped. At test time, the mod- +els were evaluated on forecasts of 40 frames beyond the +training context. For evaluation, we treat each image as +a probability distribution on the XY-plane and report the +earth mover’s distance (EMD) between the ground truth +and predicted images, averaged over the forecast horizon, +in Table 4. NCDSSM-NL and NCDSSM-LL significantly +outperform baseline models on both box and pong datasets. +Fig. 6 (Appendix) shows the variation of EMD against time +for different models. In the context window (0-2s), all mod- +els have EMD close to 0; however, in the forecast horizon +(2-6s), the EMD rises rapidly and irregularly for LatentODE +and LatentSDE but does so gradually for NCDSSM-NL +and NCDSSM-LL. This indicates that the dynamics mod- +els learned by NCDSSM-NL and NCDSSM-LL are both +accurate and robust. +Qualitatively, both NCDSSM-LL and NCDSSM-NL cor- +rectly impute the missing frames and the forecasts generated +by them are similar to ground truth. Fig. 3 shows sample +predictions for the pong dataset generated by NCDSSM-NL. +In contrast, other models only impute the missing frames +correctly, failing to generate accurate forecasts (cf. Ap- +pendix D). +6. Conclusion +In this work, we proposed a model for continuous-time +modeling of irregularly-sampled time series. NCDSSM +improves continuous-discrete SSMs with neural network- +based parameterizations of dynamics, and modern inference +and learning techniques. Through the introduction of auxil- +iary variables, NCDSSM enables efficient modeling of high- +dimensional time series while allowing accurate continuous- +discrete Bayesian inference of the dynamic states. Experi- +ments on a variety of low- and high-dimensional datasets +show that NCDSSM outperforms existing models on time +series imputation and forecasting tasks. + +Context Window +Forecast Horizon ++II ++ +Ground Truth I ++ ++II ++I +Samples ++Neural Continuous-Discrete State Space Models +Acknowledgements +This research is supported in part by by the National Re- +search Foundation (NRF), Singapore and DSO National +Laboratories under the AI Singapore Program (Award Num- +ber: AISG2-RP-2020-016). We thank Richard Kurle, Fabian +Falck, Alexej Klushyn, and Marcel Kollovieh for helpful +discussions and feedback. +References +Abou-Kandil, H., Freiling, G., Ionescu, V., and Jank, G. +Matrix Riccati equations in control and systems theory. +Birkh¨auser, 2012. +Anderson, B. D. Fixed interval smoothing for nonlinear +continuous time systems. Information and Control, 20 +(3):294–300, 1972. +Ansari, A. F., Benidis, K., Kurle, R., Turkmen, A. C., Soh, +H., Smola, A. 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Kalman filter based sensor placement for Burgers +equation. Master’s thesis, University of Waterloo, 2019. + +Neural Continuous-Discrete State Space Models +A. Proofs +A.1. Proof of Lemma 3.1 +Lemma A.1. Let A and B be two n × n matrices with square root factors A1/2 and B1/2, respectively. The matrix +C = A + B also has a square root factor, C1/2, given by +Θ, +� +C1/2 +0n×n +�⊤ = QR +�� +A1/2 +B1/2�⊤� +, +where Θ is the orthogonal Q matrix given by QR decomposition and 0n×n is an n × n matrix of zeros. +Proof. Our proof is based on Zonov (2019, Thm. 3.2). Consider the square root factor +Y = +� +A1/2 +B1/2� +. +Clearly, C = YY⊤; however, we also have C = YΘΘ⊤Y⊤, for any orthogonal matrix Θ. Thus, YΘ is also a square root +factor of C. Let Θ be an orthogonal matrix such that +YΘ = +�X +0n×n +� +, +(20) +where X is an n × n lower triangular matrix. This implies that X is a square root factor of C. +From Eq. (20), we further have the following, +Y = +�X +0n×n +� +Θ⊤, +(21) +Y⊤ = Θ +� +X⊤ +0n×n +� +, +(22) +where we post-multiply by Θ⊤ in the first step and use the fact that ΘΘ⊤ = I, and transpose both sides in the second step. +We have thus expressed Y⊤ as the product of an orthogonal matrix, Θ, and an upper triangular matrix, +�X +0n×n +�⊤. Such +a factorization can be performed by QR decomposition. Thus, we can compute the square root factor C1/2 = X via the QR +decomposition of +� +A1/2 +B1/2�⊤. +Algorithm 2 Sum of Square Root Factors +1: function SUMMATRIXSQRTS(A1/2, B1/2) +2: +, +� +C1/2 +0n×n +�⊤ = QR +�� +A1/2 +B1/2�⊤� +3: +return C1/2 +4: end function +B. Technical Details +B.1. Continuous-Discrete Bayesian Smoothing +Several approximate smoothing procedures based on Gaussian assumed density approximation have been proposed in the +literature. We refer the reader to S¨arkk¨a & Sarmavuori (2013) for an excellent review of continuous-discrete smoothers. In +the following, we discuss the Type II extended RTS smoother which is linear in the smoothing solution. According to this +smoother, the mean, ms +t, and covariance matrix, Ps +t, of the Gaussian approximation to the smoothing density, pt(zt|YT ), +follow the backward ODEs, +dms +t +dt += f(mt, t) + C(mt, t)(ms +t − mt), +(23a) +dPs +t +dt += C(mt, t)Ps +t + Ps +tC⊤(mt, t) − D(mt, t), +(23b) +where (mt, Pt) is the filtering solution given by Eq. (5), C(mt, t) = Fz(mt, t) + D(mt, t)P−1 +t +and backward means that +the ODEs are solved backwards in time from the filtering solution (ms +T = mT , Ps +T = PT ). + +Neural Continuous-Discrete State Space Models +B.2. Algorithms +In this section, we discuss the stable filtering and smoothing algorithms used in NCDSSM. +Algorithm 2 provides a utility function — SUMMATRIXSQRTS — that uses Lemma 3.1 to compute the square root +factor of the sum of two matrices with square root factors. The square root factor version of the continuous-discrete +Bayesian filtering algorithm is given in Algorithm 3. Note that the PREDICT step for linear time-invariant dynamics can be +performed analytically using matrix exponentials (S¨arkk¨a & Solin, 2019, Ch. 6). We used the analytic solver for some of our +experiments. The Type II RTS smoothing algorithm (Algorithm 4) takes the filtered distributions as input and computes +the smoothed distribution at every filtered timestep. To compute the smoothed distribution between observed timesteps, +Algorithm 3 Continuous-Discrete Bayesian Filtering +1: function UPDATE(ak, m− +k , (P− +k )1/2; H, R) +2: +R1/2 ← cholesky(R) +3: +A ← +� +R1/2 +H(P− +k )1/2 +0m×d +(P− +k )1/2 +� +4: +, +�X +0 +Y +Z +�⊤ +← QR(A⊤) +5: +Kk ← YX−1 +6: +ˆak ← Hm− +k +7: +mk ← m− +k + Kk(ak − ˆak) +8: +P1/2 +k +← Z +9: +S1/2 +k +← X +10: +return mk, P1/2 +k +, ˆak, S1/2 +k +11: end function +12: function PREDICT(mk, P1/2 +k +, tk, tk+1; f, Q, G) +13: +Φ1 ← I +14: +{ ˜mj}n +j=1 ← odeint +� dmt +dt = f(mt, t), mk, [τ1 = tk, . . . , τn = tk+1] +� +15: +{ ˜Φj}n +j=1 ← odeint +� dΦt +dt = Fz(mt, t)Φt, Φ1, [τ1 = tk, . . . , τn = tk+1] +� +▷ the two coupled ODEs above are solved together. +16: +m− +k+1 ← ˜mn +17: +(P− +k+1)1/2 ← REDUCESUMMATRIXSQRTS( +� +˜ΦnP1/2 +k +, +� η +2 ˜Φ1D1/2 +τ1 , √η ˜Φ2Dτ 1/2 +2 +, . . . , +� η +2 ˜ΦnD1/2 +τn +� +) +▷ REDUCESUMMATRIXSQRTS uses the SUMMATRIXSQRTS function in Algorithm 2, reducing it over the list. +18: +return m− +k+1, (P− +k+1)1/2 +19: end function +20: function FILTER(a0:T , t0:T ; θ) +21: +µ0, Σ0, f, Q, G, H, R ← θ +22: +m− +0 , (P− +0 )1/2 ← µ0, cholesky(Σ0) +23: +ℓ ← 0 +24: +for i ← 0, T do +25: +mi, P1/2 +i +, ˆai, S1/2 +i +← UPDATE(ai, m− +i , (P− +i )1/2; H, R) +26: +ℓ ← ℓ + log N(ai; ˆai, Si) +27: +if i = T then +28: +break +29: +end if +30: +m− +i+1, (P− +i+1)1/2 ← PREDICT(mi, P1/2 +i +, ti, ti+1; f, Q, G) +31: +end for +32: +return {mi, P1/2 +i +}T +i=0, ℓ +33: end function + +Neural Continuous-Discrete State Space Models +we cache the filtered distributions at these timesteps and provide them to the SMOOTH function together with the filtered +distributions at observed timesteps. +Algorithm 4 Continuous-Discrete Type II Extended RTS Smoothing +1: function SMOOTHSTEP(ms +k, (Ps +k)1/2, mk, P1/2 +k +, tk, tk−1; f, Q, G) +2: +Φs +1 ← I +3: +{ ˜ms +j}n +j=1 ← odeint +� +dms +t +dt += f(mk, t) + C(mk, t)(ms +t − mk), ms +k, [τ1 = tk, . . . , τn = tk−1] +� +4: +{ ˜Φs +j}n +j=1 ← odeint +� +dΦs +t +dt = C(mk, t)Φs +t, Φs +1, [τ1 = tk, . . . , τn = tk−1] +� +▷ the two coupled ODEs above are solved together. +5: +ms +k−1 ← ˜ms +n +6: +(Ps +k−1)1/2 ← REDUCESUMMATRIXSQRTS( +� +˜Φs +nP1/2 +k +, +� η +2 ˜Φs +1D1/2 +τ1 , √η ˜Φs +2Dτ 1/2 +2 +, . . . , +� η +2 ˜Φs +nD1/2 +τn +� +) +▷ REDUCESUMMATRIXSQRTS uses the SUMMATRIXSQRTS function in Algorithm 2, reducing it over the list. +7: +return ms +k−1, (Ps +k−1)1/2 +8: end function +9: function SMOOTH({mi, P1/2 +i +}T +i=0, t0:T ; θ) +10: +µ0, Σ0, f, Q, G, H, R ← θ +11: +ms +T , (Ps +T )1/2 ← mT , P1/2 +T +12: +for i ← T, 1 do +▷ note the time reversal. +13: +ms +i−1, (Ps +i−1)1/2 ← SMOOTHSTEP(ms +i, (Ps +i)1/2, mi−1, P1/2 +i−1, ti, ti−1; f, Q, G) +14: +end for +15: +return {ms +i, (Ps +i)1/2}T +i=0 +16: end function +B.3. Stable Implementation (Contd.) +Square Root Factor Measurement Update. +In Section 3.4, we discussed a square root factor version of the measurement +update step via the QR decomposition of A⊤, where, +A = +�R1/2 +H(P− +k )1/2 +0m×d +(P− +k )1/2 +� +. +(24) +Let Θ, U = QR(A⊤), where +U = +�X +0 +Y +Z +�⊤ +. +(25) +In the following, we show how P1/2 +k += Z. Our proof is based on Zonov (2019) and we refer the reader to Zonov (2019, +Appendix A) for the proof of Kk = YX−1. +Proof. Note that A is a square root factor of +�R + HP− +k H⊤ +HP− +k +(P− +k )⊤H⊤ +P− +k +� +. +(26) +Matching the terms in (26) with the terms in +UU⊤ = +� +XX⊤ +XY⊤ +YX⊤ +YY⊤ + ZZ⊤ +� +, +(27) + +Neural Continuous-Discrete State Space Models +we get the following equations, +XX⊤ = R + HP− +k H⊤, +(28a) +XY⊤ = HP− +k , +(28b) +YX⊤ = (P− +k )⊤H⊤, +(28c) +YY⊤ + ZZ⊤ = P− +k . +(28d) +From Eq. (28d), we have the following, +YY⊤ + ZZ⊤ = P− +k , +(29a) +ZZ⊤ = P− +k − YY⊤, +(29b) +ZZ⊤ = P− +k − Y(X⊤X−⊤)(X−1X)Y⊤, +(29c) +ZZ⊤ = P− +k − YX⊤(XX⊤)−1XY⊤, +(29d) +where we introduce I = (X⊤X−⊤)(X−1X) in the third step and use the property (XX⊤)−1 = X−⊤X−1 in the last step. +Substituting values from Eq. (28), we get, +ZZ⊤ = P− +k − (P− +k )⊤H⊤S−1 +k HP− +k +(30a) +ZZ⊤ = P− +k − (P− +k )⊤H⊤S−1 +k (SkS−1 +k )HP− +k +(30b) +ZZ⊤ = P− +k − P− +k H⊤S−1 +k SkS−⊤ +k +H(P− +k )⊤ +(30c) +ZZ⊤ = P− +k − KkSkK⊤ +k +(30d) +where we introduce I = SkS−1 +k +in the second step, use the fact that Sk is symmetric in the third step and substitute the +value of Kk from Eq. (6b) in last step. Note that ZZ⊤ = P− +k − KkSkK⊤ +k = Pk; therefore, Z = P1/2 +k +. +Regularizing Non-Linear Dynamics. +We now discuss the techniques we employed to regularize the latent dynamics +in NCDSSM. Particularly in the case of non-linear dynamics (NCDSSM-NL), regularization is critical for stable training. +The drift function, f, was parameterized by an MLP in all our experiments. We experimented with the tanh and softplus +non-linearities. We found that applying the non-linearity after the last layer was important when using tanh. Furthermore, +we also initialized the parameters of the last layer to 0 when using tanh. In the case of experiments with a large time interval +(e.g., MoCap and USHCN), application of spectral normalization (Miyato et al., 2018) along with the softplus non-linearity +proved critical for stable training. In the following, we present our hypothesis on why spectral normalization stabilizes +training. +According to Øksendal (2003, Section 5.2), one of the conditions for the existence of a unique solution of an SDE is the +Lipschitz continuity of the drift function, f. Applying spectral normalization regularizes the neural network to be 1-Lipschitz, +aiding its solvability using numerical methods. However, spectral normalization is even more important in the case of +NCDSSM from a practical perspective — it prevents the numerical explosion of the elements of Φt in the prediction step +(Eq. 17), as discussed below. +Consider the case of a fixed Jacobian matrix Fz in an interval [t1, t2]. In this case, the solution of Eq. (17) is given by +Φt2 = exp (Fz(t2 − t1))Φt1, +(31) +where exp (Fz(t2 − t1)) denotes the matrix exponential. For unregularized drifts, the elements of exp (Fz(t2 − t1)) can +become arbitrarily large. However, in the case of 1-Lipschitz drift functions (as provided by spectral normalization), the +spectral norm of exp (Fz) is bounded by exp(1), as shown in Lemma B.1. This controls the growth rate of the elements of +fundamental matrix, Φt. +Lemma B.1. Let g : Rm → Rm be a 1-Lipschitz function and Jg : Rm → Rm×m be its Jacobian function. Then, +∥ exp(Jg(z))∥2 ≤ exp(1) ∀ z ∈ Rm where ∥ · ∥2 denotes the spectral norm of a matrix. +Proof. The spectral norm of the Jacobian of a K-Lipschitz function is bounded by K. Thus, we have, +∥Jg(z)∥2 ≤ 1 ∀ z ∈ Rm. +(32) + +Neural Continuous-Discrete State Space Models +Using the power series representation of the matrix exponential, +exp(A) = +∞ +� +k=0 +Ak +k! , +we get the following bound on ∥ exp(Jg(z))∥2, +∥ exp(Jg(z))∥2 ≤ +∞ +� +k=0 +���� +Jg(z)k +k! +���� +2 +≤ +∞ +� +k=0 +∥Jg(z)∥k +2 +k! += exp(∥Jg(z)∥2). +(33) +Combining Eq. (33) with Eq. (32), we get, +∥ exp(Jg(z))∥2 ≤ exp(∥Jg(z)∥2) ≤ exp(1), +(34) +which completes the proof. +For the same reasons as discussed above, we initialized the transition matrices in our linear models to be random orthogonal +matrices as orthogonal matrices have spectral norm equal to 1. However, in the case of NCDSSM-LTI on the USHCN +dataset, this initialization was not sufficient during the initial phase of training and we used a random skew-symmetric +matrix instead. The matrix exponential of a skew-symmetric matrix is an orthogonal matrix. We generated a random +skew-symmetric matrix as follows, +F ∼ [N(0, 1)]m×m , +F = +�F − F⊤ +2 +� +. +Fixed Measurement Matrix. +We used a fixed rectangular identity matrix as the auxiliary measurement matrix (H in +Eq. 8) in our bouncing ball, damped pendulum and CMU MoCap (walking) experiments as it lead to improved learning of +dynamics. This parameterization forces the model to learn the static (e.g., position) and dynamic (e.g., velocity) components +in separate elements of the latent state, thereby disentangling them (Klushyn et al., 2021). +B.4. Imputation and Forecasting +In this section, we describe how to perform imputation and forecasting using a trained NCDSSM. +For imputation, the timesteps at which imputation is to be performed are provided to the FILTER function during filtering. +The filtered distributions are then passed to the SMOOTH function and (imputed) samples are drawn from the smoothed +distributions. +For forecasting, filtering is first performed over the context time series. The PREDICT function is then used up to end of the +forecast horizon, starting from the last filtered distribution. Sample forecast trajectories are then drawn from these predicted +distributions. +C. Experiment Details +C.1. Datasets +Bouncing Ball and Damped Pendulum. +The bouncing ball dataset comprises univariate time series of the position of a +ball bouncing between two fixed walls, in the absence of dissipative forces. The initial position, x0, and velocity, v0, of the +ball are chosen at random, as follows, +x0 ∼ U(−1, 1), +(35) +v0 ∼ U(0.05, 0.5) × U{−1, 1}, +(36) +where U(a, b) denotes a uniform distribution on (a, b) and U{c1, . . . , ck} denotes a uniform categorical distribution on +{c1, . . . , ck}. The observed position, yk, is a corrupted version of the true position, xk, +yk ∼ N(xk, 0.052). +(37) + +Neural Continuous-Discrete State Space Models +Collisions with the walls, located at −1 and +1, are assumed to be perfectly elastic, i.e., the sign of the velocity gets flipped +when the ball hits either of the walls. Thus, the ball exhibits piecewise-linear dynamics. We used the Euler integrator with a +step size of 0.1s to simulate the dynamics. The training, validation, and test datasets consist of 5000, 500, and 500 sequences +of length 30s each, respectively. +The damped pendulum dataset (Karl et al., 2016; Kurle et al., 2020) comprises bivariate time series of the XY-coordinates of +a pendulum oscillating in the presence of a damping force. The non-linear latent dynamics of this dataset is given by, +dθt +dt = ωt, +(38) +dωt +dt = −g +l sin(θt) − γ +mωt, +(39) +where θt and ωt are the angle and angular velocity, respectively, and g = 9.81, l = 1, m = 1, and γ = 0.25 are the +acceleration due to gravity, the length of the massless cord of the pendulum, the mass of the pendulum bob, and the damping +coefficient, respectively. The initial angle, θ0, and angular velocity, ω0, of the pendulum are chosen at random, as follows, +θ0 = π + clip (ϵ, −2, 2) , +(40) +ω0 = 4 × clip (ϵ, −2, 2) , +(41) +where ϵ ∼ N(0, 1) and clip(x, a, b) denotes clipping the value of x between a and b. The observations are Cartesian +coordinates of the pendulum’s bob with additive Gaussian noise, N(0, 0.052). We used the RK4 integrator to simulate +the latent dynamics with a step size of 0.1s. The training, validation, and test datasets consist of 5000, 1000, and 1000 +sequences of length 15s each, respectively. +CMU Motion Capture (Walking). +The CMU Motion Capture database1 comprises time series of joint angles of human +subjects performing everyday activities, e.g., walking, running, and dancing. We used walking sequences of subject 35 +from this database for our experiments. A preprocessed version of this dataset from Yildiz et al. (2019) consists of 23 +50-dimensional sequences of 300 timesteps each, split into 16 training, 3 validation and 4 test sequences. +USHCN Climate Indicators. +The USHCN Climate dataset2 consists of measurements of five climate indicators — +precipitation, snowfall, snow depth, minimum temperature, and maximum temperature — across the United States. The +preprocessed version of this dataset from De Brouwer et al. (2019) contains sporadic time series from 1,114 meteorological +stations with a total of 386,068 unique observations over 4 years, between 1996 and 2000. The timestamps are scaled to lie +in [0, 200]. The 1,114 stations are split into 5 folds of 70% training, 20% validation, and 10% test stations, respectively. +Pymunk Physical Environments. +The Pymunk physical environments datasets are video datasets of physical environ- +ments simulated using the Pymunk Physics engine. We used two environments proposed in Fraccaro et al. (2017): Box and +Pong. Each frame of these videos is a 32 × 32 binary image. The Box dataset consists of videos of a ball moving inside +a 2-dimensional box with perfectly elastic collisions with the walls of the box. The Pong dataset consists of videos of a +Pong-like environment with a ball and two paddles that move to keep the ball inside the frame. Both datasets consist of +5000 training, 100 validation, and 1000 test videos with 60 frames each. We refer the reader to Fraccaro et al. (2017) for +further details on how these datasets are generated3. +C.2. Training and Evaluation Setups +Bouncing Ball and Damped Pendulum. +We trained all the models on the first 10s/5s of the sequences (i.e., 100/50 steps) +from the training dataset for the bouncing ball/damped pendulum datasets. We randomly dropped 30%, 50%, and 80% of +the training steps for the missing-data experiments. For evaluation, we report the MSE over the missing (for imputation) and +the next 200/100 timesteps (for forecast) for the bouncing ball/damped pendulum test datasets. The MSE was averaged over +5 independent runs for 50 sample trajectories. +1The original CMU MoCap database is available at: http://mocap.cs.cmu.edu. +2The original USHCN Climate dataset is available at: https://cdiac.ess-dive.lbl.gov/ftp/ushcn daily/. +3The scripts for generating Pymunk datasets are available at: https://github.com/simonkamronn/kvae. + +Neural Continuous-Discrete State Space Models +CMU Motion Capture (Walking). +For Setup 1, we trained NCDSSM models on complete 300-timestep sequences from +the training set. During test time, we evaluated the predictive performance on the next 297 steps with a context of the +first 3 steps from the test set. For Setup 2, we trained the models on the first 200 timesteps from sequences in the training +set. During test time, we provided the models with a context of the first 100 timesteps from sequences in the test set and +evaluated their performance on the next 200 timesteps. We report the MSE averaged over 50 sample trajectories together +with 95% prediction interval based on the t-statistic for a single run, as reported in prior works. +USHCN Climate Indicators. +We trained NCDSSM models under the same setup as De Brouwer et al. (2019) using +4 years of observations from the training stations. During test time, we provided the models with the first 3 years of +observations from the test set as context and evaluated their performance on the accuracy of the next 3 measurements. The +MSE was computed between the mean of 50 sample forecast trajectories (simulating a point forecast) and the ground truth, +averaged over the 5 folds. +Pymunk Physical Environments. +We trained the models on the first 20 frames of the videos from the training dataset +with 20% of the frames randomly dropped. During test time, we provided the models with a context of 20 frames and +evaluated the forecast performance on the next 40 frames. We report the EMD between the predicted and the ground truth +frames, averaged over 16 sample trajectories. The EMD was computed using the ot.emd2 function from the Python Optimal +Transport (POT) library (Flamary et al., 2021) with the euclidean metric as the cost function. +C.3. Experiment Configurations +We ran all our experiments on 2 machines with 1 Tesla T4 GPU, 16 CPUs, and 64 GB of memory each. In this section, we +report training and hyperparameter configurations used in our experiments. +We optimized all models using the Adam optimizer with a learning rate of 0.01 for all the datasets except Pymunk physical +environments where we used 0.002. We reduced the learning rate exponentially with a decay rate of 0.9 every 500 steps for +the bouncing ball, damped pendulum, and CMU MoCap (walking) datasets, every 100 steps for the USHCN climate dataset, +and every 3000 steps for the Pymunk physical environments datasets. We trained the models for 5K, 2K, 2.5K, 150, and +100K steps with a batch size of 50, 64, 16, 100, and 32 for the bouncing ball, damped pendulum, CMU MoCap (walking), +USHCN climate indicators, and Pymunk physical environments, respectively. +For NCDSSM models, we used the following auxiliary inference and emission networks for each dataset: +• Bouncing Ball, Damped Pendulum, and USHCN Climate Indicators +– Auxiliary inference network: Identity() +– Emission network: Identity() +• CMU Motion Capture (Walking) +– Auxiliary inference network: Input(d) → Linear(64) → Softplus() → Linear (2×h) +– Emission network: Input(h) → 2×[Linear(30) → Softplus()] → Linear (d) +• Pymunk Physical Environments +– Auxiliary inference network: Input(1, 32, 32) → ZeroPad2d(padding=[0, 1, 0, 1]) → Conv2d(1, +32, kernel size=3, stride=2) → ReLU() → 2×[ZeroPad2d(padding=[0, 1, 0, 1]) → Conv2d(32, +32, kernel size=3, stride=2) → ReLU()] → Flatten → Linear(64) → Linear(2×h) +– Emission network: Input(h) → Linear(512) → 3×[Conv2d(32, 128, kernel size=3, stride=1, +padding=1) → ReLU() → PixelShuffle(upscale factor=2)] → Conv2d(32, 1, kernel size=1, +stride=1) +To ensure good initial estimation of auxiliary variables, we did not update the underlying SSM parameters for the first 100 +and 1000 training steps for the CMU MoCap (walking) and Pymunk physical environments datasets, respectively. In the +following, we list specific experiment configurations for individual experiments. +C.3.1. LATENTODE +We used the RK4 ODE solver to integrate the encoder and drift ODEs with a step size of 0.05 for all datasets. +• Bouncing Ball +– Dimension of latent state: 6 + +Neural Continuous-Discrete State Space Models +– Dimension of observations: 1 +– Encoder network: ODEGRU with a GRUCell(hidden units=10) and ODE drift function Input(10) → +Linear(30) → Tanh() → Linear(10) +– Decoder network: Input(6) → Linear(10) → Softplus() → Linear(1) +– ODE drift function: Input(6) → Linear(64) → Softplus() → Linear(6) +• Damped Pendulum +– Dimension of latent state: 6 +– Dimension of observations: 2 +– Encoder network: ODEGRU with a GRUCell(hidden units=10) and ODE drift function Input(10) → +Linear(64) → Tanh() → Linear(10) +– Decoder network: Input(6) → Linear(64) → Tanh() → Linear(2) +– ODE drift function: Input(6) → Linear(64) → Tanh() → Linear(6) +• CMU Motion Capture (Walking) +– Dimension of latent state: 10 +– Dimension of observations: 50 +– Encoder network: ODEGRU with a GRUCell(hidden units=30) and ODE drift function Input(30) → +Linear(64) → Tanh() → Linear(30) +– Decoder network: Input(10) → 2×[Linear(30) → Softplus()] → Linear(50) +– ODE drift function: Input(10) → Linear(30) → Softplus() → Linear(10) +• Pymunk Physical Environments +– Dimension of latent state: 10 +– Dimension of observations: 1024 +– Encoder network: Same CNN encoder base as in the auxiliary inference network in NCDSSM models and +ODEGRU with a GRUCell(hidden units=64) and ODE drift function Input(64) → Linear(64) → Tanh() +→ Linear(64) +– Decoder network: Same CNN decoder as in the emission network in NCDSSM models +– ODE drift function: Input(10) → Linear(64) → Tanh() → Linear(10) +C.3.2. LATENTSDE +For LatentSDE experiments, we additionally annealed the KL term in the objective function with a linear annealing schedule +from 0 to 1 over 500 steps for all datasets except Pymunk physical environments for which we annealed over 1000 steps. +As proposed in Li et al. (2020), we also provided the posterior SDEs with an additional context vector from the encoder +to incorporate information from later observations. We used the RK4 ODE solver to integrate the encoder ODEs and the +Euler-Maruyama SDE solver to integrate the prior/posterior SDEs with a step size of 0.05 for all datasets. +• Bouncing Ball +– Dimension of latent state: 6 +– Dimension of context vector: 3 +– Dimension of observations: 1 +– Encoder network: ODEGRU with a GRUCell(hidden units=10) and ODE drift function Input(10) → +Linear(64) → Tanh() → Linear(10) +– Decoder network: Input(6) → Linear(64) → Softplus() → Linear(1) +– Posterior SDE drift function: Input(6+3) → Linear(64) → Softplus() → Linear(6) +– Prior SDE drift function: Input(6) → Linear(64) → Softplus() → Linear(6) +– Posterior/Prior SDE diffusion function: 6×[Input(1) → Linear(64) → Softplus() → Linear(1)] +• Damped Pendulum +– Dimension of latent state: 6 +– Dimension of context vector: 3 +– Dimension of observations: 2 +– Encoder network: ODEGRU with a GRUCell(hidden units=10) and ODE drift function Input(10) → +Linear(64) → Tanh() → Linear(10) +– Decoder network: Input(6) → Linear(64) → Tanh() → Linear(2) +– Posterior SDE drift function: Input(6+3) → Linear(64) → Softplus() → Linear(6) +– Prior SDE drift function: Input(6) → Linear(64) → Softplus() → Linear(6) + +Neural Continuous-Discrete State Space Models +– Posterior/Prior SDE diffusion function: 6×[Input(1) → Linear(64) → Softplus() → Linear(1)] +• CMU Motion Capture (Walking) +– Dimension of latent state: 10 +– Dimension of context vector: 3 +– Dimension of observations: 50 +– Encoder network: ODEGRU with a GRUCell(hidden units=30) and ODE drift function Input(30) → +Linear(64) → Tanh() → Linear(30) +– Decoder network: Input(10) → 2×[Linear(30) → Softplus()] → Linear(50) +– Posterior SDE drift function: Input(10+3) → Linear(30) → Softplus() → Linear(10) +– Prior SDE drift function: Input(10) → Linear(30) → Softplus() → Linear(10) +– Posterior/Prior SDE diffusion function: 10×[Input(1) → Linear(30) → Softplus() → Linear(1)] +• Pymunk Physical Environments +– Dimension of latent state: 10 +– Dimension of context vector: 4 +– Dimension of observations: 1024 +– Encoder network: Same CNN encoder base as in the auxiliary inference network in NCDSSM models and +ODEGRU with a GRUCell(hidden units=64) and ODE drift function Input(64) → Linear(64) → Tanh() +→ Linear(64) +– Decoder network: Same CNN decoder as in the emission network in NCDSSM models +– Posterior SDE drift function: Input(10+4) → Linear(64) → Tanh() → Linear(10) → Tanh() +– Prior SDE drift function: Input(10) → Linear(64) → Tanh() → Linear(10) → Tanh() +– Posterior/Prior SDE diffusion function: 10×[Input(1) → Linear(64) → Softplus() → Linear(1)] +C.3.3. NCDSSM-LTI +• Bouncing Ball +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 1 +– Dimension of observations (d): 1 +– Integrator: Analytic +• Damped Pendulum +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 2 +– Dimension of observations (d): 2 +– Integrator: Analytic +• CMU Motion Capture (Walking) +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 6 +– Dimension of observations (d): 50 +– Integrator: Analytic +• USHCN Climate Indicators +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 5 +– Dimension of observations (d): 5 +– Integrator: Euler with step size 0.1 +• Pymunk Physical Environments +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 4 +– Dimension of observations (d): 1024 +– Integrator: RK4 with step size 0.05 +C.3.4. NCDSSM-NL +We set the diffusion function to G(·, t) = I for all datasets. + +Neural Continuous-Discrete State Space Models +• Bouncing Ball +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 1 +– Dimension of observations (d): 1 +– Drift function (f): Input(m) → Linear(64) → Softplus() → Linear(m) +– Integrator: RK4 with step size 0.05 +• Damped Pendulum +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 2 +– Dimension of observations (d): 2 +– Drift function (f): Input(m) → Linear(64) → Softplus() → Linear(m) +– Integrator: RK4 with step size 0.05 +• CMU Motion Capture (Walking) +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 6 +– Dimension of observations (d): 50 +– Drift function (f): Input(m) -> SN(Linear(30)) -> Softplus() -> SN(Linear(m)) +– Integrator: RK4 with step size 0.05 +• USHCN Climate Indicators +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 5 +– Dimension of observations (d): 5 +– Drift function (f): Input(m) → SN(Linear(64)) → Softplus() → SN(Linear(m)) +– Integrator: Euler with step size 0.1 +• Pymunk Physical Environments +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 4 +– Dimension of observations (d): 1024 +– Drift function (f): Input(m) → Linear(64) → Tanh() → Linear(m) → Tanh() +– Integrator: RK4 with step size 0.05 +C.3.5. NCDSSM-LL +We set the α-network to Input(m) → Linear(64) → Softplus() → Linear(K) for all datasets. +• Bouncing Ball +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 1 +– Dimension of observations (d): 1 +– Number of base matrices (K): 5 +– Integrator: RK4 with step size 0.05 +• Damped Pendulum +– Dimension of state (m): 6 +– Dimension of auxiliary variables (h): 2 +– Dimension of observations (d): 2 +– Number of base matrices (K): 5 +– Integrator: RK4 with step size 0.05 +• CMU Motion Capture (Walking) +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 6 +– Dimension of observations (d): 50 +– Integrator: RK4 with step size 0.05 +• USHCN Climate Indicators +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 5 + +Neural Continuous-Discrete State Space Models +– Dimension of observations (d): 5 +– Number of base matrices (K): 10 +– Integrator: Euler with step size 0.1 +• Pymunk Physical Environments +– Dimension of state (m): 10 +– Dimension of auxiliary variables (h): 4 +– Dimension of observations (d): 1024 +– Number of base matrices (K): 10 +– Integrator: RK4 with step size 0.05 +D. Additional Results +Table 5 shows the number of trainable parameters in each model for different experiments. NCDSSM models obtain better +performance on every dataset with significantly fewer parameters. Table 6 shows the goodness-of-fit coefficient (R2) for +ordinary least squares regression with the latent states as features, and the ground truth angle and angular velocity as targets. +NCDSSM-NL and NCDSSM-LL models obtain a high R2 coefficient showing that the latent states learned by these models +are informative about the true latent state (angle and angular velocity). +Figs. 4 and 5 show sample predictions from the best run of each model for different missing data settings on the bouncing +ball and the damped pendulum datasets, respectively. For the bouncing ball experiment, both LatentODE and LatentSDE +learn that the dataset exhibits a zig-zag pattern but are unable to accurately extrapolate it beyond the training context. +In the case of damped pendulum, LatentODE and LatentSDE perform well on the low missing data settings (0% and +30%) but completely fail on the more challenging settings of 50% and 80% missing data. In contrast, NCDSSM-NL and +NCDSSM-LL generate accurate predictions across datasets and missing data settings. Furthermore, while the predictions +shown in Figs. 4 and 5 are from the best performing runs of each model, they represent a typical run for NCDSSM-NL and +NCDSSM-LL. On the other hand, the prediction quality from LatentODE and LatentSDE models varies significantly across +random initializations. +Fig. 6 shows the variation of the EMD with time for different models on the box and pong datasets. All models have +EMD close to 0 in the context window from 0-2s; however, in the forecast horizon from 2-6s, the EMD rises gradually +for NCDSSM-NL and NCDSSM-LL but rapidly and irregularly for other models. Figs. 7 and 8 show sample predictions +from different models on the box and the pong datasets, respectively. NCDSSM-NL and NCDSSM-LL generate accurate +predictions whereas LatentODE and LatentSDE perform significantly worse. +Table 5. The number of trainable parameters in every model for different experiments. +Model +Number of Parameters +Bouncing Ball +Damped Pendulum +MoCap Walking (Setup 2) +USHCN +Pymunk Environments +LatentODE +2094 +3336 +15454 +– +204243 +LatentSDE +5461 +5557 +17187 +– +208043 +NCDSSM-LTI +63 +72 +11080 +185 +165911 +NCDSSM-NL +859 +862 +11620 +1439 +167165 +NCDSSM-LL +974 +977 +12509 +2439 +168165 +Table 6. Goodness-of-fit coefficient (R2) of ordinary least squares (OLS) regression for the best run of each model on the Pendulum +dataset. The latent states are treated as features and ground truth angle — transformed into polar coordinates: sin(angle)/ cos(angle) — +and angular velocity as targets. +Model +sin(angle)/ cos(angle) R2 (↑) (% Missing) +Angular Velocity R2 (↑) (% Missing) +0% +30% +50% +80% +0% +30% +50% +80% +LatentODE +0.000 / 0.802 +0.000 / 0.735 +0.000 / 0.744 +0.000 / 0.626 +0.001 +0.000 +0.000 +0.000 +LatentSDE +0.953 / 0.960 +0.918 / 0.957 +0.000 / 0.817 +0.000 / 0.513 +0.970 +0.962 +0.001 +0.000 +NCDSSM-LTI +0.593 / 0.537 +0.604 / 0.468 +0.477 / 0.796 +0.481 / 0.705 +0.349 +0.388 +0.162 +0.305 +NCDSSM-NL +0.984 / 0.990 +0.982 / 0.985 +0.973 / 0.976 +0.905 / 0.920 +0.986 +0.969 +0.935 +0.859 +NCDSSM-LL +0.986 / 0.989 +0.983 / 0.989 +0.972 / 0.980 +0.875 / 0.888 +0.972 +0.978 +0.955 +0.827 + +Neural Continuous-Discrete State Space Models +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +5 +10 +15 +20 +25 +Time +NCDSSM-LL +(a) 0% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +5 +10 +15 +20 +25 +Time +NCDSSM-LL +(b) 30% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +5 +10 +15 +20 +25 +Time +NCDSSM-LL +(c) 50% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +5 +10 +15 +20 +25 +Time +NCDSSM-LL +(d) 80% Missing +Figure 4. Predictions from different models on the bouncing ball dataset for the 0%, 30%, 50%, and 80% missing data settings. The +ground truth is shown using dashed lines with observed points in the context window (gray shaded region) shown as filled circles. The +vertical dashed gray line marks the beginning of the forecast horizon. Solid lines indicate median predictions with 90% prediction intervals +shaded around them. +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +2 +4 +6 +8 +10 +12 +14 +Time +NCDSSM-LL +(a) 0% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +2 +4 +6 +8 +10 +12 +14 +Time +NCDSSM-LL +(b) 30% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +2 +4 +6 +8 +10 +12 +14 +Time +NCDSSM-LL +(c) 50% Missing +LatentODE +Observations +Ground Truth +Median Prediction +LatentSDE +NCDSSM-LTI +NCDSSM-NL +0 +2 +4 +6 +8 +10 +12 +14 +Time +NCDSSM-LL +(d) 80% Missing +Figure 5. Predictions from different models on the damped pendulum dataset for the 0%, 30%, 50%, and 80% missing data settings. +The ground truth is shown using dashed lines with observed points in the context window (gray shaded region) shown as filled circles. +The vertical dashed gray line marks the beginning of the forecast horizon. Solid lines indicate median predictions with 90% prediction +intervals shaded around them. The purple and orange colors indicate observation dimensions. + +Neural Continuous-Discrete State Space Models +0 +1 +2 +3 +4 +5 +6 +Time +0 +1 +2 +EMD +Box +LatentODE +LatentSDE +NCDSSM-LTI +NCDSSM-NL +NCDSSM-LL +0 +1 +2 +3 +4 +5 +6 +Time +0 +2 +4 +6 +EMD +Pong +LatentODE +LatentSDE +NCDSSM-LTI +NCDSSM-NL +NCDSSM-LL +Figure 6. Variation of EMD over time for the Box (left) and Pong (right) datasets. The EMD rises gradually with time for NCDSSM-LL +and NCDSSM-NL but rapidly and irregularly for other models. +(a) Box LatentODE +(b) Box LatentSDE +(c) Box NCDSSM-LTI +(d) Box NCDSSM-NL +(e) Box NCDSSM-LL +Figure 7. Sample predictions from different models on the Box dataset. The top row in each figure is the ground truth with some missing +observations in the context window (before the dashed grey line). The next five rows show trajectories sampled from each model. Best +viewed zoomed-in on a computer. + +Neural Continuous-Discrete State Space Models +(a) Pong LatentODE +(b) Pong LatentSDE +(c) Pong NCDSSM-LTI +(d) Pong NCDSSM-NL +(e) Pong NCDSSM-LL +Figure 8. Sample predictions from different models on the Pong dataset. The top row in each figure is the ground truth with some missing +observations in the context window (before the dashed grey line). The next five rows show trajectories sampled from each model. Best +viewed zoomed-in on a computer. + diff --git a/bNFIT4oBgHgl3EQfmCv-/content/tmp_files/load_file.txt b/bNFIT4oBgHgl3EQfmCv-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08a5afca43a850574ea6780eafaa5fdbc6a076ae --- /dev/null +++ b/bNFIT4oBgHgl3EQfmCv-/content/tmp_files/load_file.txt @@ -0,0 +1,1621 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf,len=1620 +page_content='Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series Abdul Fatir Ansari 1 † Alvin Heng 2 Andre Lim 2 Harold Soh 2 Abstract Learning accurate predictive models of real-world dynamic phenomena (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', climate, biological) remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' One key issue is that the data generated by both natural and ar- tificial processes often comprise time series that are irregularly sampled and/or contain missing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' In this work, we propose the Neural Continuous-Discrete State Space Model (NCDSSM) for continuous-time modeling of time series through discrete-time observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' NCDSSM employs auxiliary variables to disen- tangle recognition from dynamics, thus requiring amortized inference only for the auxiliary vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Leveraging techniques from continuous- discrete filtering theory, we demonstrate how to perform accurate Bayesian inference for the dy- namic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' We propose three flexible parame- terizations of the latent dynamics and an efficient training objective that marginalizes the dynamic states during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Empirical results on mul- tiple benchmark datasets across various domains show improved imputation and forecasting perfor- mance of NCDSSM over existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Introduction State space models (SSMs) provide an elegant framework for modeling time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Combinations of SSMs with neural networks have proven effective for various time series tasks such as segmentation, imputation, and forecasting (Kr- ishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Fraccaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Rangapuram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Kurle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Ansari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' However, most existing models are limited to the discrete time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=', uniformly sampled) setting, whereas data from various phys- ical and industrial systems in the real world are sometimes only available at irregular (often sparse) intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Such sys- †Work done while at National University of Singapore, prior to joining Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' 1AWS AI Labs 2School of Computing, National University of Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNFIT4oBgHgl3EQfmCv-/content/2301.11308v1.pdf'} +page_content=' Correspondence to: Abdul Fatir Ansari 0 for ǫ′ > ǫM and I(ǫ′) = 0 for +ǫ′ ⩽ ǫM, which is the reason that the integration in dispersion relation (8) ranges from ǫM +to +∞. We emphatically introduce the T-matrix element T(c′;b)a(ǫ′) as follows. +In our theory T(c′;b)a(ǫ′) must involve bound state, so this matrix element can not be +calculated as ordinary S-matrix element. Using the Heisenberg picture, we can obtain the +total matrix element between a final state |Q′ +1...Q′ +n′, Q1...Qn out⟩ and a specified initial +bound state |P in⟩ +−iR(c′;b)a(ǫ′) =⟨Q′ +1...Q′ +n′, Q1...Qn out|P in⟩ +=i2n′ � +d4z1...d4zn′f ∗ +Q′ +1(z1)...f ∗ +Q′ +n′(zn′)S′−1 +z1 ...S′−1 +zn′ +× ⟨Q1...Qn out|Tφ(z1)...ψ(zi)... ¯ψ(zj)...φ(zn′)|P in⟩. +(10) +Here φ and ψ represent boson and fermion field operators in the Heisenberg picture, respec- +tively. The functions f are solutions to the corresponding free field equations of motion, and +S′ represents free field propagator. Of great interest is the matrix element of a time-order +product of Heisenberg field operators between bound states in Eq. +(10). Mandelstam’s +approach is generalized to evaluate the bound state matrix element with respect to ǫ′ +⟨Q1...Qn out|Tφ(z1)...ψ(zi)... ¯ψ(zj)...φ(zn′)|P in⟩ += +� +d4y1d4y2...d4y2n−1d4y2nd4x1d4x2 +× ¯χQ1(y1, y2)...¯χQn(y2n−1, y2n)T(y1...y2n; z1...zi...zj...zn′; x1, x2)χP(x1, x2), +(11) +where T(y1...y2n; z1...zi...zj...zn′; x1, x2) is the two-particle irreducible Green’s function, ¯χ and +χ are BS wave functions for the final and initial bound states, respectively. The function T +can, in principle, be evaluated by means of perturbation theory. It is necessary to emphasize +that the general matrix element (11) is calculated with respect to ǫ′, and the energy in T is +equal to the final state energy ǫ′ extending from ǫM to +∞, i.e., ǫM < ǫ′ < ∞. Therefore the +final state may be a virtual state, while the traditional Feynman diagram represents only +the physical case ǫ′ = M0. In this paper we introduce a Feynman diagram to represent the +virtual states, called virtual Feynman diagram, shown in Figure 1. The crosses in virtual +5 + +' +n +z +j +z +i +z +1 +z +2 +y +1 +y +2 +x +1 +x +FIG. +1: +General +matrix +element +between +two +bound +states +⟨Q out|Tφ(z1)...ψ(zi)... ¯ψ(zj)...φ(zn′)|P in⟩ with respect to ǫ′. +The filled blob represents the +two-particle irreducible Green’s function, and the unfilled ellipses represent BS amplitudes. The +final state energy extends from ǫM to +∞, while the initial state energy is specified. The crosses +mean that the final state is a virtual state, which is different from the traditional Feynman +diagram. +Feynman diagram mean that the energy in T is equal to the final state energy ǫ′ extending +from ǫM to +∞ while the bare mass M0 of initial bound state is specified. When ǫ′ = M0, +the crosses in virtual Feynman diagram disappear and it becomes the traditional Feynman +diagram. Removing delta-function factor (2π)4δ(4)(Q′ +1 + ... + Qn − P ǫ′) in R(c′;b)a(ǫ′), we +obtain T(c′;b)a(ǫ′). +To illustrate this, we imagine that the initial bound state (MS) is composed of two +heavy vector mesons (V M and V M) and the final state contains a heavy meson (HM) and +a light meson (LM). If a bound state with spin j is created by two massive vector fields, +its BS wave function can be defined as χj +P (λτ)(x1, x2) = ⟨0|TAλ(x1)A† +τ(x2)|P, j⟩ and we have +given the general form for this BS wave function χj +λτ(P, p) in the momentum representation, +where p is the relative momentum of two vector fields [8, 10]. This BS wave function should +satisfy the equation +χj +λτ(P, p) = − +� +d4p′ +(2π)4∆F λθ(p′ +1)Vθθ′,κ′κ(p, p′; P)χj +θ′κ′(P, p′)∆F κτ(p′ +2), +(12) +where Vθθ′,κ′κ is the interaction kernel, p′ +1 and p′ +2 are the momenta carried by two vector +fields, ∆F λθ(p′ +1) and ∆F κτ(p′ +2) are the propagators for the spin 1 fields. Owing to the effective +interaction Lagrangian at low energy QCD (1), we have to consider that the heavy meson is +a bound state composed of a quark and an antiquark and investigate the interaction of light +6 + +meson with quarks in heavy meson. Through the heavy meson form factor describing the +heavy meson structure, we have obtained the interaction kernel between two heavy vector +mesons (V M and V M) derived from one light meson (σ, ω, ρ, φ) exchange in Refs. [8, 16]. +In our previous works [8, 14], BS equation (12) has been solved and the bare mass M0 and +BS wave function χj +λτ(P, p) for the bound state composed of two heavy vector mesons have +been obtained. In this paper, we are interested only in mass shift for molecular state and +do not repeat the procedure for solving BS equation. +Taking into account the internal structure of heavy mesons (V M, V M and HM) and +retaining the lowest order value of T [9], we can obtain the T-matrix element with respect +to ǫ′ in the momentum representation +T(c′;b)a(ǫ′) = +ig′ +0ε̺′ +µ (Q′)ε̺ +ν(Q) +(2π)9/2� +8EH(Q)EL(Q′)E(P) +� d4kd4p +(2π)8 Tr[SD +F (p2)¯ΓH +ν (Q, q)SC +F(p1) +× ΓV +λ (p′ +1, k)SA +F (p3)γµSB +F(p4)Γ +¯V +τ (p′ +2, k′)χj +λτ(P, p)], +(13) +where p1, p3, p4, p2 are the momenta of four quarks; p′ +1 and p′ +2 are the momenta of two heavy +vector mesons; q, k and k′ are the relative momenta between quark and antiquark in heavy +mesons, respectively; ε(p) is the polarization vector of vector meson with momentum p, +ΓH(K, k) represents BS amplitude of heavy meson, SF(p) is the quark propagator and its +superscript is a flavor label, shown as Figure 2. In our approach, the initial bound state is +considered as a four-quark state [9], so the generalized BS amplitude of initial bound state +should be ΓV +λ (p′ +1, k)χj +λτ(P, p)Γ ¯V +τ (p′ +2, k′), which has been specified. In Figure 2(a), the energy +in T is equal to the energy of final state which is a virtual state, and then the quark momenta +in left-hand side of crosses depend on the final state energy and the momenta in right-hand +side depend on the initial state energy, i.e., p1 −p2 −p3 +p4 = Q+Q′ = P ǫ′ and p′ +1 −p′ +2 = P. +In the rest frame, we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) and ǫM < ǫ′ < ∞. When +ǫ′ = M0, the crosses in Figure 2(a) disappear and Figure 2(a) becomes the traditional +Feynman diagram, shown as Figure 2(b). From Figure 2(b), we have calculated the matrix +element T(c′;b)a(M0) with bare mass for meson-meson bound state and obtained the decay +width Γ(M0) with bare mass [9]. +Finally, we can expect that Ga(ǫ) has a pole on the second Riemann sheet from Eq. (5) +ǫ0 ∼= M0 + (2π)3[D(M0) − iI(M0)] = M − iΓ(M0)/2, +(14) +where ∆M = (2π)3D(M0) is the shift for energy level of resonance and M = M0+(2π)3D(M0) +7 + +VM +VM +) +, +( +p +P +j +) +' +, +' +( +2 +k +p +V +) +, +( +q +Q +H +) +, +' +( +1 +k +p +V +4 +p +3 +p +2 +p +1 +p +' +2 +p +' +1 +p +P +Q +' +Q +L M +H M +MS +(a) +VM +VM +) +, +( +p +P +j +) +' +, +' +( +2 +k +p +V +) +, +( +q +Q +H +) +, +' +( +1 +k +p +V +4 +p +3 +p +2 +p +1 +p +' +2 +p +' +1 +p +P +Q +' +Q +L M +H M +MS +(b) +FIG. 2: Matrix element between bound states in the momentum representation. The momenta +in the final state satisfy Q + Q′ = P ǫ′ and the momentum of the initial state is P. The solid +lines denote quark propagators. In the rest frame, we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) +and ǫM < ǫ′ < ∞. In diagram (a) the final state is a virtual state and the crosses mean that +the momenta of quark propagators depend on the final state energy; in diagram (b) the crosses +disappear when ǫ′ = M0. +is the physical mass for resonance. The bare mass M0 of two-body bound state is obtained +by solving BS equation (2), which should not be the mass of physical resonance. Γ(M0) with +bare mass also should not be the width of physical resonance, which should depend on its +physical mass M. Replacing M0 in the momentum of initial bound state by M and setting +ǫ′ = M, we can calculate the matrix element T(c′;b)a(M) from Eq. (10) and obtain the decay +width Γ for physical resonance. +Up to now, a systematical and accurate theoretical approach in the framework of rela- +tivistic quantum field theory to investigate resonance has been established. In this paper, we +only explore exotic meson resonance which is considered as an unstable meson-meson molec- +ular state. The extension of our approach to more general resonances is straightforward, +while the interaction Lagrangian may be modified. +III. +EXAMPLE +As an illustration, we investigate exotic state χc0(3915) [17–19], once named X(3915). +In experiments two strong decay modes of χc0(3915) have been observed: J/ψω and D+D−. +8 + +Here, we assume that the isoscalar χc0(3915) is a mixed state of two unstable molecular +states D∗0 ¯D∗0 and D∗+D∗− with spin-parity quantum numbers 0+. Firstly, we consider the +mixed state of two bound states D∗0 ¯D∗0 and D∗+D∗−, and this bare state can be denoted +as 1/ +√ +2|D∗0 ¯D∗0⟩+1/ +√ +2|D∗+D∗−⟩. In Refs. [8, 10, 14], we have obtained the bare mass M0 +and BS wave function χ0+ +λτ (P, p). In this paper, our attention is focused on the mass shift +due to all decay channels and the decay width of physical resonance. +Let us list all decay channels which are fully open or only just ”virtual”. The narrow +state χc0(3915) was discovered in 2005 [17] and for a long time a series of experiments +only observed one strong decay mode of χc0(3915): J/ψω denoted as c′ +1. In 2020 LHCb +Collaboration observed another decay channel D+D− [19] denoted as c′ +2. Though the neutral +channel D0 ¯D0 still has not been observed, this neutral channel should exist for the isospin +conservation, which is denoted as c′ +3. Because the total energy ǫ′ of the final state extends +from ǫM to +∞, we obtain one virtual channel D∗ ¯D∗ derived from the interaction Lagrangian +(1), denoted as c′ +4. Since bound state lies below the threshold, i.e., M0 < MD∗ + M ¯D∗, the +virtual channel c′ +4 can not occur inside the physical world. Then we can apply Eqs. (10) +and (11) to evaluate the T-matrix element T(c′;b)a(ǫ′) for arbitrary decay channel. +In Figure 2, V M and V M become D∗ and ¯D∗, respectively; HM becomes J/ψ and +LM becomes ω; and decay channel J/ψω can be exhibited by these two Feynman diagrams. +Applying Eq. (9), we obtain the function +I1(ǫ′) =1 +2 +� +d3Qd3Q′(2π)4δ(4)(Q + Q′ − P ǫ′) +� +spins +|T(c′ +1;b)a(ǫ′)|2, +(15) +where T(c′ +1;b)a(ǫ′) is the bound state matrix element with respect to ǫ′. These heavy mesons +J/ψ, D∗ and ¯D∗ are considered as quark-antiquark bound states, and T(c′ +1;b)a(ǫ′) has been +given by Eq. +(13), where flavor labels C = D and A = B represent c-quark and light +quark, respectively. The meson-quark coupling constant g′ +0 becomes gω and g2 +ω = 2.42/2 was +obtained within QCD sum rules approach [9]. BS amplitudes of heavy vector mesons J/ψ +and D∗ have the form ΓV +λ (K, k) = (γλ + Kλγ · K/M2 +V )exp(−k2/ω2 +V ), where ωJ/ψ=0.826GeV +and ωD∗=1.50GeV [9, 13]. These momenta in Figure 2(a) become p1 = (Q + Q′)/2 + p + k, +p2 = (Q′ − Q)/2 + p + k, p3 = k, p4 = Q′ + k, p′ +1 = p + P/2, p′ +2 = p − P/2, Q + Q′ = P ǫ′ = +(0, 0, 0, iǫ′) and P = (0, 0, 0, iM0). Using Eq. (13), we can calculate the T-matrix element +T(c′ +1;b)a(ǫ′) with respect to arbitrary energy ǫ′ for channel c′ +1. From Eq. (15), we obtain the +9 + +function I1(ǫ′) for channel c′ +1 and dispersion relation (8) becomes +D1(M0) = −P +π +� ∞ +ǫc′ +1,M +I1(ǫ′) +ǫ′ − M0 +dǫ′, +(16) +where ǫc′ +1,M = MJ/ψ + Mω. The T-matrix element T(c′ +1;b)a(ǫ′) and the function I1(ǫ′) for +channel c′ +1 are calculated over the real interval ǫc′ +1,M < ǫ′ < ∞, and we obtain the mass shift +∆M1 = (2π)3D1(M0) due to channel c′ +1. +For decay channel D+D−, we obtain the function I2(ǫ′) from Eq. (9) +I2(ǫ′) =1 +2 +� +d3Q1d3Q2(2π)4δ(4)(Q1 + Q2 − P ǫ′) +� +spins +|T(c′ +2;b)a(ǫ′)|2, +(17) +where T(c′ +2;b)a(ǫ′) represents the bound state matrix element with ǫ′. Considering the lowest +order term of T, we obtain T(c′ +2;b)a(ǫ′) represented graphically by Figure 3, where p1 − p2 − +p3 + p4 = p1 − p2 − q3 + q4 = Q1 + Q2 = P ǫ′, p′ +1 − p′ +2 = P, and the crosses mean that the +momenta of quark propagators and the momentum w of the exchanged light meson depend +on Q1 and Q2. The T-matrix element with respect to ǫ′ for channel c′ +2 becomes +T(c′ +2;b)a(ǫ′) = +−ig2 +(2π)9/2� +8ED+(Q1)ED−(Q2)E(P) +� d4kd4k′d4p +(2π)12 +Tr[Sd +F(q3) +× ¯ΓD+(Q1, q)Sc +F(p1)ΓD∗ +λ (p′ +1, k)Sl +F(p3)Odl(p3, q3)]χ0+ +λτ (P, p)∆F(w) +× Tr[Sl +F(p4)Γ +¯D∗ +τ (p′ +2, k′)Sc +F(p2)¯ΓD−(Q2, q′)Sd +F(q4)Odl(q4, p4)], +(18) +where q, q′, k and k′ are the relative momenta between quark and antiquark in heavy mesons, +respectively; the meson-quark coupling constants g were obtained within QCD sum rules +approach [8, 16], Odl(p, q) represents the meson-quark vertex, ∆F(w) is the light meson +propagator, the superscript of quark propagator SF(p) is flavor label, and l = u, d represents +the u, d-antiquark in heavy vector meson D∗0 or D∗+, respectively. BS amplitude of heavy +pseudoscalar meson D+ has the form ΓD+(K, k) = iγ5exp(−k2/ω2 +D), where ωD=1.50GeV +[13]. The meson-quark vertex Odl(p, q) is unit matrix for one-σ exchange; and it becomes +γµ for one light vector meson exchange. From Eqs. (17), (8) and (14), we obtain the mass +shift ∆M2 due to channel c′ +2 +∆M2 = (2π)3D2(M0) = −P +π +� ∞ +ǫc′ +2,M +(2π)3I2(ǫ′) +ǫ′ − M0 +dǫ′, +(19) +where ǫc′ +2,M = MD+ + MD−. The T-matrix element T(c′ +2;b)a(ǫ′) and the function I2(ǫ′) for +channel c′ +2 are calculated over the real interval ǫc′ +2,M < ǫ′ < ∞. Following the same procedure +10 + +1 +Q +D +- +D +* +D +* +D +2 +p +w +2 +Q +4 +q +3 +q +4 +p +3 +p +1 +p +' +2 +p +' +1 +p +P +M S +) +, +( +1 +q +Q +D +) +, +' +( +1 +* +k +p +D +) +' +, +' +( +2 +* +k +p +D +) +' +, +( +2 +q +Q +D +) +, +( +p +P +FIG. 3: Matrix element for decay channel D+D−. The momenta in the final state satisfy Q1+Q2 = +P ǫ′, the momentum of the initial state is P, and we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) and +ǫc′ +2,M < ǫ′ < ∞. w represents the momentum of the exchanged light meson. The crosses mean that +the momenta of quark propagators and the momentum w of the exchanged light meson depend on +the final state energy. +as for channels c′ +1 and c′ +2, we can calculate the mass shifts ∆M3 and ∆M4 due to the channel +c′ +3 and virtual channel c′ +4, respectively. +Since the isospin conservation, we have the constituent quark masses mu = md = +0.33GeV, mc = 1.55GeV [6] and the meson masses Mσ = 0.45GeV, Mω = 0.782GeV, +Mρ = 0.775GeV, Mφ = 1.019GeV, MD∗0 = MD∗+ = 2.007GeV, MD0 = MD+ = 1.865GeV, +MJ/ψ = 3.097GeV [20]. +By doing the numerical calculation, we obtain the mass shifts +∆Mi(i = 1, 2, 3, 4) due to three open decay channels J/ψω, D+D−, D0 ¯D0 and one virtual +channel D∗ ¯D∗, respectively. Subsequently, the mass M for physical resonance χc0(3915) can +be applied to calculate its decay width. Replacing M0 by M in Eq. (13) and setting ǫ′ = M, +we calculate the matrix element T(c′ +1;b)a(M) and obtain that the width for physical decay +model χc0(3915) → J/ψω is Γ1 = 2(2π)3I1(M). Replacing M0 by M in Eq. (18) and setting +ǫ′ = M, we calculate the matrix element T(c′ +2;b)a(M) and obtain that the width for physical +decay model χc0(3915) → D+D− is Γ2 = 2(2π)3I2(M). For the isospin conservation, it is +easy to obtain the width Γ3 for physical decay model χc0(3915) → D0 ¯D0. Our numerical +results are presented in Table I, and the mass M and full width Γ are in good agreement +with experimental data. Furthermore, the calculated D+D− width Γ2 is very small com- +pared with the calculated J/ψω width Γ1, and then we can explain why the decay model +χc0(3915) → D+D− had not been observed in experiments for a long time. Therefore, this +work provides a further verification for the molecular hypothesis of χc0(3915) and predicts the +exact values of these strong decay widths Γ1(χc0(3915) → J/ψω), Γ2(χc0(3915) → D+D−) +11 + +Quantity +M0 +∆M1 +∆M2 +∆M3 +∆M4 +M +Γ1 +Γ2 +Γ3 +Γ +this work 3953.7 −24.0 +−1.4 +−1.4 +−4.6 +3922.3 +22.3 1.5 1.5 +25.3 +PDG[20] +3921.7±1.8 +18.8±3.5 +TABLE I: Mass M and width Γ for physical resonance χc0(3915). M0 is the bare mass of mixed +state of two bound states D∗0 ¯D∗0 and D∗+D∗−, ∆Mi is the calculated shift due to ith decay +channel. (Dimensioned quantities in MeV.) +and Γ3(χc0(3915) → D0 ¯D0). In this paper we emphatically illuminate the physical meaning +of resonance theory in quantum field theory, and the details in computational process will +be shown in our future article. +IV. +CONCLUSION +We recognize that resonance can not be completely treated as a stationary bound state +and provide a reasonable and feasible scheme to describe resonance in the framework of +relativistic quantum field theory. +Based on BS wave function, we provide a description +of the prepared state and investigate the temporal evolution of two-body bound state as +determined by the total Hamiltonian. According to dispersion relation, the total matrix +elements for all decay channels should be calculated with respect to arbitrary energy, and +these matrix elements are expressed in terms of the Heisenberg picture. +Mandelstam’s +approach is generalized to calculate the matrix element between bound states with arbitrary +energy, which is exhibited in virtual Feynman diagram. Finally, the mass and decay width +for physical resonance are obtained. In this paper, we illustrate resonance theory in quantum +field theory by reference to the example of exotic meson which is considered as an unstable +meson-meson molecular state, and obviously our work can be extended to more general +resonances and creates a new paradigm for investigating hadron resonances. +Acknowledgments +This work was supported by the National Natural Science Foundation of China under +Grants No. 11705104 and No. 11801323; Shandong Provincial Natural Science Foundation, +China under Grants No. ZR2016AQ19 and No. ZR2016AM31; and SDUST Research Fund +12 + +under Grant No. 2018TDJH101. +[1] E.S. Swanson, Short range structure in the X(3872), Phys. Lett. B 588 (2004) 189. +[2] N.A. T¨ornqvist, Isospin breaking of the narrow charmonium state of Belle at 3872 MeV as a +deuson, Phys. Lett. B 590 (2004) 209. +[3] X. Liu, S.L. Zhu, Y (4143) is probably a molecular partner of Y (3930), Phys. Rev. D 80 (2009) +017502. +[4] L. Maiani, F. Piccinini, A.D. Polosa, V. Riquer, Diquark-antidiquark states with hidden or +open charm and the nature of X(3872), Phys. Rev. D 71 (2005) 014028. +[5] L. Maiani, A.D. Polosa, V. Riquer, Indications of a Four-Quark Structure for the X(3872) and +X(3876) Particles from Recent Belle and BABAR Data, Phys. Rev. Lett. 99 (2007) 182003. +[6] D. Ebert, R.N. Faustov, V.O. Galkin, Masses of heavy tetraquarks in the relativistic quark +model, Phys. Lett. B 634 (2006) 214. +[7] T. Branz, T. Gutsche, V.E. Lyubovitskij, Hadronic molecule structure of the Y (3940) and +Y (4140), Phys. Rev. D 80 (2009) 054019. +[8] X. Chen, X. L¨u, Mass of Y (3940) in Bethe-Salpeter equation for quarks, Eur. Phys. J. C 75 +(2015) 98. +[9] X. Chen, X. L¨u, Decay width of hadronic molecule structure for quarks, Phys. Rev. D 97 +(2018) 114005. +[10] X. Chen, X. L¨u, R. Shi, X. Guo, Q. Wang, Radiative decay of hadronic molecule state for +quarks, Phys. Rev. D 101 (2020) 014009. +[11] P. Maris, C.D. Roberts, P.C. Tandy, Pion mass and decay constant, Phys. Lett. B 420 (1998) +267. +[12] M.A. Ivanov, Y.L. Kalinovsky, C.D. Roberts, Survey of heavy-meson observables, Phys. Rev. +D 60 (1999) 034018. +[13] M.A. Ivanov, J.G. K¨orner, S.G. Kovalenko, C.D. Roberts, B-meson to light-meson transition +form factors, Phys. Rev. D 76 (2007) 034018. +[14] X. Chen, X. L¨u, R. Shi, X. Guo, Calculation of mass of Y (4140) by introducing mixed molecule +state in quark model, Nucl. Phys. B 909 (2016) 243. +[15] M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, (1964). +13 + +[16] X. Chen, R. Liu, R. Shi, X. L¨u, Bethe-Salpeter wave functions for the bound states composed +of two vector fields of arbitrary spin and their application, Phys. Rev. D 87 (2013) 065013. +[17] S.-K. Choi, et al., (Belle Collaboration), Observation of a near-threshold ωJ/ψ mass enhance- +ment in exclusive B → KωJ/ψ decays, Phys. Rev. Lett. 94 (2005) 182002. +[18] A. Vinokurova, et al., (Belle Collaboration), Search for B decays to final states with the ηc +meson, JHEP 06 (2015) 132. +[19] R. Aaij, et al., (LHCb Collaboration), Amplitude analysis of the B+ → D+D−K+ decay, +Phys. Rev. D 102 (2020) 112003. +[20] R.L. Workman, et al., (Particle Data Group), Review of particle physics, Prog. Theor. Exp. +Phys. 2022 (2022) 083C01. +14 + diff --git a/bdAyT4oBgHgl3EQfwPlq/content/tmp_files/load_file.txt b/bdAyT4oBgHgl3EQfwPlq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc04cd9f4146fb4ae3d9a8179df5ab9553c5b641 --- /dev/null +++ b/bdAyT4oBgHgl3EQfwPlq/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf,len=467 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='00645v1 [hep-ph] 2 Jan 2023 Resonance in quantum field theory Xiaozhao Chen1, ∗ and Xiaofu L¨u2, 3, 4 1Department of Fundamental Courses, Shandong University of Science and Technology, Taian, 271019, China 2Department of Physics, Sichuan University, Chengdu, 610064, China 3Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing 100080, China 4CCAST (World Laboratory), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Box 8730, Beijing 100080, China (Dated: January 3, 2023) Abstract In the framework of relativistic quantum field theory, the solution of Bethe-Salpeter equation for bound state can not describe resonance, so we develop Bethe-Salpeter equation to investigate resonance which is regarded as an unstable state created by two Heisenberg field operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Based on Bethe-Salpeter wave function, we consider the temporal evolution of two-body bound state determined by the total Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The total matrix element for arbitrary decay channel is expressed in terms of the Heisenberg picture, and Mandelstam’s approach is generalized to calculate the matrix element between bound states with respect to arbitrary energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Some innovations to Feynman diagram are made so that the key features of dispersion relation can be more clearly exhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' This resonance theory in quantum field theory is applied to investigate exotic particle which is considered as an unstable meson-meson molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' PACS numbers: 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Yx, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Rt, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Ki ∗Electronic address: chen˙xzhao@sina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' corresponding author 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' INTRODUCTION Many exotic particles have been discovered in experiment and many possible alternative interpretations beyond quark-antiquark state have been proposed in theory [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Among these interpretations, Bethe-Salpeter (BS) equation is frequently used to investigate the properties of exotic particles which are considered as two-body bound states [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In quantum field theory, BS equation is a nonperturbative method [11–13], which should be only applied to deal with two-body bound state in the strict sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' However, in experiments exotic particles are unstable states, so these exotic particles are resonances which can not be completely treated as stationary two-body bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' More importantly, present field theory seldom involves the issue concerning unstable two-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this work, resonance is regarded as an unstable state created by two Heisenberg field operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' We develop BS equation to deal with resonance in the framework of relativistic quantum field theory and illustrate this theory based on BS equation for exotic meson resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In the previous works about hadronic molecule states [1–3, 8–10], exotic particles were considered as meson-meson bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Solving BS equations for meson-meson bound states, the authors of these works obtained the masses and BS wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The ”bare” mass of meson-meson bound state was regarded as mass of exotic meson resonance and the mass shift for molecular state due to decay channels has seldom been considered [1–3, 8– 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' As well-known, all decay channels of resonance should contribute to its physical mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Therefore, it is necessary to seek a development of BS equation for dealing with resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper, exotic meson resonance is considered as an unstable meson-meson molecular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Based on BS wave function for meson-meson bound state, we can provide a description for the prepared state and then study the temporal evolution of meson-meson molecular state determined by the total Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Using dispersion relation, the Heisenberg picture and Mandelstam’s approach, we obtain the shift for energy level of resonance and then the physical mass is used to calculate its decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' An innovative Feynman diagram is introduced, in which the key features of dispersion relation can be exhibited clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' RESONANCE THEORY IN QUANTUM FIELD THEORY Let us begin with the interaction Lagrangian for the coupling of light quark fields to light meson fields as in effective theory at low energy QCD [9] LI = ig0 ¯Qγ5PQ + ig′ 0 ¯QγµVµQ + gσ ¯Q′Q′σ, (1) where ¯Q = (¯u, ¯d, ¯s), ¯Q′ = (¯u, ¯d), g represents the corresponding meson-quark coupling constant, P and V are the octet pseudoscalar and nonet vector meson matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From this Lagrangian, we have investigated the light meson interaction with quarks in heavy mesons and obtained the interaction of heavy meson with light meson through the heavy meson form factor [8, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Using path integrals, one can obtain a homogeneous integral equation for arbitrary bound state composed of two mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In symbolic notation BS equation may be written as (S(1)−1S(2)−1 + V)χ = 0, (2) where χ represents BS wave function, the kernel V is the sum of all irreducible graphs, S(1) and S(2) represent meson propagators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Solving this BS equation, one can obtain the bare mass M0 and BS wave function χP(x1, x2) for this meson-meson bound state with momentum P = (P, i � P2 + M2 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' We emphasize that the kernel V is defined in two- body channel so V is not complete interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The kernel plays a central role for making two-body system to be a stable state, but it can not provide any motive for decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Since resonance decays spontaneously into other particles, we can suppose that at the times t1 = 0 and t2 = 0 this unstable state has been prepared to decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' This prepared state (ps) can be described by the ground-state BS wave function which has the form X ps a = χP(x1, t1 = 0, x2, t2 = 0) = 1 (2π)3/2 1 � 2E(P) eiP·(η1x1+η2x2)χP(x1 − x2), (3) where E(p) = � p2 + m2 and η1 +η2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Then the time evolution of this system determined by the total Hamiltonian H has the explicit form X (t) = e−iHtX ps a = 1 2πi � C2 dǫe−iǫt 1 ǫ − H X ps a , (4) where G(ǫ) = (ǫ − H)−1 is the Green’s function and the contour C2 runs from icr + ∞ to icr − ∞ in energy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The Green’s function can be represented by scattering matrix [15] Ga(ǫ) = (χps a , G(ǫ)χps a ) = 1 ǫ − M0 − (2π)3Ta(ǫ), (5) 3 where χps a represents (2π)−3/2[2E(P)]−1/2χP(x1 − x2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The proof of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (5) has been given by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' This work will give Ta(ǫ) in the framework of relativistic quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In field theory the operator T(ǫ) is just the scattering matrix with energy ǫ, and Ta(ǫ) is the T-matrix element between two bound states, which should be defined as ⟨a out|a in⟩ = ⟨a in|a in⟩ − i(2π)4δ(4)(P − P)Ta(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Because of the analyticity of Ta(ǫ), we define Ta(ǫ) = D(ǫ) − iI(ǫ), (6) where ǫ approaches the real axis from above, D and I are the real and imaginary parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' When there is only one decay channel, we can use the unitarity of Ta(ǫ) to obtain [15] 2I(ǫ) = � b (2π)4δ(3)(Pb − P)δ(Eb − ǫ)|Tba(ǫ)|2, (7) where the final 4-vector momentum is Pb = (Pb, iEb) and the T-matrix element Tba(ǫ) is defined as ⟨b out|a in⟩ = −i(2π)4δ(3)(Pb − P)δ(Eb − ǫ)Tba(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The delta-function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (7) means that the energy ǫ in scattering matrix is equal to the total energy Eb of the final state, and � b represents summing over all final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' For Eb = ǫ, we also denote the total energy of the final state by ǫ and I(ǫ) becomes a function of the final state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Using dispersion relation for the function Ta(ǫ), we obtain D(ǫ) = −P π � ∞ ǫM I(ǫ′) ǫ′ − ǫdǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (8) The symbol P means that this integral is a principal value integral and the variable of integration is the total energy ǫ′ of the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The function I(ǫ′) should be calculated over the real interval ǫM < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' As usual the momentum of initial bound state a is set as P = (0, 0, 0, iM0) in the rest frame and ǫM denotes the sum of all particle masses in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Let us suppose that there are several decay channels and the final state b may contain n composite particles and n′ elementary particles in decay channel c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (7), we have I(ǫ′) =1 2 � c′ � d3Q′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='d3Q′ n′d3Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='d3Qn(2π)4δ(4)(Q′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' + Qn − P ǫ′) � spins |T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′)|2, (9) where Q′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Q′ n′, Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Qn are the momenta of final particles, P ǫ′ = (0, 0, 0, iǫ′), T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) is the T-matrix element with respect to ǫ′, � spins represents summing over final spins and 4 averaging over initial spins, and � c′ represents summing over all open and virtual decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (9) the energy in scattering matrix is equal to the total energy ǫ′ of the final state b which extends from ǫM to +∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=', ǫM < ǫ′ < ∞, while the bare mass M0 and BS amplitude of initial bound state a have been specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' It is obvious that the final state may be a ”virtual” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (9), we have I(ǫ′) > 0 for ǫ′ > ǫM and I(ǫ′) = 0 for ǫ′ ⩽ ǫM, which is the reason that the integration in dispersion relation (8) ranges from ǫM to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' We emphatically introduce the T-matrix element T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In our theory T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) must involve bound state, so this matrix element can not be calculated as ordinary S-matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Using the Heisenberg picture, we can obtain the total matrix element between a final state |Q′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Q′ n′, Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Qn out⟩ and a specified initial bound state |P in⟩ −iR(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) =⟨Q′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Q′ n′, Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Qn out|P in⟩ =i2n′ � d4z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='d4zn′f ∗ Q′ 1(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='f ∗ Q′ n′(zn′)S′−1 z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='S′−1 zn′ × ⟨Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Qn out|Tφ(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='ψ(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' ¯ψ(zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='φ(zn′)|P in⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (10) Here φ and ψ represent boson and fermion field operators in the Heisenberg picture, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The functions f are solutions to the corresponding free field equations of motion, and S′ represents free field propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Of great interest is the matrix element of a time-order product of Heisenberg field operators between bound states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Mandelstam’s approach is generalized to evaluate the bound state matrix element with respect to ǫ′ ⟨Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='Qn out|Tφ(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='ψ(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' ¯ψ(zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='φ(zn′)|P in⟩ = � d4y1d4y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='d4y2n−1d4y2nd4x1d4x2 × ¯χQ1(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='¯χQn(y2n−1, y2n)T(y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='y2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zn′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' x1, x2)χP(x1, x2), (11) where T(y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='y2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='zn′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' x1, x2) is the two-particle irreducible Green’s function, ¯χ and χ are BS wave functions for the final and initial bound states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The function T can, in principle, be evaluated by means of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' It is necessary to emphasize that the general matrix element (11) is calculated with respect to ǫ′, and the energy in T is equal to the final state energy ǫ′ extending from ǫM to +∞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=', ǫM < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Therefore the final state may be a virtual state, while the traditional Feynman diagram represents only the physical case ǫ′ = M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper we introduce a Feynman diagram to represent the virtual states, called virtual Feynman diagram, shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=" The crosses in virtual 5 ' n z j z i z 1 z 2 y 1 y 2 x 1 x FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' 1: General matrix element between two bound states ⟨Q out|Tφ(z1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='ψ(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' ¯ψ(zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='φ(zn′)|P in⟩ with respect to ǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The filled blob represents the two-particle irreducible Green’s function, and the unfilled ellipses represent BS amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The final state energy extends from ǫM to +∞, while the initial state energy is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The crosses mean that the final state is a virtual state, which is different from the traditional Feynman diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Feynman diagram mean that the energy in T is equal to the final state energy ǫ′ extending from ǫM to +∞ while the bare mass M0 of initial bound state is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' When ǫ′ = M0, the crosses in virtual Feynman diagram disappear and it becomes the traditional Feynman diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Removing delta-function factor (2π)4δ(4)(Q′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' + Qn − P ǫ′) in R(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′), we obtain T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' To illustrate this, we imagine that the initial bound state (MS) is composed of two heavy vector mesons (V M and V M) and the final state contains a heavy meson (HM) and a light meson (LM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' If a bound state with spin j is created by two massive vector fields, its BS wave function can be defined as χj P (λτ)(x1, x2) = ⟨0|TAλ(x1)A† τ(x2)|P, j⟩ and we have given the general form for this BS wave function χj λτ(P, p) in the momentum representation, where p is the relative momentum of two vector fields [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' This BS wave function should satisfy the equation χj λτ(P, p) = − � d4p′ (2π)4∆F λθ(p′ 1)Vθθ′,κ′κ(p, p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' P)χj θ′κ′(P, p′)∆F κτ(p′ 2), (12) where Vθθ′,κ′κ is the interaction kernel, p′ 1 and p′ 2 are the momenta carried by two vector fields, ∆F λθ(p′ 1) and ∆F κτ(p′ 2) are the propagators for the spin 1 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Owing to the effective interaction Lagrangian at low energy QCD (1), we have to consider that the heavy meson is a bound state composed of a quark and an antiquark and investigate the interaction of light 6 meson with quarks in heavy meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Through the heavy meson form factor describing the heavy meson structure, we have obtained the interaction kernel between two heavy vector mesons (V M and V M) derived from one light meson (σ, ω, ρ, φ) exchange in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In our previous works [8, 14], BS equation (12) has been solved and the bare mass M0 and BS wave function χj λτ(P, p) for the bound state composed of two heavy vector mesons have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper, we are interested only in mass shift for molecular state and do not repeat the procedure for solving BS equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Taking into account the internal structure of heavy mesons (V M, V M and HM) and retaining the lowest order value of T [9], we can obtain the T-matrix element with respect to ǫ′ in the momentum representation T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) = ig′ 0ε̺′ µ (Q′)ε̺ ν(Q) (2π)9/2� 8EH(Q)EL(Q′)E(P) � d4kd4p (2π)8 Tr[SD F (p2)¯ΓH ν (Q, q)SC F(p1) × ΓV λ (p′ 1, k)SA F (p3)γµSB F(p4)Γ ¯V τ (p′ 2, k′)χj λτ(P, p)], (13) where p1, p3, p4, p2 are the momenta of four quarks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' p′ 1 and p′ 2 are the momenta of two heavy vector mesons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' q, k and k′ are the relative momenta between quark and antiquark in heavy mesons, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' ε(p) is the polarization vector of vector meson with momentum p, ΓH(K, k) represents BS amplitude of heavy meson, SF(p) is the quark propagator and its superscript is a flavor label, shown as Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In our approach, the initial bound state is considered as a four-quark state [9], so the generalized BS amplitude of initial bound state should be ΓV λ (p′ 1, k)χj λτ(P, p)Γ ¯V τ (p′ 2, k′), which has been specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In Figure 2(a), the energy in T is equal to the energy of final state which is a virtual state, and then the quark momenta in left-hand side of crosses depend on the final state energy and the momenta in right-hand side depend on the initial state energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=', p1 −p2 −p3 +p4 = Q+Q′ = P ǫ′ and p′ 1 −p′ 2 = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In the rest frame, we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) and ǫM < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' When ǫ′ = M0, the crosses in Figure 2(a) disappear and Figure 2(a) becomes the traditional Feynman diagram, shown as Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From Figure 2(b), we have calculated the matrix element T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(M0) with bare mass for meson-meson bound state and obtained the decay width Γ(M0) with bare mass [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Finally, we can expect that Ga(ǫ) has a pole on the second Riemann sheet from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=" (5) ǫ0 ∼= M0 + (2π)3[D(M0) − iI(M0)] = M − iΓ(M0)/2, (14) where ∆M = (2π)3D(M0) is the shift for energy level of resonance and M = M0+(2π)3D(M0) 7 VM VM ) , ( p P j ) ' , ' ( 2 k p V ) , ( q Q H ) , ' ( 1 k p V 4 p 3 p 2 p 1 p ' 2 p ' 1 p P Q ' Q L M H M MS (a) VM VM ) , ( p P j ) ' , ' ( 2 k p V ) , ( q Q H ) , ' ( 1 k p V 4 p 3 p 2 p 1 p ' 2 p ' 1 p P Q ' Q L M H M MS (b) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' 2: Matrix element between bound states in the momentum representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The momenta in the final state satisfy Q + Q′ = P ǫ′ and the momentum of the initial state is P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The solid lines denote quark propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In the rest frame, we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) and ǫM < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In diagram (a) the final state is a virtual state and the crosses mean that the momenta of quark propagators depend on the final state energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' in diagram (b) the crosses disappear when ǫ′ = M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' is the physical mass for resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The bare mass M0 of two-body bound state is obtained by solving BS equation (2), which should not be the mass of physical resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Γ(M0) with bare mass also should not be the width of physical resonance, which should depend on its physical mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Replacing M0 in the momentum of initial bound state by M and setting ǫ′ = M, we can calculate the matrix element T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(M) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (10) and obtain the decay width Γ for physical resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Up to now, a systematical and accurate theoretical approach in the framework of rela- tivistic quantum field theory to investigate resonance has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper, we only explore exotic meson resonance which is considered as an unstable meson-meson molec- ular state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The extension of our approach to more general resonances is straightforward, while the interaction Lagrangian may be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' EXAMPLE As an illustration, we investigate exotic state χc0(3915) [17–19], once named X(3915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In experiments two strong decay modes of χc0(3915) have been observed: J/ψω and D+D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' 8 Here, we assume that the isoscalar χc0(3915) is a mixed state of two unstable molecular states D∗0 ¯D∗0 and D∗+D∗− with spin-parity quantum numbers 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Firstly, we consider the mixed state of two bound states D∗0 ¯D∗0 and D∗+D∗−, and this bare state can be denoted as 1/ √ 2|D∗0 ¯D∗0⟩+1/ √ 2|D∗+D∗−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' [8, 10, 14], we have obtained the bare mass M0 and BS wave function χ0+ λτ (P, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper, our attention is focused on the mass shift due to all decay channels and the decay width of physical resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Let us list all decay channels which are fully open or only just ”virtual”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The narrow state χc0(3915) was discovered in 2005 [17] and for a long time a series of experiments only observed one strong decay mode of χc0(3915): J/ψω denoted as c′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In 2020 LHCb Collaboration observed another decay channel D+D− [19] denoted as c′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Though the neutral channel D0 ¯D0 still has not been observed, this neutral channel should exist for the isospin conservation, which is denoted as c′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Because the total energy ǫ′ of the final state extends from ǫM to +∞, we obtain one virtual channel D∗ ¯D∗ derived from the interaction Lagrangian (1), denoted as c′ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Since bound state lies below the threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=', M0 < MD∗ + M ¯D∗, the virtual channel c′ 4 can not occur inside the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Then we can apply Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (10) and (11) to evaluate the T-matrix element T(c′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) for arbitrary decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In Figure 2, V M and V M become D∗ and ¯D∗, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' HM becomes J/ψ and LM becomes ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' and decay channel J/ψω can be exhibited by these two Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (9), we obtain the function I1(ǫ′) =1 2 � d3Qd3Q′(2π)4δ(4)(Q + Q′ − P ǫ′) � spins |T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′)|2, (15) where T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) is the bound state matrix element with respect to ǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' These heavy mesons J/ψ, D∗ and ¯D∗ are considered as quark-antiquark bound states, and T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) has been given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (13), where flavor labels C = D and A = B represent c-quark and light quark, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The meson-quark coupling constant g′ 0 becomes gω and g2 ω = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='42/2 was obtained within QCD sum rules approach [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' BS amplitudes of heavy vector mesons J/ψ and D∗ have the form ΓV λ (K, k) = (γλ + Kλγ · K/M2 V )exp(−k2/ω2 V ), where ωJ/ψ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='826GeV and ωD∗=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='50GeV [9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' These momenta in Figure 2(a) become p1 = (Q + Q′)/2 + p + k, p2 = (Q′ − Q)/2 + p + k, p3 = k, p4 = Q′ + k, p′ 1 = p + P/2, p′ 2 = p − P/2, Q + Q′ = P ǫ′ = (0, 0, 0, iǫ′) and P = (0, 0, 0, iM0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (13), we can calculate the T-matrix element T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) with respect to arbitrary energy ǫ′ for channel c′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (15), we obtain the 9 function I1(ǫ′) for channel c′ 1 and dispersion relation (8) becomes D1(M0) = −P π � ∞ ǫc′ 1,M I1(ǫ′) ǫ′ − M0 dǫ′, (16) where ǫc′ 1,M = MJ/ψ + Mω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The T-matrix element T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) and the function I1(ǫ′) for channel c′ 1 are calculated over the real interval ǫc′ 1,M < ǫ′ < ∞, and we obtain the mass shift ∆M1 = (2π)3D1(M0) due to channel c′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' For decay channel D+D−, we obtain the function I2(ǫ′) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (9) I2(ǫ′) =1 2 � d3Q1d3Q2(2π)4δ(4)(Q1 + Q2 − P ǫ′) � spins |T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′)|2, (17) where T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) represents the bound state matrix element with ǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Considering the lowest order term of T, we obtain T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) represented graphically by Figure 3, where p1 − p2 − p3 + p4 = p1 − p2 − q3 + q4 = Q1 + Q2 = P ǫ′, p′ 1 − p′ 2 = P, and the crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on Q1 and Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The T-matrix element with respect to ǫ′ for channel c′ 2 becomes T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) = −ig2 (2π)9/2� 8ED+(Q1)ED−(Q2)E(P) � d4kd4k′d4p (2π)12 Tr[Sd F(q3) × ¯ΓD+(Q1, q)Sc F(p1)ΓD∗ λ (p′ 1, k)Sl F(p3)Odl(p3, q3)]χ0+ λτ (P, p)∆F(w) × Tr[Sl F(p4)Γ ¯D∗ τ (p′ 2, k′)Sc F(p2)¯ΓD−(Q2, q′)Sd F(q4)Odl(q4, p4)], (18) where q, q′, k and k′ are the relative momenta between quark and antiquark in heavy mesons, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' the meson-quark coupling constants g were obtained within QCD sum rules approach [8, 16], Odl(p, q) represents the meson-quark vertex, ∆F(w) is the light meson propagator, the superscript of quark propagator SF(p) is flavor label, and l = u, d represents the u, d-antiquark in heavy vector meson D∗0 or D∗+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' BS amplitude of heavy pseudoscalar meson D+ has the form ΓD+(K, k) = iγ5exp(−k2/ω2 D), where ωD=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='50GeV [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The meson-quark vertex Odl(p, q) is unit matrix for one-σ exchange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' and it becomes γµ for one light vector meson exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (17), (8) and (14), we obtain the mass shift ∆M2 due to channel c′ 2 ∆M2 = (2π)3D2(M0) = −P π � ∞ ǫc′ 2,M (2π)3I2(ǫ′) ǫ′ − M0 dǫ′, (19) where ǫc′ 2,M = MD+ + MD−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The T-matrix element T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(ǫ′) and the function I2(ǫ′) for channel c′ 2 are calculated over the real interval ǫc′ 2,M < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=" Following the same procedure 10 1 Q D D D D 2 p w 2 Q 4 q 3 q 4 p 3 p 1 p ' 2 p ' 1 p P M S ) , ( 1 q Q D ) , ' ( 1 k p D ) ' , ' ( 2 k p D ) ' , ( 2 q Q D ) , ( p P FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' 3: Matrix element for decay channel D+D−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The momenta in the final state satisfy Q1+Q2 = P ǫ′, the momentum of the initial state is P, and we have P = (0, 0, 0, iM0), P ǫ′ = (0, 0, 0, iǫ′) and ǫc′ 2,M < ǫ′ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' w represents the momentum of the exchanged light meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' The crosses mean that the momenta of quark propagators and the momentum w of the exchanged light meson depend on the final state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' as for channels c′ 1 and c′ 2, we can calculate the mass shifts ∆M3 and ∆M4 due to the channel c′ 3 and virtual channel c′ 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Since the isospin conservation, we have the constituent quark masses mu = md = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='33GeV, mc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='55GeV [6] and the meson masses Mσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='45GeV, Mω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='782GeV, Mρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='775GeV, Mφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='019GeV, MD∗0 = MD∗+ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='007GeV, MD0 = MD+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='865GeV, MJ/ψ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='097GeV [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' By doing the numerical calculation, we obtain the mass shifts ∆Mi(i = 1, 2, 3, 4) due to three open decay channels J/ψω, D+D−, D0 ¯D0 and one virtual channel D∗ ¯D∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Subsequently, the mass M for physical resonance χc0(3915) can be applied to calculate its decay width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Replacing M0 by M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (13) and setting ǫ′ = M, we calculate the matrix element T(c′ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(M) and obtain that the width for physical decay model χc0(3915) → J/ψω is Γ1 = 2(2π)3I1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Replacing M0 by M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (18) and setting ǫ′ = M, we calculate the matrix element T(c′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='b)a(M) and obtain that the width for physical decay model χc0(3915) → D+D− is Γ2 = 2(2π)3I2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' For the isospin conservation, it is easy to obtain the width Γ3 for physical decay model χc0(3915) → D0 ¯D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Our numerical results are presented in Table I, and the mass M and full width Γ are in good agreement with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Furthermore, the calculated D+D− width Γ2 is very small com- pared with the calculated J/ψω width Γ1, and then we can explain why the decay model χc0(3915) → D+D− had not been observed in experiments for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Therefore, this work provides a further verification for the molecular hypothesis of χc0(3915) and predicts the exact values of these strong decay widths Γ1(χc0(3915) → J/ψω), Γ2(χc0(3915) → D+D−) 11 Quantity M0 ∆M1 ∆M2 ∆M3 ∆M4 M Γ1 Γ2 Γ3 Γ this work 3953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='7 −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='4 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='4 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='6 3922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='3 PDG[20] 3921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='7±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='8±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content='5 TABLE I: Mass M and width Γ for physical resonance χc0(3915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' M0 is the bare mass of mixed state of two bound states D∗0 ¯D∗0 and D∗+D∗−, ∆Mi is the calculated shift due to ith decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' (Dimensioned quantities in MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=') and Γ3(χc0(3915) → D0 ¯D0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper we emphatically illuminate the physical meaning of resonance theory in quantum field theory, and the details in computational process will be shown in our future article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' CONCLUSION We recognize that resonance can not be completely treated as a stationary bound state and provide a reasonable and feasible scheme to describe resonance in the framework of relativistic quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Based on BS wave function, we provide a description of the prepared state and investigate the temporal evolution of two-body bound state as determined by the total Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' According to dispersion relation, the total matrix elements for all decay channels should be calculated with respect to arbitrary energy, and these matrix elements are expressed in terms of the Heisenberg picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Mandelstam’s approach is generalized to calculate the matrix element between bound states with arbitrary energy, which is exhibited in virtual Feynman diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Finally, the mass and decay width for physical resonance are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' In this paper, we illustrate resonance theory in quantum field theory by reference to the example of exotic meson which is considered as an unstable meson-meson molecular state, and obviously our work can be extended to more general resonances and creates a new paradigm for investigating hadron resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdAyT4oBgHgl3EQfwPlq/content/2301.00645v1.pdf'} 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diff --git a/btAzT4oBgHgl3EQfZvwc/content/2301.01355v1.pdf b/btAzT4oBgHgl3EQfZvwc/content/2301.01355v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a70a280105860dcd4c678d3bb7b214fa3a1983b7 --- /dev/null +++ b/btAzT4oBgHgl3EQfZvwc/content/2301.01355v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:17e943ae35d76d242cabcf87e5dae3a3a7afbc61f3e551dc23e7af8b469e20eb +size 1473479 diff --git a/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/2301.13349v1.pdf.txt b/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/2301.13349v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..589b7853684e07cfcbf4abfe7b7cd604a2e1c323 --- /dev/null +++ b/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/2301.13349v1.pdf.txt @@ -0,0 +1,2971 @@ +Unconstrained Dynamic Regret via Sparse Coding +Zhiyu Zhang +Boston University +zhiyuz@bu.edu +Ashok Cutkosky +Boston University +ashok@cutkosky.com +Ioannis Ch. Paschalidis +Boston University +yannisp@bu.edu +Abstract +Motivated by time series forecasting, we study Online Linear Optimization (OLO) under the coupling +of two problem structures: the domain is unbounded, and the performance of an algorithm is measured +by its dynamic regret. Handling either of them requires the regret bound to depend on certain complexity +measure of the comparator sequence – specifically, the comparator norm in unconstrained OLO, and the path +length in dynamic regret. In contrast to a recent work [JC22] that adapts to the combination of these two +complexity measures, we propose an alternative complexity measure by recasting the problem into sparse +coding. Adaptivity can be achieved by a simple modular framework, which naturally exploits more intricate +prior knowledge of the environment. Along the way, we also present a new gradient adaptive algorithm for +static unconstrained OLO, designed using novel continuous time machinery. This could be of independent +interest. +1 +Introduction +Time series forecasting is a fundamental problem in science and engineering. To design forecasting strategies, a +classical procedure is to model the time series based on batched data, either statistically or empirically, and +then deploy such models online. The effectiveness of this procedure critically relies on certain stationarity +of the environment, thus may fail under distribution shifts. The present work addresses this issue from the +perspective of online learning – we design an online fine-tuning framework such that given any oracle forecaster, +the fine-tuned predictions are equipped with robustness guarantees that do not rely on statistical assumptions at +all. +Concretely, we study the following variant of Online Convex Optimization (OCO). In the t-th round, +1. We query an oracle forecaster A for its prediction wt ∈ Rd, determine a fine-tuning adjustment xt ∈ Rd, +and then predict their sum xt + wt. +2. The environment reveals a true value zt ∈ Rd and a convex G-Lipschitz1 loss function lt : Rd → R, +minimized at zt. We suffer the loss lt(xt + wt). +Our goal is to achieve low regret against any alternative sequence of predictions y1, . . . , yT ∈ Rd selected in +hindsight, where T is the time horizon. The y1:T sequence does not have to be the true time series z1:T , which +will be clear shortly. +Since for any subgradient gt ∈ ∂lt(xt + wt) we have lt(xt + wt) − lt(yt) ≤ ⟨gt, xt + wt − yt⟩, for the rest of +the paper we will assume only observing gt (instead of lt and zt), and define the regret in the formulation of +Online Linear Optimization (OLO) [Haz16, Ora19]. Let ut = yt − wt be the “ideal” fine-tuning adjustment had +we known the comparing sequence y1:T beforehand. Then, +RegretT (u1:T ) := +T +� +t=1 +⟨gt, xt − ut⟩ = +T +� +t=1 +⟨gt, xt + wt − yt⟩ . +(1) +As an example, if y1:T = z1:T , then bounding the regret subsumes bounding the forecasting error �T +t=1 lt(xt +wt). +Through the lens of OLO, we call Eq.(1) the unconstrained dynamic regret. Compared to the most standard +setting of OLO, the challenge here is due to the coupling of two problem structures. +1With respect to ∥·∥2. +1 +arXiv:2301.13349v1 [cs.LG] 31 Jan 2023 + +• The domain Rd is unbounded. +• The comparator sequence u1:T is time-varying. +Even under only one of these conditions, it appears that the environment is given too much power: no matter +how we predict, there always exist some g1:T and u1:T sequences inducing large regret. Circumventing this issue +relies on comparator adaptivity – instead of only depending on the time horizon T, the regret bound also depends +on certain complexity measures of u1:T . For the time series application, this allows us to write2 +T +� +t=1 +lt(xt + wt) ≤ inf +y1:T +� T +� +t=1 +lt(yt) + RegretT (y1:T − w1:T ) +� +, +(2) +where the minimizing argument y1:T trades off its cumulative loss and the complexity of u1:T = y1:T − w1:T . +Choosing the complexity measure introduces inductive bias into the associated algorithm, which is ubiquitous +in high dimensional statistics and machine learning. The rationale is that no algorithm works well universally +for all problem instances, therefore a meaningful goal is to find the suitable inductive bias for the considered +application, and design the corresponding optimal algorithm for that. +Specifically for our setting, prior works mostly studied the two problem structures separately, as reviewed +in Section 1.2. For static regret (ut = u) on unconstrained domains, the standard complexity measure is the +comparator norm ∥u∥ [MO14, OP16, CO18], whereas for dynamic regret on bounded domains, one typically +considers the path length �T −1 +t=1 ∥ut − ut+1∥ [Zin03, ZLZ18]. A recent work [JC22] studied unconstrained dynamic +regret by combining these two complexity measures, resulting in the regret bound (simplified) +RegretT (u1:T ) = ˜O +� +�G +� +� +� +� +� T +� +t=1 +∥ut∥2 +� �T −1 +� +t=1 +∥ut+1 − ut∥2 +�� +� . +However, by closely examining its inductive bias, such a bound may not be the most natural one for “non- +converging” environments. In order to achieve low regret (hence low cumulative loss via Eq.(2)), it is implicitly +assumed that the residual sequence u1:T = y1:T −w1:T is small and almost constant. The latter poses a somewhat +stringent requirement on the oracle forecaster A. For example, if u1:T is periodic, as often encountered in time +series with seasonality, then the regret bound is in general linear in T. Moreover, the regret bound is achieved +by a heavily customized mirror descent algorithm, which deviates from classical frameworks and relies on rather +sophisticated algebra. +In this paper, we will take a conceptually different sparse coding approach. The obtained regret bound adapts +to a new complexity measure of u1:T , which naturally exploits more intricate prior knowledge of the environment. +1.1 +Contribution +The contributions of this paper are twofold. +• Our first contribution is a simple framework that achieves a new type of unconstrained dynamic regret +bounds (Section 2). In a broad sense, it is based on two ideas. +1. We consider the sequence space RdT that contains x1:T , g1:T and u1:T , rather than the default domain +Rd that contains per-round quantities. This is a fundamental view for the batch analysis of sequential +data, such as in signal processing [Mal08, VKG14] and time series modeling [BD16, SS17], but (in our +opinion) under-explored in the online learning literature.3 Static online learning can be considered as +a special case. +2. We use advances in static unconstrained OLO to aggregate dynamic base algorithms. In contrast +to expert-based model selection approaches, this enables learning linear combinations of the base +algorithms, rather than their convex combinations. +2Such bounds are called oracle inequalities in statistical learning. +3Possibly due to the emphasis on static regret by the community: the sequence u1:T collapses into a time-invariant u. +2 + +Combining the two ideas converts our problem into Online Linear Regression (OLR). If the comparator +u1:T can be linearly represented by a certain collection of feature vectors (i.e., a dictionary) in RdT , then +our regret bound adapts to (i) the energy of u1:T ; and (ii) the sparsity of its representation, without +knowing either conditions beforehand. This brings two advantages. +1. Our approach is built upon close connections to signal processing, thus can benefit from prior works +there. For example, a major research topic4 in signal processing is finding the appropriate (typically +redundant) dictionary for specific applications, such that the considered signal admits a sparse +representation. We allow taking such a dictionary as prior knowledge and adapting to its quality. +2. Instead of requiring heavy customization like [JC22], many static unconstrained OLO algorithm, +given the dictionary, can be used as a black box to solve OLR. Therefore, our approach automatically +inherits a wide range of favorable properties from the static regret setting, such as Lipschitz constant +adaptivity [Cut19a], scale-freeness [MK20] and generalized loss-regret tradeoffs [ZCP22a]. +Overall, the proposed approach is not a replacement, but a complement to [JC22]. They represent different +inductive bias, thus should be selected based on the specific application at hand. Nonetheless, simply +adding them can always theoretically guarantee the best of both worlds. +• Our second contribution is a new static unconstrained OLO algorithm, which can be used as a subroutine +of the sparse coding framework (Section 3). +To explain what it does, let us consider again the time series application. Intuitively, given an oracle +forecaster A, we have to determine how much we trust it. This is essentially a tradeoff: if we want +low regret on comparators y1:T that are close to w1:T , we have to sacrifice the regret with respect to +far-away comparators, and vice versa. In the setting of static regret, our prior work [ZCP22a] proposed +a continuous-time-inspired algorithm with the optimal tradeoff,5 but the bound is not simultaneously +adaptive to the gradient variance. Such gradient adaptivity has been a hallmark of practical algorithms, as +popularized by AdaGrad [DHS11]. +In this paper, we propose an algorithm that closes this gap. The key technique is a new discretization +argument that quantifies the deviation of the discrete time algorithm from its ideal, continuous time +counterpart. Plugging it into the sparse coding framework, we obtain a dynamic regret bound that adapts +to not only the sparsity of the comparator (on the transform domain), but also the sparsity of the observed +gradients (on the time domain). +1.2 +Related work +Our paper addresses the connection between unconstrained online learning and dynamic regret. Although they +both embody the idea of comparator adaptivity, unified studies have been scarce. +Unconstrained OLO +To obtain static regret bounds in OLO, Online Gradient Descent (OGD) [Zin03] is +often the default approach. With learning rate η, it guarantees O(η−1 ∥u∥2 +2 +ηT) regret with respect to any static +comparator u ∈ Rd. Without the prior knowledge of ∥u∥2, it is impossible to tune η optimally. To address this +issue, a series of works (also called parameter-free online learning) [SM12, MO14, OP16, CO18, FRS18, MK20, +ZCP22a] developed vastly different strategies to achieve the oracle optimal rate O(∥u∥ +√ +T) up to logarithmic +factors. Most recent works are based on a dual space analysis and an elegant loss-regret duality [MO14], with +the model selection approach from [FKMS17, CLW21, JC22] being a notable exception. +In these regret bounds, the complexity of u is measured by the comparator norm ∥u∥, or more generally, +∥u − w∥ given a prior w. L1 and L2 norm bounds were presented in [SM12], while general Banach norm bounds +were developed by [FRS18, CO18]. Historically, the L1 norm has renowned connections to sparsity, as suggested by +LASSO [Tib96], compressed sensing [CRT06], and several works in online learning [KW95, SM12, Ger13, vdH19]. +However, we are not aware of any prior use of such regret bounds in characterizing the structural simplicity of +nonstationary environments. +4As the title of [Mal08] suggests. Often framed as representation learning. +5Defined as achieving O( +√ +T) regret without the doubling trick, c.f., Section 3.1. +3 + +Our second contribution is dedicated to static unconstrained OLO itself, thus requires a more detailed review +of existing results. This is deferred to Section 3.1 for cleaner exposition. +Dynamic regret +Although the field of online learning primarily focused on the static regret, comparing against +dynamic sequences has been studied by several lines of works. The closest topic to ours is the universal dynamic +regret, where the regret bound adapts to the complexity of the comparator u1:T on a bounded domain with diameter +D. Typically, the complexity measure is the path length PT,p = �T −1 +t=1 ∥ut − ut+1∥p [HW01] or its generalization, +e.g., norm squared [KMBAY15]. The optimal bound for OLO is O(G +� +DTPT,2) [Zin03, HW15, JRSS15, ZLZ18]. +With curved losses, the accelerated rate ˜O(T 1/3P 2/3 +T,1 ) is achievable [BW21, BW22]. +As expected, one cannot go beyond linear dynamic regret in the worst case. The hope is that for “converging” +environments where reasonable comparators have short path lengths, the overall regret bound can be sublinear +in T. Except [JC22, LZZZ22], a shared limitation is the requirement of a bounded domain. A practical solution +is to estimate the range of the problem offline, but since the diameter D is used to select the hyperparameter, +wrong estimates will deteriorate the empirical performance of the algorithm. +Besides the universal dynamic regret, there are other notions of dynamic regret that do not induce oracle +inequalities like Eq.(2), e.g., (i) the restricted dynamic regret [YZJY16, ZYY+17, BW19, BW20, BZW21], which +depends on the complexity of certain offline optimal comparator; and (ii) regret bounds that depend on the +functional variation �T −1 +t=1 maxx |lt(x) − lt+1(x)| [BGZ15, CWW19]. They are both incompatible with OLO on +unbounded domains. +Notably, we emphasize the difference between our work and a dynamic model approach from [HW15, ZLZ18]. +On a bounded domain X, their algorithms can take N dynamic models Φt,n : X → X, n ∈ [1 : N] as input. The +regret bound has a similar form as path length bounds [Zin03], but replaces the path length with the error of +the best dynamic model on the comparator, i.e., minn +�T −1 +t=1 ∥ut+1 − Φt,n(ut)∥. Our dictionary also represents +certain dynamic prior knowledge, but a key difference is that instead of using the best dictionary element to +model the comparator, we use the best linear combination of the dictionary. This allows handling unconstrained +domain through subspace modeling. +Online regression +Our framework builds on online regression, which, in its nonparametric form, has been +connected to the path length characterization of dynamic regret [RS14, GG15]. Prior works are mostly restricted +to the square loss, and efficient computation can be a challenge [BW21]. +For the special case of Online Linear Regression (OLR) with square loss, the celebrated VAW forecaster +[AW01, Vov01] guarantees O(N log T) regret against any unbounded coefficient vector ˆu ∈ RN, where N is +the dimension of the feature space. Such a fast rate becomes vacuous when N > T [GY14], therefore [Ger13] +proposed a sparsity regret bound ˜O(∥ˆu∥0) and an accompanying inefficient algorithm as its high dimensional +generalization. Efficient computation was addressed by [GW18], but the obtained result only applies to bounded +ˆu. In some sense, such sparsity regret bounds are the square loss analogue of the L1-norm parameter-free bounds +in OLO. They are also closely related to sparsity oracle inequalities in statistics, as reviewed by [Ger13]. +Parametric time series models +Besides the dynamic regret approach to time series forecasting, significant +research effort has been devoted to parametric strategies with stronger inductive bias, such as the ARMA model, +state space models, and more recent deep learning models. Online learning has been applied to such models +as well [AHMS13, AHZ15, AM16, KM16, HLS+18], leading to forecasting guarantees under mild statistical +assumptions. Taking the autoregressive (AR) model for example, we will show that learning it can be converted +to an instance of the sparse coding framework. +Other sparsity topics in OL +Finally, we review other sparsity-related topics in online learning, which do +not fit into the scope of this paper. [LLZ09, Xia09, DSSST10, SST11] considered using online learning to solve +batch L1 regularized problems. The goal is to achieve sparse predictions instead of sparsity adaptive regret +bounds. [Kal14, FKK16, KKLP17] studied online sparse regression, where only a subset of features are available +in each round. The challenge is to handle bandit feedback in OLR. +4 + +1.3 +Notation +For two integers a ≤ b, [a : b] is the set of all integers c such that a ≤ c ≤ b. Treating all vectors as column vectors, +span(A) denotes the column space of a matrix A. For a function Φ : R × R → R, assuming differentiability, +let ∂1Φ and ∂2Φ be its first order partial derivatives with respect to the two arguments. Similarly, ∂11Φ, ∂12Φ +and ∂22Φ denote second order partial derivatives. f ∗ is the Fenchel conjugate of a function f. log represents +natural logarithm when the base is omitted, and log+(·) := 0 ∨ log(·). KL(·||·) is the KL divergence. ΠX(x) is +the Euclidean projection from x to a closed convex set X. +We define the imaginary error function as erfi(x) = +� x +0 exp(u2)du. Note that it is scaled by √π/2 from the +usual definition, thus can also be queried from standard software packages like SciPy. Let erfi−1 be its inverse +function. Specialized notations for the sparse coding framework are detailed in Section 2.1. +2 +Sparsity adaptive dynamic regret +In this section we present our sparse coding framework for unconstrained dynamic regret. The basic setting is +described in Section 2.1. Focusing on the sequence space RdT , Section 2.2 presents our main result. Section 2.3 +discusses our framework from a generalized primal-dual perspective. +2.1 +Setting +We start by formally introducing our setup. For sequences x1:T , g1:T and u1:T , we will flatten everything and +treat them as dT dimensional vectors, concatenating per-round quantities in Rd. They are called signals. +Our framework requires online access to a dictionary matrix H ∈ RdT ×N, whose columns are N nonzero +feature vectors. We write H in an equivalent block form as [ht,n]1≤t≤T,1≤n≤N, where each block ht,n ∈ Rd×1. +The accompanied linear transform u = Hˆu relates a signal u ∈ RdT to a coefficient vector ˆu ∈ RN. Adopting the +convention in signal processing, we will call RdT the time domain, and RN the transform domain. In general, +symbols without hat refer to time domain quantities, while their transform domain counterparts are denoted +with hat. +With such notations, we consider the following interaction protocol, which could be termed multivariate OLR +with linear loss. In the t-th round, our algorithm observes a d-by-N feature matrix Ht := [ht,n]1≤n≤N, makes a +prediction xt ∈ Rd, receives a loss gradient gt ∈ Rd satisfying ∥gt∥2 ≤ G, and then suffers the loss ⟨gt, xt⟩. The +performance metric is the dynamic regret defined in Eq.(1), where the comparator u1:T is unconstrained in RdT . +2.2 +Main result +Overall, our strategy is to apply a static unconstrained OLO algorithm on the direction of each feature vector, +and then aggregate their predictions. Concretely, let us start with a single feature vector. +Size 1 dictionary +Consider an index n ∈ [1 : N], which is associated to the feature h1:T,n := [h1,n, . . . , hT,n] ∈ +RdT . We suppress the index n and write it as h1:T = [h1, . . . , hT ]. For any comparator u1:T ∈ span(h1:T ), there +exists ˆu ∈ R such that u1:T = h1:T ˆu. The cumulative loss of u1:T can be rewritten as +⟨g1:T , u1:T ⟩ = ⟨g1:T , h1:T ⟩ ˆu = +T +� +t=1 +⟨gt, ht⟩ ˆu, +which is the loss of the coefficient ˆu on surrogate losses ⟨gt, ht⟩. To compete with u1:T ∈ span(h1:T ), it suffices to +run a 1D static regret algorithm that competes with ˆu ∈ R. Formally, we present this procedure as Algorithm 1. +By further assuming bounded ∥ht∥2, Algorithm 1 could take any static unconstrained OLO algorithm as a +black box. However, since the feature ht is revealed before picking xt, we can use a better black box that adapts +to the scale of ht, even if ∥ht∥2 is unbounded. This is crucial for our purpose, as it allows the dynamic regret +bound to adapt to the energy of the comparator, E(u1:T ) := ∥u1:T ∥2 +2. As an example, we present such a black +box as Algorithm 5 in Appendix A.2, which generalizes a recent result [ZCP22a] to the setting with time-varying +but known Lipschitz constants. +5 + +Algorithm 1 Sparse coding with size 1 dictionary. +Require: An algorithm A for static 1D unconstrained OLO, and a nonzero feature vector h1:T ⊂ RdT revealed +online. +1: for t = 1, 2, . . . , T do +2: +Receive ht ∈ Rd, and pass G ∥ht∥2 to A as the Lipschitz constant of its next (t-th) loss. +3: +Query A for its t-th output, and assign it to ˆxt ∈ R. +4: +Predict xt = ˆxtht ∈ Rd, and receive a loss gradient gt ∈ Rd. +5: +Compute ˆgt = ⟨gt, ht⟩, and send it to A as its t-th surrogate loss gradient. +6: end for +Although not simultaneously adaptive to the magnitude of gt, Algorithm 5 enjoys other appealing properties +in the static setting, such as the optimal loss-regret tradeoff (reviewed in Section 3.1) and the optimal leading +constant. Its analysis goes through a non-gradient-adaptive discretization argument (the Discrete Itˆo formula +[HLPR20]), which sets the stage for our improved technique later. Plugging it into Algorithm 1 yields Lemma 2.1 +below. Proofs for this subsection are deferred to Appendix A.3. +Lemma 2.1. Let ˆε > 0 be an arbitrary hyperparameter for Algorithm 5. Applying it as a subroutine, for all +T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees +RegretT (u1:T ) ≤ GεT + +√ +2 ∥u1:T ∥2 G +�� +log +� +1 + ∥u1:T ∥2 +√ +2εT +� ++ 1 +� +, +where εT = ˆε ∥h1:T ∥2. The subscript emphasizes that εT depends on T. +General dictionary +With the single direction learner above, let us turn to the general setting with N features. +We run N copies of Algorithm 1 in parallel, aggregate their predictions, and the regret bound sums Lemma 2.1, +similar to [Cut19b] in the static setting. An extra twist is that each feature is associated with a different +hyperparameter: it introduces a prior on the transform domain, which is essential for the overparameterized +regime with N ≫ dT. In summary, the pseudocode is presented as Algorithm 2, and the regret bound is +Theorem 1. +Algorithm 2 Sparse coding with general dictionary. +Require: A dictionary H = [ht,n], where ht,n ∈ Rd. Constants ˆε1, . . . , ˆεN > 0. +1: For all n ∈ [1 : N], initialize a copy of Algorithm 1 as An. It runs Algorithm 5 as a subroutine, with +hyperparameter ˆεn. +2: for t = 1, 2, . . . , T do +3: +Receive Ht = [ht,n]1≤n≤N. For all n, send ht,n to An, and query its prediction wt,n. +4: +Predict xt = �N +n=1 wt,n. +5: +Receive loss gradient gt, and send it to A1, . . . , AN as loss gradients. +6: end for +Theorem 1. For all T ∈ N+ and u1:T ∈ RdT , Algorithm 2 guarantees +RegretT (u1:T ) ≤ 2GET + +√ +2GUT +�� +log+ +UT +√ +2ET ++ +� +KL(q||π) + 2 +� ++ G +T +� +t=1 +∥ut,0∥2 , +where +1. For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N +n=1 u1:T,n; +2. ET = �N +n=1 ˆεn ∥h1:T,n∥2 and UT = �N +n=1 ∥u1:T,n∥2; +3. π and q are N dimensional probability vectors defined by πn = ˆεn ∥h1:T,n∥2 /ET , and qn = ∥u1:T,n∥2 /UT . +6 + +To interpret this result, we start from the simplest case. If the size N = d, the dictionary Ht = Id (the d +dimensional identity matrix), and the hyperparameters satisfy �N +n=1 ˆεn = ε, then against any static comparator +(ut = u ∈ Rd), Theorem 1 guarantees +RegretT (u1:T ) ≤ 2εG +√ +T + +√ +2 ∥u∥1 G +√ +T +�� +log+ +∥u∥1 +√ +2ε + +� +KL(q||π) + 2 +� +, +(3) +where πn = ˆεn/ε, and q = u/∥u∥1. Intuitively, since ε can be tuned arbitrarily low, the first term on the RHS is +typically negligible. Within the rest, the KL term gives the bound a Bayesian flavor:6 we use a prior π to guess +the posterior distribution q, i.e., how the “strength” of the comparator is spread across different feature vectors. +Simply picking π as the uniform distribution results in KL(q||π) ≤ log N, and the bound recovers the standard +˜O(∥u∥1 +√ +T) bound in static unconstrained OLO [Ora19, Section 9.3]. +Next, we enter the dynamic realm. Assume feature vectors are orthogonal, and the comparator u1:T ∈ span(H). +Within Theorem 1, we are free to set u1:T,0 = 0, and let u1:T,n be the projection of u1:T onto span(h1:T,n). Due +to orthogonality, the projection preserves the energy of the comparator, i.e, +E(u1:T ) = +T +� +t=1 +∥ut∥2 +2 = +N +� +n=1 +∥u1:T,n∥2 +2 . +By further defining SH(u1:T ) := (�N +n=1 ∥u1:T,n∥2)2/�N +n=1 ∥u1:T,n∥2 +2, we have +UT = +� +SH(u1:T )E(u1:T ). +Note that SH(u1:T ) is a classical sparsity measure of {u1:T,n}1≤n≤N [HR09]: if there are only N0 ≤ N nonzero +vectors within this collection, then SH(u1:T ) ≤ N0 due to the Cauchy-Schwarz inequality. Therefore, through +the complexity measure UT , Theorem 1 adapts to (i) the energy of u1:T ; and (ii) the sparsity of its rep- +resentation, without knowing either condition beforehand. With low enough ET , the bound has the order +˜O( +� +SH(u1:T )E(u1:T )) ≤ ˜O( +√ +NT): the easier the comparator is (low energy, and sparse on H), the lower the +bound becomes. +So far we have only considered the underparameterized regime (N ≤ dT) where feature vectors can be +orthogonal. However, recent trends in signal processing have emphasized overparameterization (N ≫ dT) as a +key to obtain sparser representations. Theorem 1 can be nicely interpreted in this context as well: since it applies +to any decomposition of u1:T , as long as u1:T can be represented by a subset ˜H of orthogonal features within +H, the regret bound adapts to S ˜ +H(u1:T ), i.e., the sparsity of u1:T measured on ˜H. In other words, Theorem 1 +adapts to the quality of the optimal (comparator-dependent) sub-dictionary ˜H. Note that: +• Algorithm 2 runs N base algorithms in parallel. For efficient computation with large N, the dictionary +itself has to be sparse, which is called the local property in signal processing [Mal08]. See Appendix A.4 for +a comparison between Fourier and wavelet dictionaries. +• Theorem 1 suffers a large-N penalty through the KL term. In practice, one may pick a good prior π, +instead of the uniform distribution, to reduce this root-logarithmic overhead. +Power law phenomenon +To further demonstrate the quantitative benefit, let us consider an empirically +justified setup. In signal processing, the study of sparsity has been partially motivated by the power law [Pri21]: +for many real world signals, even with a standard Fourier or wavelet dictionary, the n-th largest transform +domain coefficient has magnitude roughly proportional to n−α, where α ∈ (0.5, 1). Suppose d = 1, and the +comparator u1:T exhibits the power law through an orthogonal transformation of RT . Then, when T is large, +SH(u1:T ) = (�T +n=1 n−α)2 +�T +n=1 n−2α +≈ 2α − 1 +(1 − α)2 (T)2−2α = O +� +T 2−2α� +. +With E(u1:T ) = O(T), we obtain a sublinear ˜O(T 1.5−α) dynamic regret bound. +6Analogous to comparator adaptive bounds in the expert problem [LS15, KVE15, CLW21, NBC+21]. +7 + +Example +Finally, we note that the strength of our framework lies in the incorporation of domain knowledge +through the dictionary H. In Appendix A.4, we discuss several concrete examples, including classical Fourier +and wavelet dictionaries, the autoregressive dictionary defined by time series, and dictionaries learned by online +learning algorithms. As an added bonus, different unconstrained dynamic regret bounds, such as [JC22] and +the different instances of Theorem 1, can be combined by simply summing their corresponding predictions +(Appendix A.5). +2.3 +Primal-dual interpretation +Concluding this section, we discuss our framework from a primal-dual perspective. In static OLO [Ora19], the +primal space refers to the domain Rd, while the dual space refers to the space of linear maps on Rd, or intuitively, +where we store a sufficient statistic of the observed information. The same algorithm can have different but +equivalent analysis on the primal space and the dual space, e.g, the Follow the Regularized Leader (FTRL) versus +the potential method. Our framework generalizes the static setting, thus can be understood in a similar way. +Specifically, we consider an analogous primal-dual relation between the time domain RdT and the transform +domain RN. From this angle, Algorithm 2 runs as follows, c.f., Figure 1. +Environment +Transform domain ℝ𝑁 +𝑆𝑡 = 𝑆𝑡−1 − ො𝑔𝑡 +ො𝑔𝑡 = ℋ𝑡 +𝑇𝑔𝑡 +Potential on ℝ𝑁 +ො𝑥𝑡+1 = 𝜕Φ(𝑆𝑡) +𝑥𝑡+1 = ℋ𝑡+1 ො𝑥𝑡+1 +Figure 1: Algorithm 2 as potential method with general sufficient statistic. +• At the end of the t-th round, we multiply gt ∈ Rd by the d-by-N feature matrix Ht. The sufficient statistic +in RN is updated as St = St−1 − HT +t gt. +• By evaluating the gradient of a potential function Φ at St, we obtain a transform domain prediction +ˆxt+1 ∈ RN (analogous to Line 3 of Algorithm 1). +• After the next feature matrix Ht+1 is revealed, we define the t + 1-th prediction as xt+1 = Ht+1ˆxt+1 ∈ Rd. +Crucially, instead of storing the sum of loss gradients (as in static OLO), we store a N dimensional filtered +version of the gradient sequence g1:T . Roughly speaking, N captures the complexity of the comparator class, +against which our algorithm guarantees sublinear regret. +As for the proof strategy, we have focused on aggregating regret bounds on the time domain. Alternatively, +one could use a transform domain analysis to obtain the same result, generalizing the standard workflow in static +unconstrained OLO [MO14]. The key is a loss-regret duality for sequences.7 +Lemma 2.2. If there exists a function fT : RdT → R such that the prediction sequence x1:T guarantees a loss +upper bound ⟨g1:T , x1:T ⟩ ≤ −fT (g1:T ), then for all u1:T ∈ RdT , +RegretT (u1:T ) ≤ f ∗ +T (−u1:T ). +In general, the function fT can be fully nonlinear, so we consider a more structured function class where the +nonlinearity acts on a linear sketch of the input, i.e., fT (x) = Φ(HT x) for some nonlinear potential function +Φ : RN → R. By picking xt = Htˆxt, constructing the loss upper bound is converted into a coin-betting problem +with decision ˆxt ∈ RN, where existing theoretical results are available [Cov66, OP16, ZCP22a]. +7We present two other versions in Appendix A.6, which are more closely tied to path-length-based dynamic regret bounds. +8 + +3 +A better static algorithm +As shown in Figure 1, the sparse coding framework consists of two components: (i) choosing a dictionary H that +captures the dynamics of the environment; and (ii) designing a good potential function (or static unconstrained +OLO algorithm) with low quantitative regret bound. We now present our second contribution, which addresses +the latter. Section 3.1 surveys the background of this topic, while our new algorithm, including its implication +for the sparse coding framework, is presented in Section 3.2. For static comparators ut = u ∈ Rd, we will write +the regret as RegretT (u). +3.1 +Loss-regret tradeoff +An important topic in static unconstrained OLO is the loss-regret tradeoff. Due to a celebrated no free lunch +theorem [Cov66], all such algorithms are required to trade off their cumulative loss RegretT (0) with their +leading regret term, i.e., RegretT (u) for large ∥u∥. Roughly speaking, such a tradeoff represents how much +we trust the initialization of the algorithm. Most prior works [MO14, OP16] are natively designed with O(1) +loss and O(∥u∥ +� +T log(∥u∥ T)) regret, while in principle, the optimal tradeoff corresponds to O( +√ +T) loss and +O(∥u∥ +� +T log(∥u∥)) regret, which matches the minimax optimal O( +√ +T) rate on bounded domains (with respect +to T alone). Although different loss-regret tradeoffs are mutually convertible through the doubling trick [SS11], +doing so significantly downgrades the empirical performance of the algorithm,8 thus should (ideally) be avoided +in theory as well. +A recent work of ours [ZCP22a] achieved the optimal tradeoff in an “anytime” manner without doubling +tricks. However, compared to other frontiers in this field (e.g., [MK20]), the regret bound does not simultaneously +adapt to the observed gradient variance VT . The importance of such gradient adaptivity has been demonstrated +in practice [DHS11], but from a technical perspective, it is challenging to add this property to non-gradient- +adaptive unconstrained algorithms, as both the algorithm and the analysis need to be modified with considerable +sophistication. Existing techniques [CO18, MK20, JC22] are closely tied to the suboptimal loss-regret tradeoff,9 +and their extensions to our objective are unclear. +At the center of the optimal tradeoff [ZCP22a] is a nonstandard erfi potential function, which solves a Partial +Differential Equation (PDE) that characterizes the continuous time (CT) limit of the learning game. In a broader +context, the interplay between discrete time (DT) online learning and its CT limit has received growing attention +[KS10, Zhu14, BEZ20, DK20, HLPR20, KKW20, PLH22, ZCP22b], as the latter is often easier to analyze and +gain intuition from. However, a bottleneck here is the discretization of CT-derived algorithms – the standard +technique is the Discrete Itˆo formula [HLPR20], which by construction is not gradient adaptive. Therefore, +although gradient adaptivity has been studied in CT before, e.g., [Fre09] and [HLPR20, Appendix B.4], the +obtained benefits have not been extended to the DT online learning problem we consider. +In this section, we improve [ZCP22a] by simultaneously achieving gradient adaptivity and the optimal +loss-regret tradeoff, without doubling tricks. The key technique is a new discretization argument that further +induces gradient adaptivity, which could be of separate interest. +3.2 +Main result +Concretely, we consider OLO with domain Rd and a known Lipschitz constant G. Compared to the setting of +Section 2.1, we now focus on the static regret RegretT (u). Our approach requires two potential functions defined +as follows, where the parameters satisfy ε > 0, α > 0 and z > k > 0. +φ(x, y) = ε√αx +� +2 +� +y +√ +4αx +0 +erfi(u)du − 1 +� +, +Φ(V, S) = φ(V + z + kS, S). +(4) +φ is the “basic” potential function from [ZCP22a], which solves the Backward Heat Equation (BHE) ∂1φ+α∂22φ = +0. Note that φ can be evaluated efficiently using Lemma A.1. Φ is the potential function we actually apply, +which is constructed from φ with a change of variable. +8And incurs a multiplicative constant in the bound. +9With O(√VT log VT ) dependence on VT alone, worse than the optimal O(√VT ) rate achieved by adaptive OGD on bounded +domains. +9 + +Overall, our algorithm has a hierarchical structure. The key component is the 1D base algorithm (Algorithm 3), +where for clarity, all the algorithmic quantities are denoted with tilde. We enforce the requirement − �t +i=1 ˜li ≥ −1 +to make sure ˜St ≥ −1, thus the gradient computation in Line 3 is well-defined. Then, the meta-algorithm +(Algorithm 4) applies two standard techniques [CO18, Cut20] on top of the base algorithm: the first reduces the +domain of the base algorithm from R to R+, while the second extends it from R+ to Rd. +Algorithm 3 1D base algorithm +Require: The potential function Φ defined in Eq.(4). Constants ε > 0, α > 0 and z > k > 0. Surrogate loss +gradients ˜l1:T satisfying ˜lt ∈ [−1, 1] and − �t +i=1 ˜li ≥ −1 for all t. +1: Initialize ˜V0 = 0, ˜S0 = 0. +2: for t = 1, 2, . . . , T do +3: +Predict ˜zt = ∂2Φ( ˜Vt−1, ˜St−1). +4: +Receive the surrogate loss gradient ˜lt. +5: +Let ˜Vt = ˜Vt−1 + ˜l2 +t , and ˜St = ˜St−1 − ˜lt. +6: end for +Algorithm 4 Meta algorithm on Rd. +1: Define A1d as a copy of Algorithm 3. Define AB as OGD on the d-dimensional unit L2 norm ball, with +adaptive learning rate ηt = +� +2/�t +i=1 ∥gi∥2 +2. Initialization of AB is arbitrary. +2: for t = 1, 2, . . . do +3: +Query A1d for its prediction ˜zt ∈ R. Let zt = ΠR+(zt). +4: +Query AB for its prediction wt ∈ Rd. +5: +Predict xt = ztwt, receive the loss gradient gt ∈ Rd. +6: +Send gtG−1 as the surrogate loss to AB. +7: +Define lt = +� +gtG−1, wt +� +, and +˜lt = +� +lt, +lt˜zt ≥ ltzt, +0, +else. +8: +Send ˜lt as the surrogate loss to A1d. +9: end for +Before analyzing its performance, Proposition 3 in Appendix B.3 shows that the surrogate loss ˜lt defined in +the meta-algorithm indeed satisfies − �t +i=1 ˜li ≥ −1, therefore the entire hierarchical procedure is well-posed. +Then, with the gradient variance defined as VT = �T +t=1 ∥gt∥2 +2, we present the regret bound as Theorem 2. +Theorem 2. With ε > 0, α = 1, k = 2 and z = 16, Algorithm 4 guarantees for all T ∈ N+ and u ∈ Rd, +RegretT (u) ≤ ε +� +VT + 2G ¯S + ∥u∥2 +� +¯S + 2 +� +2VT +� +, +where +¯S = 8G +� +1 + +� +log(2 ∥u∥2 ε−1 + 1) +�2 ++ 2 +� +VT + 16G2 +� +1 + +� +log(2 ∥u∥2 ε−1 + 1) +� +. +Let us make this bound a bit more interpretable. Using asymptotic orders, we can simplify it into (see +Appendix B.3 for the derivation) +RegretT (u) ≤ ε +�� +VT + 6G +� ++ ∥u∥2 O +�� +VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) +� +, +which is simultaneously valid in two regimes: (i) ∥u∥2 ≫ ε and VT ≫ G2; and (ii) u = 0, i.e., RegretT (0) ≤ +ε +�√VT + 6G +� +. Note that the logarithmic residual term (outside the root) is standard in gradient adaptive uncon- +strained OLO. Therefore, with a O(√VT ) maximum loss bound, Algorithm 4 guarantees a O(∥u∥2 +√VT log ∥u∥2) +10 + +regret bound10 – this matches the minimax optimal rate O(√VT ) on bounded domains, achieved by adaptive +OGD [DHS11]. Compared to prior works, we improve [ZCP22a] by achieving second order gradient adaptivity, +and [CO18, MK20, JC22] by a better asymptotic rate on VT . +Sketch of the analysis +We now sketch our analysis of the base algorithm (Algorithm 3), including the key +idea of discretization. At one point we consider two-case cases. Alternate expressions for the second case are +provided in red. Overall, the analysis has a similar procedure as typical potential methods: we first upper-bound +the cumulative loss �T +t=1 ˜lt˜zt, and then obtain the regret bound through a loss-regret duality [MO14]. The loss +upper bound follows from a telescopic sum on the one-step bound: +˜lt˜zt = ˜lt∂2Φ( ˜Vt−1, ˜St−1) ≤ Φ( ˜Vt−1, ˜St−1) − Φ( ˜Vt, ˜St). +Proving it is the main technical challenge of our analysis (Lemma B.3), as in most prior works. +To this end, we aim to show that for all V ≥ 0, S ≥ −1 and c ∈ [−1, 1] satisfying S + c ≥ −1, +fV,S(c) := Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S) ≤ 0. +It is clear that fV,S(0) = f ′ +V,S(0) = 0. Therefore, to prove fV,S(c) ≤ 0, a sufficient condition is the concavity of +fV,S on the considered input domain. By calculating the Hessian and using |c| ≤ 1, +f ′′ +V,S(c) ≤ 2∂1Φ(V + c2, S + c) + 4∂11Φ(V + c2, S + c) + 4 +��∂12Φ(V + c2, S + c) +�� + ∂22Φ(V + c2, S + c). +Furthermore, due to Eq.(4), the derivatives of Φ are concisely related to the derivatives of φ: if ∂12Φ(V + c2, S + +c) ≤(≥) 0, then +f ′′ +V,S(c) ≤ 2∂1φ + [k −(+) 2]2∂11φ + 2[k −(+) 2]∂12φ +� +�� +� +:=∆ ++∂22φ, +(5) +where the derivatives on the RHS are evaluated at the input pair (V + c2 + z + k(S + c), S + c). +Now, the key observation is that the RHS of Eq.(5) has a striking similarity to the Backward Heat Equation +∂1φ + α∂22φ = 0, which the basic potential function φ satisfies. This motivates us to view ∆ as the discretization +error. Ideally, if ∆ ≤ 0, then f ′′ +V,S(c) ≤ 0 by simply picking α = 1/2. The reality is only slightly more complicated: +• We pick k = 2 to eliminate the harder case within the two. +• As for the other case, it only occurs when S + c is at most constant-away from 0. Picking a large enough +constant offset z, we have ∆ ≤ ∂22φ, therefore f ′′ +V,S(c) ≤ 0 follows from α = 1. +In summary, through a change of variable, we show how to utilize the CT property (i.e., the BHE) of potential +functions in the verification of DT algorithms. The discretization error is more finely characterized compared to +the Discrete Itˆo formula (surveyed in Appendix A.2), which results in additional gradient adaptivity. Moreover, +since the BHE succinctly captures a family of adaptive potentials [ZCP22a], the argument above could be +applicable to other loss-regret tradeoffs as well. Such generality and simplicity may provide benefits over existing +techniques without CT connections [CO18, MK20, JC22]. +Application to dynamic regret +Finally, we apply this static algorithm to bound the dynamic regret, through +our sparse coding framework. Slightly different from Section 2, we impose an additional assumption, ∥ht,n∥2 ≤ 1. +As shown in Lemma B.8, with asymptotic simplification, the dynamic bound against any u1:T ∈ span(H) becomes +RegretT (u1:T ) ≤ +N +� +n=1 +ˆεn +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht,n⟩2 + 6G +� +� + ˜O +� +� +N +� +n=1 +� +� +� +� +T +� +t=1 +⟨gt, ut,n⟩2 +� +� , +where the first sum is a cumulative loss term tuned to be small, and {ut,n} is an arbitrary decomposition of the +comparator satisfying �N +n=1 ut,n = u1:T . Compared to Theorem 1 which guarantees a similar form +RegretT (u1:T ) ≤ 2G +N +� +n=1 +ˆεn +� +� +� +� +T +� +t=1 +∥ht,n∥2 +2 + ˜O +� +�G +N +� +n=1 +� +� +� +� +T +� +t=1 +∥ut,n∥2 +2 +� +� , +10Loosely assimilating the residual term for clarity. +11 + +our improved approach further achieves gradient adaptivity. In other words, the obtained algorithm adapts +to not only the sparsity of the comparator (on the transform domain), but also the sparsity of the observed +gradients (on the time domain). +4 +Conclusion +In this paper, we presented two complementary results for unconstrained OLO. +• Through a sparse coding framework, one can convert static unconstrained OLO algorithms to the dynamic +setting, and the regret bound adapts to both the energy and the sparsity of the comparator sequence. This +is closely connected to representation learning, thus may lead to deeper integration of the two research +areas. +• We propose an algorithm that simultaneously achieves the gradient variance adaptivity and the optimal loss- +regret tradeoff. The key technique is a new discretization argument, which could facilitate the continuous +time analysis of online learning in general. +References +[AHMS13] Oren Anava, Elad Hazan, Shie Mannor, and Ohad Shamir. 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Advances in Neural Information Processing Systems, 30, 2017. +16 + +Appendix +Organization +Appendix A presents details on our sparse coding framework (Section 2). Appendix B presents +details on the improved static unconstrained OLO algorithm (Section 3). +A +Detail on sparse coding +The sparse coding framework requires running a static OLO algorithm as a subroutine. We present some basic +facts on the erfi function in Appendix A.1, and a non-gradient-adaptive static OLO subroutine in Appendix A.2. +Appendix A.3 contains the proof of our sparsity adaptive regret bound. Appendix A.4 discusses several concrete +choices of the dictionary. Appendix A.5 shows how to combine algorithms with different unconstrained dynamic +regret guarantees. Appendix A.6 complements the primal-dual interpretation of our framework from Section 2.3. +A.1 +Fact on the erfi function +First of all, we will use the following facts on the erfi function. Note that in this paper, we scale it from its usual +definition, c.f., Section 1.3. +Lemma A.1. For all x ∈ R, +� x +0 +erfi(u)du = xerfi(x) − 1 +2 exp(x2) + 1 +2. +The proof follows from a simple integration by parts, therefore omitted. +Lemma A.2. For all x ≥ 1, erfi(x) ≥ exp(x2)/2x. +Proof of Lemma A.2. Let f(x) = erfi(x) − exp(x2)/2x. f(1) = erfi(1) − e/2 > 0. For all x ≥ 1, +f ′(x) = +1 +2x2 exp(x2) > 0. +Lemma A.3 (From Theorem 4 of [ZCP22a]). For all x ≥ 0, erfi−1(x) ≤ 1 + +� +log(x + 1). +A.2 +Unconstrained OL with varying Lipschitzness +We present a non-gradient-adaptive, static 1D unconstrained OLO algorithm as Algorithm 5. It is designed to +exploit time-varying, but known Lipschitz constants on the loss functions. The regret bound is Lemma A.4. +Algorithm 5 1D Static unconstrained OLO with time-varying Lipschitzness. +Require: A hyperparameter ˆε > 0. A sequence of Lipschitz constants G1:T such that each loss gradient ˆgt ∈ R +satisfies |ˆgt| ≤ Gt. +1: Initialize V0 = S0 = 0. Define a potential function as +Φ(V, S) = ˆε +√ +V +� +2 +� +S +√ +2V +0 +erfi(x)dx − 1 +� +. +(6) +Note that +� +erfi(x)dx can be evaluated using Lemma A.1. +2: for t = 1, 2, . . . do +3: +Receive the t-th Lipschitz constant Gt, and let Vt = Vt−1 + G2 +t. +4: +If Gt = 0, predict ˆxt = 0. Otherwise, predict +ˆxt = +1 +2Gt +[Φ (Vt, St−1 + Gt) − Φ (Vt, St−1 − Gt)] . +5: +Observe the loss gradient ˆgt ∈ R, and let St = St−1 − ˆgt. +6: end for +17 + +Lemma A.4. For all T ∈ N+ and ˆu ∈ R, Algorithm 5 guarantees +T +� +t=1 +⟨ˆgt, ˆxt − ˆu⟩ ≤ ˆε +� +� +� +� +T +� +t=1 +G2 +t + +√ +2 |ˆu| +� +� +� +� +T +� +t=1 +G2 +t +�� +log +� +1 + |ˆu| +√ +2ˆε +� ++ 1 +� +. +The proof of Lemma A.4 generalizes the argument of [HLPR20, ZCP22a] by allowing arbitrary time-varying +gap parameters. It also demonstrates the existing technique (the Discrete Itˆo formula) for discretizing continuous- +time-derived algorithms. This is in contrast to our improved technique in Section 3. +First, consider a function Φ : R × R → R. In light of standard partial derivatives ∂1Φ, ∂2Φ and ∂22Φ, given a +gap parameter δ > 0, we define discrete derivatives (denoted with bars) as +¯∂δ +1Φ(V, S) = 1 +δ2 +� +Φ(V, S) − Φ(V − δ2, S) +� +, +¯∂δ +2Φ(V, S) = 1 +2δ [Φ(V, S + δ) − Φ(V, S − δ)] , +¯∂δ +22Φ(V, S) = 1 +δ2 [Φ(V, S + δ) + Φ(V, S − δ) − 2Φ(V, S)] . +If δ = 0, define ¯∂δ +1Φ(V, S) = ¯∂δ +2Φ(V, S) = ¯∂δ +22Φ(V, S) = 0. +The Discrete Itˆo formula [HLPR20, Lemma 3.13 and 3.14] has been shown useful in connecting discrete time +online learning algorithms with their continuous time counterparts. We generalize it as follows. +Lemma A.5 (Discrete Itˆo formula with general gap). Consider any function Φ : R≥0 × R → R, convex in its +second argument. For all V ≥ 0, S, c ∈ R and δ ≥ 0 satisfying |c| ≤ δ, we have +Φ(V + δ2, S + c) − Φ(V, S) ≤ c¯∂δ +2Φ(V + δ2, S) + δ2 +� +¯∂δ +1Φ(V + δ2, S) + 1 +2 +¯∂δ +22Φ(V + δ2, S) +� +. +(7) +Proof of Lemma A.5. The case of δ = 0 trivially holds. As for δ > 0, applying the discrete derivatives, +LHS = Φ(V + δ2, S + c) − 1 +2 +� +Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) +� ++ 1 +2 +� +Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) +� +− Φ(V, S) += Φ(V + δ2, S + c) − 1 +2 +� +Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) +� ++ δ2 +� +¯∂δ +1Φ(V + δ2, S) + 1 +2 +¯∂δ +22Φ(V + δ2, S) +� +. +Comparing it with our objective, it remains to show +Φ(V + δ2, S + c) − 1 +2 +� +Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) +� +≤ c¯∂δ +2Φ(V + δ2, S) += c +2δ +� +Φ(V + δ2, S + δ) − Φ(V + δ2, S − δ) +� +. +Regrouping the terms, it suffices to show +Φ(V + δ2, S + c) ≤ δ + c +2δ Φ(V + δ2, S + δ) + δ − c +2δ Φ(V + δ2, S − δ), +which follows from the convexity of Φ. +To apply the Discrete Itˆo formula to Algorithm 5, we need to fix a minor issue: the function Φ(V, S) from +Eq.(6) is not well-defined for V = 0. Without loss of generality, we will impose Φ(0, S) = 0 for all S. Notice +18 + +that Φ from Eq.(6) is convex in S, and the prediction in Algorithm 5 is precisely the discrete derivative, i.e., +ˆxt = ¯∂Gt +2 Φ(Vt, St−1). Plugging in V ← Vt−1, S ← St−1, δ ← Gt and c ← −ˆgt into Lemma A.5, we have +Φ(Vt, St) − Φ(Vt−1, St−1) ≤ −ˆgtˆxt + G2 +t +� +¯∂Gt +1 Φ(Vt, St−1) + 1 +2 +¯∂Gt +22 Φ(Vt, St−1) +� +. +The second term on the RHS can be seen as a perturbation on an otherwise clean recursive inequality. The +form of this perturbation term also closely resembles the Backward Heat Equation (BHE) ∂1Φ + 1 +2∂22Φ = 0, +which, as shown in [ZCP22a], is satisfied by existing potential functions in unconstrained OLO, including Eq.(6). +This explains why the Discrete Itˆo formula is useful: to convert continuous-time-derived algorithms to discrete +time, it suffices to characterize the discretization error on the BHE. As long as the discretization error (the +perturbation term above) is upper-bounded, we can still control the cumulative loss of the algorithm by a +telescopic sum, i.e., +T +� +t=1 +ˆgtˆxt ≤ Φ(0, 0) − Φ(VT , ST ) + +T +� +t=1 +G2 +t +� +¯∂Gt +1 Φ(Vt, St−1) + 1 +2 +¯∂Gt +22 Φ(Vt, St−1) +� +. +(8) +Specifically for Algorithm 5, we bound the perturbation term as follows. It uses a key result from [HLPR20]. +Lemma A.6. For all t ∈ N+, Algorithm 5 guarantees +G2 +t +� +¯∂Gt +1 Φ(Vt, St−1) + 1 +2 +¯∂Gt +22 Φ(Vt, St−1) +� +≤ 0. +Proof of Lemma A.6. Let us define f(x) = 2xerfi(x) − exp(x2). Due to the definition of Φ in Eq.(6) and the +simplification of +� +erfi(x)dx in Lemma A.1, for all V > 0 and S ∈ R, +Φ(V, S) = ˆε +√ +V f +� +S +√ +2V +� +. +When V = 0, we have defined Φ(0, S) = 0. +Now, consider the quantities in Algorithm 5. To proceed, there are two cases: (i) Vt−1 > 0; (ii) Vt−1 = 0. If +Vt−1 > 0, plugging in the discrete derivatives, +G2 +t +� +¯∂Gt +1 Φ(Vt, St−1) + 1 +2 +¯∂Gt +22 Φ(Vt, St−1) +� += 1 +2Φ(Vt, St−1 + Gt) + 1 +2Φ(Vt, St−1 − Gt) − Φ(Vt−1, St−1) += 1 +2 ˆε +� +Vt +� +f +�St−1 + Gt +√2Vt +� ++ f +�St−1 − Gt +√2Vt +� +− 2 +� +Vt−1 +Vt +f +� +St−1 +� +2Vt−1 +�� +. +Due to [HLPR20, Lemma 3.10], for all x ∈ R and z ∈ [0, 1), +f +�x + z +√ +2 +� ++ f +�x − z +√ +2 +� +≤ 2 +� +1 − z2f +� +x +� +2(1 − z2) +� +. +Taking x = St−1/√Vt and z = Gt/√Vt proves the first case. +As for the second case (Vt−1 = 0), note that St−1 = 0 and Vt = G2 +t. Then, +G2 +t +� +¯∂Gt +1 Φ(Vt, St−1) + 1 +2 +¯∂Gt +22 Φ(Vt, St−1) +� += 1 +2Φ(G2 +t, Gt) + 1 +2Φ(G2 +t, −Gt). +If Gt = 0, then it holds trivially that RHS = 0. Otherwise, +RHS = ˆε +2Gt +� +f +� 1 +√ +2 +� ++ f +� +− 1 +√ +2 +�� +≤ 0, +due to straightforward evaluation of f. This completes the proof of the second case. +19 + +With Lemma A.5 and A.6, it becomes fairly standard to prove the guarantee of Algorithm 5, i.e., Lemma A.4. +See, for example, [Ora19, Chapter 9] for the overall proof strategy. +Proof of Lemma A.4. Plugging Lemma A.6 into Eq.(8), we obtain a cumulative loss bound +T +� +t=1 +ˆgtˆxt ≤ −Φ(VT , ST ). +Due to a standard loss-regret duality [Ora19, Theorem 9.6], the regret can be bounded by +T +� +t=1 +⟨ˆgt, ˆxt − ˆu⟩ ≤ Φ∗ +VT (ˆu), +where Φ∗ +VT denotes the Fenchel conjugate of the function ΦVT (·) := Φ(VT , ·). Finally, due to the proof of [ZCP22a, +Theorem 4], +Φ∗ +VT (ˆu) ≤ ˆε +� +VT + |ˆu| +� +2VT +�� +log +� +1 + |ˆu| +√ +2ˆε +� ++ 1 +� +. +A.3 +Proof of main results +This subsection presents the omitted proofs for Section 2.2. +Lemma 2.1. Let ˆε > 0 be an arbitrary hyperparameter for Algorithm 5. Applying it as a subroutine, for all +T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees +RegretT (u1:T ) ≤ GεT + +√ +2 ∥u1:T ∥2 G +�� +log +� +1 + ∥u1:T ∥2 +√ +2εT +� ++ 1 +� +, +where εT = ˆε ∥h1:T ∥2. The subscript emphasizes that εT depends on T. +Proof of Lemma 2.1. We start by rewriting the dynamic regret as +RegretT (u1:T ) = +T +� +t=1 +⟨gt, xt − ut⟩ = +T +� +t=1 +⟨gt, htˆxt − htˆut⟩ = +T +� +t=1 +⟨ˆgt, ˆxt − ˆut⟩ . +The static regret on the RHS can be bounded from Lemma A.4, with the Lipschitz constant Gt = G ∥ht∥2. +Concretely, +RegretT (u1:T ) ≤ Gˆε +� +� +� +� +T +� +t=1 +∥ht∥2 +2 + +√ +2G |ˆu| +� +� +� +� +T +� +t=1 +∥ht∥2 +2 +�� +log +� +1 + |ˆu| +√ +2ˆε +� ++ 1 +� += Gˆε ∥h1:T ∥2 + +√ +2G |ˆu| ∥h1:T ∥2 +� +� +� +� +� +�log +� +1 + |ˆu| ∥h1:T ∥2 +√ +2ˆε ∥h1:T ∥2 +� ++ 1 +� +� += GεT + +√ +2 ∥u1:T ∥2 G +�� +log +� +1 + ∥u1:T ∥2 +√ +2εT +� ++ 1 +� +. +Theorem 1. For all T ∈ N+ and u1:T ∈ RdT , Algorithm 2 guarantees +RegretT (u1:T ) ≤ 2GET + +√ +2GUT +�� +log+ +UT +√ +2ET ++ +� +KL(q||π) + 2 +� ++ G +T +� +t=1 +∥ut,0∥2 , +where +20 + +1. For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N +n=1 u1:T,n; +2. ET = �N +n=1 ˆεn ∥h1:T,n∥2 and UT = �N +n=1 ∥u1:T,n∥2; +3. π and q are N dimensional probability vectors defined by πn = ˆεn ∥h1:T,n∥2 /ET , and qn = ∥u1:T,n∥2 /UT . +Proof of Theorem 1. To begin with, we apply a dynamic analogue of [Cut19b] to sum the regret bound of single +direction learners. Any comparator u1:T can be decomposed into the directions of feature vectors plus an +unconstrained residual. Therefore, for all decomposition u1:T = �N +n=0 u1:T,n such that u1:T,n ∈ span(h1:T,n) for +all n ∈ [1 : T], we have +RegretT (u1:T ) = ⟨g1:T , x1:T − u1:T ⟩ = ⟨−g1:T , u1:T,0⟩ + +N +� +n=1 +⟨g1:T , w1:T,n − u1:T,n⟩ . +On each direction we apply Lemma 2.1. Moreover, ⟨−g1:T , u1:T,0⟩ ≤ G �T +t=1 ∥ut,0∥2. It leads to +RegretT (u1:T ) ≤ G +N +� +n=1 +� +� +�ˆεn ∥h1:T,n∥2 + +√ +2 ∥u1:T,n∥2 +� +� +� +� +� +�log +� +1 + +∥u1:T,n∥2 +√ +2ˆεn ∥h1:T,n∥2 +� ++ 1 +� +� +� +� +� + G +T +� +t=1 +∥ut,0∥2 += GET + +√ +2GUT + +√ +2GUT +N +� +n=1 +qn +� +log +� +1 + +qnUT +√ +2πnET +� ++ G +T +� +t=1 +∥ut,0∥2 . +Now consider the third term. Without loss of generality, assume qn > 0 for all n, and UT > 0. Applying +log(1 + x) ≤ log x + x−1, +N +� +n=1 +qn +� +log +� +1 + +qnUT +√ +2πnET +� +≤ +N +� +n=1 +qn +�√ +2πnET +qnUT ++ log +UT +√ +2ET ++ log qn +πn += +N +� +n=1 +√qn +�√ +2πnET +UT ++ qn log +UT +√ +2ET ++ qn log qn +πn +≤ +�√ +2ET +UT ++ log +UT +√ +2ET ++ KL(q||π) +(Cauchy-Schwarz) +≤ +�√ +2ET +UT ++ +� +log+ +UT +√ +2ET ++ +� +KL(q||π). +Finally, note that √UT ET ≤ (UT + ET )/2. Combining everything completes the proof. +A.4 +Example +The idea of the sparse coding framework is closely related to signal processing and representation learning, where +a fundamental objective is to find a dictionary that sparsely represents the signal structure. Through a few +examples, we show that it ties several distinct applications together. +Fourier dictionary +Many prediction tasks exhibit natural periodicity, such as the daily temperature, the +seasonal sale of a product, and the load on a power grid. Here, trigonometric feature vectors are a reasonable +choice. Taking d = 1 for example, with a known base frequency ω and an order K ∈ N, one can define a size 2K +dictionary from (for all k ∈ [1 : K]) +ht,2k−1 = cos(kωt), +ht,2k = sin(kωt). +It is also optional to add an all-one feature to track the constant offset of u1:T . +Alternatively, if T is fixed, we may set N = T and define H as the Discrete Fourier Transform (DFT) matrix. +Since we only consider real inputs, the complex DFT dictionary can be simplified into the real form above with +ω = 2π/T, which is intuitively suitable for tasks with unknown periodicity. +21 + +Wavelet dictionary +Wavelets are powerful tools to handle multi-scale signal structures, and specifically in +our framework, “shifting” environments. With d = 1 and N = T, we consider the simplest Haar wavelet, where +the dictionary H is set as the transpose of the (un-normalized) Haar matrix. The precise definition is standard, +but out of our scope. However, the idea can be clearly illustrated in the special case with T = 8: +H = +� +����������� +1 +1 +1 +0 +1 +0 +0 +0 +1 +1 +1 +0 +−1 +0 +0 +0 +1 +1 +−1 +0 +0 +1 +0 +0 +1 +1 +−1 +0 +0 +−1 +0 +0 +1 +−1 +0 +1 +0 +0 +1 +0 +1 +−1 +0 +1 +0 +0 +−1 +0 +1 +−1 +0 +−1 +0 +0 +0 +1 +1 +−1 +0 +−1 +0 +0 +0 +−1 +� +����������� +. +It is an orthogonal basis of RdT . Projecting a signal onto features on the left is equivalent to downsampling, while +the removed local details are captured by features on the right. Compared to using the dense DFT matrix, such +local property simplifies the computation in our framework, as the base algorithm An in Algorithm 2 trivially +outputs wt,0 = 0 when the input feature ht,0 = 0. Therefore, in each round, the Haar-wavelet-based algorithm +only maintains O(log T) black-box 1D algorithms, as opposed to O(T) in the Fourier-based algorithm. +Dictionary from time series +Specifically for time series forecasting, we can learn classical parametric +strategies, such as the autoregressive (AR) model, by choosing H properly. As shown in [AHMS13], learning +it is a fundamental task for learning the more general ARMA models. If a time series z1:T is generated by a +(noiseless) AR(k) model, then with parameters α1:k, it satisfies zt = �k +i=1 αizt−i. +We consider the time series setup from the beginning of the paper, with d = 1 and w1:T = 0. Setting +N = p and ht,n = zt−n, Theorem 1 translates to a forecasting regret bound against any prediction sequence y1:T +generated by an AR(k) model. In particular, the bound adapts to the magnitude and sparsity of the comparator +model parameter, which induces an oracle inequality similar to Eq.(2). This improves the non-adaptive approach +from [AHMS13]. +Learned dictionary +Since H is only queried online, we may generate H itself using an online learning +algorithm. If the base learner guarantees a regret bound against certain normalized comparators, then our +approach can enhance it by adapting to the actual scale of the comparator, which is unbounded a priori. +Concretely, consider N = 1, i.e., Algorithm 1. For any (unknown) ˆu ∈ R, +RegretT (u1:T ) = ˆu +T +� +t=1 +� +gt, ht − ut +ˆu +� ++ +T +� +t=1 +⟨gt, ht⟩ (ˆxt − ˆu). +The first sum can be bounded by the guarantee of h1:T – this is the ideal adaptive bound we aim for. In this +regard, the second sum is the overhead of such adaptivity, similar to the objective in Lemma 2.1. Specific +applications of this technique can be found in [CO18] and [JC22, Appendix I], while here we show that in general, +it can be viewed as an instance of the sparse coding framework. +A.5 +Model selection by summation +For unconstrained dynamic regret, an appealing property is that different regret bounds can be simply aggregated +by summation. This is essentially the idea of Theorem 1 itself. From this angle, our sparse coding framework +and the path length bound from [JC22] are mutually complementary. +Let A1 and A2 be two algorithms, each guaranteeing an unconstrained dynamic regret bound fi(u1:T ) for all +u1:T ∈ RdT , i = 1 or 2. Consider a master algorithm A that simply predicts the sum of their predictions. Then, +for any decomposition of the comparator u1:T = u(1) +1:T + u(2) +1:T , +RegretT (u1:T ) ≤ f1(u(1) +1:T ) + f2(u(2) +1:T ). +22 + +For example, take A1 as the sparse coding algorithm, and A2 as the algorithm from [JC22]. Then, within +Theorem 1, we can replace the trivial characterization of ut,0 by a path length bound on ut,0. The sacrifice is +only a slightly larger cumulative loss term, i.e., 2GET in Theorem 1. The result can also be plugged into the +oracle inequality Eq.(2). +A.6 +Detail on the primal-dual interpretation +Supplementing the primal-dual discussion in Section 2.3, we present several versions of the loss-regret duality +on the sequence space RdT . The first one generalizes a classical argument in static unconstrained OLO [MO14, +Theorem 1]. The other two are to our knowledge new, and are more closely tied to the path length characterization +of the comparator. Define st = �t +i=1 gi, i.e., the sum of past gradients. +Lemma 2.2. If there exists a function fT : RdT → R such that the prediction sequence x1:T guarantees a loss +upper bound ⟨g1:T , x1:T ⟩ ≤ −fT (g1:T ), then for all u1:T ∈ RdT , +RegretT (u1:T ) ≤ f ∗ +T (−u1:T ). +Proof of Lemma 2.2. This follows from a standard Fenchel duality argument. +RegretT (u1:T ) = ⟨g1:T , x1:T − u1:T ⟩ ≤ ⟨g1:T , −u1:T ⟩ − fT (g1:T ) ≤ sup +x∈RdT ⟨x, −u1:T ⟩ − fT (x) = f ∗ +T (−u1:T ). +Lemma A.7. Recall that we defined st = �t +i=1 gi. If there exists fT : RdT → R such that x1:T guarantees +⟨g1:T , x1:T ⟩ ≤ −fT (s1:T ), then for all u1:T ∈ RdT , +RegretT (u1:T ) ≤ f ∗ +T (u2 − u1, u3 − u2, . . . , uT − uT −1, −uT ). +Proof of Lemma A.7. We start by rewriting the comparator loss. +⟨g1:T , u1:T ⟩ = ⟨sT , uT ⟩ + +T +� +t=1 +⟨gt, ut − uT ⟩ = ⟨sT , uT ⟩ + +T +� +t=1 +T −1 +� +i=t +⟨gt, ui − ui+1⟩ += ⟨sT , uT ⟩ + +T −1 +� +i=1 +i +� +t=1 +⟨gt, ui − ui+1⟩ = ⟨sT , uT ⟩ + +T −1 +� +t=1 +⟨st, ut − ut+1⟩ . +Given the loss upper bound, +RegretT (u1:T ) ≤ +T −1 +� +t=1 +⟨st, ut+1 − ut⟩ + ⟨sT , −uT ⟩ − fT (s1:T ) +≤ +sup +x1:T ∈RdT +�T −1 +� +t=1 +⟨xt, ut+1 − ut⟩ + ⟨xT , −uT ⟩ − fT (x1:T ) +� += f ∗ +T (u2 − u1, u3 − u2, . . . , uT − uT −1, −uT ). +By reversing the index, we have the following lemma. +Lemma A.8. If there exists fT : RdT → R such that x1:T guarantees ⟨g1:T , x1:T ⟩ ≤ −fT (sT , sT − s1, . . . , sT − +sT −1), then for all u1:T ∈ RdT , +RegretT (u1:T ) ≤ f ∗ +T (−u1, u1 − u2, u2 − u3, . . . , uT −1 − uT ). +Proof of Lemma A.8. Similar to the proof above, +⟨g1:T , u1:T ⟩ = ⟨sT , u1⟩ + +T +� +t=1 +⟨gt, ut − u1⟩ = ⟨sT , u1⟩ + +T +� +t=1 +t−1 +� +i=1 +⟨gt, ui+1 − ui⟩ += ⟨sT , u1⟩ + +T −1 +� +i=1 +T +� +t=i+1 +⟨gt, ui+1 − ui⟩ = ⟨sT , u1⟩ + +T −1 +� +t=1 +⟨sT − st, ut+1 − ut⟩ . +23 + +RegretT (u1:T ) ≤ ⟨sT , −u1⟩ + +T −1 +� +t=1 +⟨sT − st, ut − ut+1⟩ − fT (sT , sT − s1, . . . , sT − sT −1) +≤ +sup +x1:T ∈RdT +� +⟨x1, −u1⟩ + +T +� +t=2 +⟨xt, ut−1 − ut⟩ − fT (x1:T ) +� += f ∗ +T (−u1, u1 − u2, u2 − u3, . . . , uT −1 − uT ). +B +Detail on the improved static algorithm +This section presents the second contribution of the paper, an improved static unconstrained OLO algorithm. +Appendix B.1 contains the derivatives of our potential functions, which will be useful in the proof. Appendix B.2 +analyzes the base algorithm (Algorithm 3), with its regret bound presented as Lemma B.6. Appendix B.3 presents +the analysis of the meta algorithm, resulting in Theorem 2, the main theorem of this section. Appendix B.4 +discusses the application of this static algorithm to the dynamic regret problem, through the sparse coding +framework. +B.1 +Facts of the potential function +For the two potential functions defined in Section 3.2, we compute their derivatives as follows. This will be +useful later on. +∂1φ(x, y) = −ε√α +2√x exp +� y2 +4αx +� +, +∂2φ(x, y) = εerfi +� +y +√ +4αx +� +, +∂11φ(x, y) = ε√α +4x3/2 +� y2 +2αx + 1 +� +exp +� y2 +4αx +� +, +∂12φ(x, y) = − +εy +4√αx3/2 exp +� y2 +4αx +� +, +∂22φ(x, y) = +ε +2√αx exp +� y2 +4αx +� +. +Due to the change of variable, the derivatives of Φ can be concisely represented as +∂1Φ(V, S) = ∂1φ(V + z + kS, S), +∂2Φ(V, S) = k∂1φ(V + z + kS, S) + ∂2φ(V + z + kS, S), +∂11Φ(V, S) = ∂11φ(V + z + kS, S), +∂12Φ(V, S) = k∂11φ(V + z + kS, S) + ∂12φ(V + z + kS, S), +∂22Φ(V, S) = k2∂11φ(V + z + kS, S) + 2k∂12φ(V + z + kS, S) + ∂22φ(V + z + kS, S). +B.2 +Analysis of the base algorithm +The key component of our approach is the base algorithm (Algorithm 3). Within its analysis, the most crucial +part is the characterization of the one step change of the potential (Lemma B.3). This subsection is outlined +as follows. We first present two simple lemmas on the property of our potential function Φ. Then, we prove +the key lemma (Lemma B.3), which leads to a cumulative loss upper bound (Lemma B.4). As in the standard +analysis of potential methods, converting the loss upper bound to the regret bound relies on computing the +Fenchel conjugate of Φ – this is the focus of Lemma B.5. Finally, Lemma B.6 combines everything into the +regret bound of the base algorithm. +To begin with, we first show that Φ(V, S) is convex in S, just like more standard potential functions. +24 + +Lemma B.1. If ε > 0, α > 0 and z > k > 0, the potential function Φ(V, S) satisfies ∂22Φ(V, S) ≥ 0 for all +V ≥ 0 and S ≥ −1. +Proof of Lemma B.1. Define the shorthands x = V + z + kS and y = S. For all V ≥ 0 and S ≥ −1, we have +x > 0, therefore +∂22Φ(V, S) = k2∂11φ(x, y) + 2k∂12φ(x, y) + ∂22φ(x, y) += ε√α +4x3/2 exp +� y2 +4αx +� �k2y2 +2αx + k2 − 2ky +α ++ 2x +α +� += ε√α +4x3/2 exp +� y2 +4αx +� �k2y2 +2αx + k2 + 2(V + z) +α +� +≥ 0. +Next, we show that the base algorithm makes strictly negative prediction ˜zt when St−1 is negative. This will +be exploited by the meta-algorithm to ensure that the surrogate losses received by the base algorithm satisfy the +St ≥ −1 constraint. +Lemma B.2. If ε > 0, α > 0 and z > k > 0, the potential function Φ(V, S) satisfies ∂2Φ(V, S) < 0 for all +V ≥ 0 and −1 ≤ S ≤ 0. +Proof of Lemma B.2. Let us check ∂2Φ(V, 0) < 0. Indeed, +∂2Φ(V, 0) = k∂1φ(V + z, 0) + ∂2φ(V + z, 0) = − +εk√α +2 +√ +V + z < 0. +Moreover, ∂22Φ(V, S) ≥ 0 due to Lemma B.1. Therefore, we have ∂2Φ(V, S) < 0 for all V ≥ 0 and −1 ≤ S ≤ 0. +The key lemma in our analysis is the following, which says a suitable combination of parameters yields a +one-step loss bound on the potential function Φ, as long as the second argument of Φ is always larger than −1. +Such a lemma is typically the central component in the classical potential analysis. +Lemma B.3 (Key lemma: one step potential bound). With α = 1, k = 2, z = 16 and an arbitrary ε > 0, the +potential function Φ(V, S) satisfies +Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S) ≤ 0, +for all V ≥ 0, S ≥ −1 and c ∈ [−1, 1] ∩ [−1 − S, ∞). +Proof of Lemma B.3. Let us view our objective as a function of c, +fV,S(c) := Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S). +Taking the derivatives, +f ′ +V,S(c) = 2c∂1Φ(V + c2, S + c) + ∂2Φ(V + c2, S + c) − ∂2Φ(V, S), +f ′′ +V,S(c) = 2∂1Φ(V + c2, S + c) + 4c2∂11Φ(V + c2, S + c) + 4c∂12Φ(V + c2, S + c) + ∂22Φ(V + c2, S + c) +≤ 2∂1Φ(V + c2, S + c) + 4∂11Φ(V + c2, S + c) + 4 +��∂12Φ(V + c2, S + c) +�� + ∂22Φ(V + c2, S + c). +(9) +Note that fV,S(0) = f ′ +V,S(0) = 0. Therefore, to prove fV,S(c) ≤ 0, it suffices to show f ′′ +V,S(c) ≤ 0 for all considered +values of V , S and c. The RHS of Eq.(9) has a striking similarity to the Backward Heat Equation – in fact, after +a change of variable, the resulting expressions, namely Eq.(10) and Eq.(11) below, will resemble a BHE on φ +(∂1φ + α∂22φ = 0) plus a perturbation. The main goal of this proof is to control such perturbations by properly +choosing α, k and z. +Concretely, due to the absolute value in Eq.(9), we will analyze two cases. Technically, the first case is harder, +therefore we pick k to simplify its analysis. The second case requires S + c to be around zero – this is an “edge” +case and relatively easier to handle. +25 + +Case 1: ∂12Φ(V + c2, S + c) ≤ 0. +Substituting the derivatives of Φ by the derivatives of φ, we have +f ′′ +V,S(c) ≤ 2∂1φ + (k − 2)2∂11φ + 2(k − 2)∂12φ + ∂22φ +��� +(V +c2+z+k(S+c),S+c). +(10) +The RHS means we evaluate all the derivative functions at (V + c2 + z + k(S + c), S + c). Plugging in our specific +choice of k and α, +f ′′ +V,S(c) ≤ 2∂1φ + ∂22φ +��� +(V +c2+z+2(S+c),S+c) +(k = 2) +≤ 2∂1φ + 2∂22φ +��� +(V +c2+z+2(S+c),S+c) +(∂22φ ≥ 0) += 0. +(φ satisfies the BHE with α = 1.) +Case 2: ∂12Φ(V + c2, S + c) ≥ 0. +Similar to the first case, +f ′′ +V,S(c) ≤ 2∂1φ + (k + 2)2∂11φ + 2(k + 2)∂12φ + ∂22φ +��� +(V +c2+z+k(S+c),S+c). +(11) +Consider the k-dependent “perturbation” terms in Eq.(11), i.e., (k + 2)2∂11φ + 2(k + 2)∂12φ. Our goal is to +upper bound it by ∂22φ, such that an upper bound of f ′′ +V,S(c) follows from the BHE. Plugging in the derivatives +of φ from Appendix B.1, for all inputs (x, y), +(k + 2)2∂11φ + 2(k + 2)∂12φ − ∂22φ +��� +(x,y) = +ε +4√αx3/2 exp +� y2 +4αx +� � +(k + 2)2 +� y2 +2x + α +� +− 2(k + 2)y − 2x +� +. +We aim to show the bracket on the RHS is negative at x = V + c2 + z + k(S + c) and y = S + c. Also plugging +in our choice of α = 1 and k = 2, this amounts to showing +♦ := +2(S + c)2 +V + c2 + z + 2S + 2c + 4 − 3(S + c) − 1 +2(V + c2 + z) ≤ 0. +The idea is that we can pick a large enough z to make it hold. Concretely, +• If S + c > 0, then +♦ ≤ 2(S + c)2 +2S + 2c + 4 − 3(S + c) − 1 +2z ≤ 4 − 1 +2z, +and it suffices to pick z ≥ 8. +• If S + c ≤ 0, then since c ∈ [−1 − S, ∞), we have S + c ≥ −1. As long as z > 2, +♦ ≤ +2 +z − 2 + 7 − 1 +2z. +It suffices to pick z ≥ 16. +In summary, z = 16 ensures ♦ ≤ 0. Due to the BHE on φ, +f ′′ +V,S(c) ≤ 2∂1φ + 2∂22φ +��� +(V +c2+z+2(S+c),S+c) = 0. +Combining the two cases completes the proof. +Based on Lemma B.3, we immediately obtain a cumulative loss bound of the base algorithm. The proof is a +straightforward telescopic sum, therefore omitted. +Lemma B.4 (Cumulative loss bound). With α = 1, k = 2 and z = 16, Algorithm 3 guarantees for all T ∈ N+, +T +� +t=1 +˜lt˜zt ≤ Φ(0, 0) − Φ( ˜VT , ˜ST ). +26 + +As for the regret bound, similar to the standard duality argument [Ora19, Chapter 9], we need the Fenchel +conjugate of the potential function Φ. With any V ≥ 0, define +Φ∗ +V (u) := +sup +S∈[−1,∞) +uS − Φ(V, S) +as the conjugate of Φ(V, S) with respect to S. Slightly different from the standard definition where the supremum +is over R, here the supremum is over [−1, ∞), since the surrogate losses in the base algorithm satisfy ˜St ≥ −1 +for all t. To proceed, we will only consider the dual variable satisfying u ≥ 0. +Lemma B.5 (Conjugate). With ε > 0, α > 0 and z > k > 0, for all u ≥ 0, +Φ∗ +V (u) := +sup +S∈[−1,∞) +uS − Φ(V, S) +≤ u ¯S + ε +� +α(V + z + k ¯S), +where +¯S = 4αk +� +1 + +� +log(2uε−1 + 1) +�2 ++ +� +4α(V + z) +� +1 + +� +log(2uε−1 + 1) +� +. +Proof of Lemma B.5. We first show that the supremum over S in the Fenchel conjugate is attainable by some +S∗ ∈ [0, ∞). To this end, define a function f(S) = uS − Φ(V, S). f is continuous, with f ′(S) = u − ∂2Φ(V, S). +Moreover, due to Lemma B.1, f is concave on [−1, ∞). The existence of S∗ then follows from analyzing the +boundary. +• For all S ∈ [−1, 0], we have f ′(S) ≥ 0. The reason is u ≥ 0, and ∂2Φ(V, S) ≤ 0 due to Lemma B.2. +• For sufficiently large S, we aim to show f ′(S) < 0. +Let us begin by writing down ∂2Φ(V, S), from +Appendix B.1. +∂2Φ(V, S) = εerfi +� +S +� +4α(V + z + kS) +� +− +εk√α +2 +√ +V + z + kS exp +� +S2 +4α(V + z + kS) +� +. +Now consider large S that satisfies S ≥ +� +4α(V + z + kS). +Due to an estimate of the erfi function +(Lemma A.2), +erfi +� +S +� +4α(V + z + kS) +� +≥ +� +α(V + z + kS) +S +exp +� +S2 +4α(V + z + kS) +� +. +Moreover, +� +α(V + z + kS) +S +− +k√α +√ +V + z + kS = +√α(V + z) +S +√ +V + z + kS ≥ 0. +Therefore, +∂2Φ(V, S) = +� +ε +2erfi +� +S +� +4α(V + z + kS) +� +− +εk√α +2 +√ +V + z + kS exp +� +S2 +4α(V + z + kS) +�� ++ ε +2erfi +� +S +� +4α(V + z + kS) +� +≥ ε +2erfi +� +S +� +4α(V + z + kS) +� +. +(12) +For sufficiently large S, we have RHS > u, hence f ′(S) < 0. +Summarizing the above, we have shown that there exists S∗ ∈ [0, ∞) such that +Φ∗ +V (u) := +sup +S∈[−1,∞) +uS − Φ(V, S) = uS∗ − Φ(V, S∗). +Moreover, S∗ should satisfy the first order condition f ′(S∗) = 0, i.e., u = ∂2Φ(V, S∗). Our goal next is to upper +bound S∗ by a function of u. Again, we analyze two cases. +27 + +Case 1. +If S∗ satisfies S∗ < +� +4α(V + z + kS∗), then by regrouping the terms, we have (S∗)2 − 4αkS∗ − +4α(V + z) < 0. Solving this quadratic inequality, +S∗ ≤ 1 +2 +� +4αk + +� +(4αk)2 + 16α(V + z) +� += 2αk + +� +4α2k2 + 4α(V + z) +≤ 4αk + +� +4α(V + z). +Case 2. +If S∗ satisfies S∗ ≥ +� +4α(V + z + kS∗), then same as the earlier analysis in the present proof, Eq.(12), +we have +u ≥ ε +2erfi +� +S∗ +� +4α(V + z + kS∗) +� +. +For conciseness, define the notation p = erfi−1(2uε−1). Then, (S∗)2 − 4αkp2S∗ − 4αp2(V + z) < 0. Solving the +quadratic inequality, +S∗ ≤ 1 +2 +� +4αkp2 + +� +(4αkp2)2 + 16αp2(V + z) +� +≤ 2αkp2 + +� +4α2k2p4 + 4αp2(V + z) +≤ 4αkp2 + +� +4α(V + z)p. +Now we can combine the above two cases. Specifically, p ≤ 1+ +� +log(2uε−1 + 1) due to Lemma A.3. Therefore, +S∗ ≤ 4αk +� +1 + +� +log(2uε−1 + 1) +�2 ++ +� +4α(V + z) +� +1 + +� +log(2uε−1 + 1) +� +. +Define the RHS as ¯S. Then, from the definition of the Fenchel conjugate, +Φ∗ +V (u) = uS∗ − Φ(V, S∗) += uS∗ − ε +� +α(V + z + kS∗) +� +2 +� +S∗ +√ +4α(V +z+kS∗) +0 +erfi(u)du − 1 +� +≤ u ¯S + ε +� +α(V + z + k ¯S). +Plugging in ¯S completes the proof. +Finally, we assemble the cumulative loss bound (Lemma B.4) and the conjugate of the potential (Lemma B.5) +into the regret bound of the base algorithm. +Lemma B.6 (Regret of the base algorithm). With ε > 0, α = 1, k = 2 and z = 16, Algorithm 3 guarantees for +all T ∈ N+ and ˜u ≥ 0, +T +� +t=1 +˜lt(˜zt − ˜u) ≤ ε +� +˜VT + 2 ¯S + ˜u ¯S, +where +¯S = 8 +� +1 + +� +log(2˜uε−1 + 1) +�2 ++ 2 +� +˜VT + 16 +� +1 + +� +log(2˜uε−1 + 1) +� +. +Proof of Lemma B.6. Due to the standard loss-regret duality [Ora19, Theorem 9.6], starting from the cumulative +28 + +loss bound (Lemma B.4), the regret can be bounded by +T +� +t=1 +˜lt(˜zt − ˜u) ≤ ST ˜u + Φ(0, 0) − Φ( ˜VT , ˜ST ) +≤ Φ(0, 0) + +sup +S∈[−1,∞) +� +S˜u − Φ( ˜VT , S) +� += Φ(0, 0) + Φ∗ +˜VT (˜u) +≤ −4ε + ε +� +˜VT + 16 + 2 ¯S + ˜u ¯S +≤ ε +� +˜VT + 2 ¯S + ˜u ¯S. +Plugging in ¯S from Lemma B.5 completes the proof. +B.3 +Proof of the main result +This subsection presents the theoretical guarantees of the meta algorithm (Algorithm 4). We first show that +when combined with the base algorithm (Algorithm 3), the whole procedure is well-posed, in the sense that the +surrogate loss ˜lt satisfies − �t +i=1 ˜li ≥ −1 for all t. +Proposition 3 (Well-posedness). The surrogate loss ˜lt defined in Algorithm 4 satisfies − �t +i=1 ˜li ≥ −1 for all t. +Proof of Proposition 3. First, notice that |˜lt|≤ |lt| = +��� +gtG−1, wt +��� ≤ 1. +Next, we prove by induction. Consider − �t−1 +i=1 ˜li, which is defined as ˜St−1 in the base algorithm (Algorithm 3). +Suppose ˜St−1 ≥ −1, which trivially holds at t = 1. Let us analyze two cases. +• If ˜St−1 ≥ 0, then − �t +i=1 ˜li = ˜St−1 − ˜lt ≥ ˜St−1 − |˜lt|≥ −1. +• If −1 ≤ ˜St−1 < 0, then due to Lemma B.2, the prediction ˜zt of the base algorithm satisfies ˜zt < 0. The +meta algorithm projects it to zt = 0. Then, due to our definition of ˜lt in the meta algorithm, +˜lt = +� +lt, +lt ≤ 0, +0, +else, +which is non-positive. Therefore, − �t +i=1 ˜li = ˜St−1 − ˜lt ≥ ˜St−1 ≥ −1. +An induction completes the proof. +Next we present the main result, the static regret bound of Algorithm 4. Here we define the gradient variance +VT = �T +t=1 ∥gt∥2 +2. +Theorem 2. With ε > 0, α = 1, k = 2 and z = 16, Algorithm 4 guarantees for all T ∈ N+ and u ∈ Rd, +RegretT (u) ≤ ε +� +VT + 2G ¯S + ∥u∥2 +� +¯S + 2 +� +2VT +� +, +where +¯S = 8G +� +1 + +� +log(2 ∥u∥2 ε−1 + 1) +�2 ++ 2 +� +VT + 16G2 +� +1 + +� +log(2 ∥u∥2 ε−1 + 1) +� +. +Proof of Theorem 2. Since the meta algorithm simply applies two existing black-box reductions [CO18, Cut20], +the proof is straightforward given Lemma B.6. First, due to a polar decomposition theorem [CO18, Theorem 2], +the regret can be decomposed into the regret of AB with respect to u/∥u∥2, plus the regret of zt with respect to +∥u∥2. Then, the latter is upper-bounded by the regret of ˜zt – this is because our definition of zt and ˜lt follows +29 + +the procedure of [Cut20, Theorem 2], where a convex constraint can be added to an unconstrained algorithm +without changing its regret bound. In summary, we have +RegretT (u) ≤ G +T +� +t=1 +lt(zt − ∥u∥2) + ∥u∥2 +T +� +t=1 +⟨gt, wt − u/∥u∥2⟩ +≤ G +T +� +t=1 +˜lt(˜zt − ∥u∥2) + ∥u∥2 +T +� +t=1 +⟨gt, wt − u/∥u∥2⟩ . +The two regret terms on the RHS represent the regret bound of A1d and AB, respectively. +Now, the first term follows from Lemma B.6, where ˜VT = �T +t=1 ˜l2 +t ≤ �T +t=1 l2 +t = �T +t=1 +� +gtG−1, wt +�2 ≤ VT /G2. +As for the regret of AB, due to [Ora19, Theorem 4.14], +T +� +t=1 +⟨gt, wt − u/∥u∥2⟩ ≤ 2 +� +2VT . +Combining these two components completes the proof. +Finally, let us use asymptotic orders to make this bound a bit more interpretable. Consider the regime +of large ∥u∥2 and VT , i.e., ∥u∥2 ≫ ε and VT ≫ G2. We preserve the dependence of ε, as it is an arbitrary +hyperparameter. In contrast, α, z and k are absolute constants, therefore subsumed by the big-Oh. +Using log(1 + x) ≤ x, we can crudely bound ¯S by +¯S ≤ 8G +� +1 + +� +2 ∥u∥2 ε−1 +�2 ++ 2 +� +VT + 16G2 +� +1 + +� +2 ∥u∥2 ε−1 +� += 8G + 2 +� +VT + 16G2 + o +� +∥u∥2 ε−1� +VT +� +. +Plugging this crude bound of ¯S into the first term of the regret bound, we have +RegretT (u) ≤ ε +� +VT + 16G2 + 4G +� +VT + 16G2 + ε +� +o +� +G ∥u∥2 ε−1� +VT +� ++ ∥u∥2 +� +¯S + 2 +� +2VT +� +≤ ε( +� +VT + 16G2 + 2G) + o +�√ +G ∥u∥2 V 1/4 +T +� ++ ∥u∥2 +� +¯S + 2 +� +2VT +� +≤ ε( +� +VT + 6G) + o +�√ +G ∥u∥2 V 1/4 +T +� ++ ∥u∥2 +� +¯S + 2 +� +2VT +� +. +Next, notice that ¯S = O +�� +VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) +� +. Using it to replace the remaining ¯S above, +RegretT (u) ≤ ε +�� +VT + 6G +� ++ o +�√ +G ∥u∥2 V 1/4 +T +� ++ ∥u∥2 O +�� +VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) +� +. +The second term can be assimilated into the third term. The result becomes +RegretT (u) ≤ ε +�� +VT + 6G +� ++ ∥u∥2 O +�� +VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) +� +, +(13) +where O(·) subsumes absolute constants. +Note that this bound is not only valid for large ∥u∥2, but also valid when u = 0 (this can be directly verified +from Theorem 2). Therefore, we can use it to characterize the loss-regret tradeoff. It is the same loss-regret +tradeoff as [ZCP22a], but with time T replaced by the gradient variance VT . +Towards the optimal leading constant +Without considering gradient adaptivity, [ZCP22a] showed that +the optimal leading term in the regret bound (including the multiplicative constant) is ∥u∥2 G +� +2T log(∥u∥2 ε−1), +c.f., Eq.(3). In Theorem 2, if we ignore the logarithmic residue log(∥u∥2 ε−1) outside the square root,11 then the +11Which does not depend on VT . +30 + +leading term is 2 ∥u∥2 +� +VT log(∥u∥ ε−1). In the worst case with VT = G2T, the constant of the latter has a +√ +2 +gap with respect to the lower bound. This is essentially due to our analysis, where α is picked as 1 instead of 1/2 +to handle the second case in the proof sketch (Section 3.2). One can use a smaller α (corresponding to smaller +leading constant in the regret) in exchange for a larger z (the additive constant on VT ). However, achieving the +lower bound +√ +2 without blowing up the additive term remains to be studied in future works. +B.4 +Application to dynamic regret +Given the improved static algorithm, we now apply its 1D version to the sparse coding framework. For clarity, +we will adopt the asymptotic regret bound, Eq.(13), and loosely assimilate the residual term. Since it is applied +on the transform domain, we will denote transform domain quantities with hat, according to our convention in +Section 2.1. Then, analogous to Lemma A.4 applied in the main paper, our improved static algorithm, given an +arbitrary hyperparameter ˆε > 0, guarantees for all T ∈ N+ and ˆu ∈ R, +T +� +t=1 +⟨ˆgt, ˆxt − ˆu⟩ ≤ ˆε +� +� +� +� +� +� +T +� +t=1 +ˆg2 +t + 6 ˆG +� +� + |ˆu| O +� +� +� +� +� +� +T +� +t=1 +ˆg2 +t log(|ˆu| ε−1) +� +� , +(14) +where ˆG is the Lipschitz constant for the surrogate loss ˆgt. +In the sparse coding framework, we will assume the dictionary satisfies ∥ht∥2 ≤ 1, which holds for Fourier +and (un-normalized) wavelet dictionaries (Appendix A.4). Then, let us apply the improved static algorithm to a +single feature vector (Algorithm 1). Note that instead of setting the surrogate Lipschitz constant as G ∥ht∥2 +(Line 2 of Algorithm 1), we now set it as G. The resulting dynamic regret bound is the following, which is +analogous to Lemma 2.1. +Lemma B.7. Let ˆε > 0 be an arbitrary hyperparameter for our improved static algorithm (the 1D version of +Algorithm 4). Applying it as a subroutine, for all T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees +RegretT (u1:T ) ≤ εT + +� +� +� +� +T +� +t=1 +⟨gt, ut⟩2O +� +�log +� +�ε−1 +T +� +� +� +� +T +� +t=1 +⟨gt, ut⟩2 +� +� +� +� , +where +εT = ˆε +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht⟩2 + 6G +� +� . +In particular, the big-Oh bound holds in two regimes: (i) �T +t=1 ⟨gt, ut⟩2 ≫ ˆε2 �T +t=1 ⟨gt, ht⟩2 and �T +t=1 ⟨gt, ht⟩2 ≫ +G2; and (ii) u1:T = 0. Compared to Lemma 2.1, the better underlying static algorithm essentially improves +G2 ∥h1:T ∥2 +2 in the dynamic regret bound with �T +t=1 ⟨gt, ht⟩2, and G2 ∥u1:T ∥2 +2 with �T +t=1 ⟨gt, ut⟩2. +Proof of Lemma B.7. Similar to the proof of Lemma 2.1, the dynamic regret of Algorithm 1 equals Eq.(14) on +the transform domain. In particular, ˆu satisfies u1:T = ˆuh1:T . The surrogate Lipschitz constant ˆG in Eq.(14) +31 + +equals G, the actual Lipschitz constant for the dynamic regret problem. With �T +t=1 ⟨gt, ht⟩2 ≫ G2, +RegretT (u1:T ) ≤ ˆε +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht⟩2 + 6G +� +� + |ˆu| O +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht⟩2 log(|ˆu| ˆε−1) +� +� += εT + |ˆu| +� +� +� +� +T +� +t=1 +⟨gt, ht⟩2O +� +�log +� +�ε−1 +T |ˆu| +� +� +� +� +T +� +t=1 +⟨gt, ht⟩2 +� +� +� +� += εT + +� +� +� +� +T +� +t=1 +⟨gt, |ˆu| |ht|⟩2O +� +�log +� +�ε−1 +T +� +� +� +� +T +� +t=1 +⟨gt, |ˆu| |ht|⟩2 +� +� +� +� += εT + +� +� +� +� +T +� +t=1 +⟨gt, ut⟩2O +� +�log +� +�ε−1 +T +� +� +� +� +T +� +t=1 +⟨gt, ut⟩2 +� +� +� +� . +Next, let us consider general size N dictionaries, analogous to Theorem 1. Still, we assume that for all t and +n, ∥ht,n∥2 ≤ 1. +Lemma B.8. Consider Algorithm 2, with its static subroutine replaced by the 1D version of Algorithm 4. For +all T ∈ N+ and u1:T ∈ RdT , it guarantees +RegretT (u1:T ) ≤ ET + UT · O +� +log UT +ET ++ KL(q||π) +� ++ G +T +� +t=1 +∥ut,0∥2 , +where +1. For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N +n=1 u1:T,n; +2. ET and UT are non-negative numbers defined by +ET = +N +� +n=1 +ˆεn +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht,n⟩2 + 6G +� +� , +UT = +N +� +n=1 +� +� +� +� +T +� +t=1 +⟨gt, ut,n⟩2; +3. π and q are N dimensional probability vectors defined by +πn = ˆεn +ET +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht,n⟩2 + 6G +� +� , +qn = +1 +UT +� +� +� +� +T +� +t=1 +⟨gt, ut,n⟩2. +If π is the uniform distribution, and u1:T ∈ span(H), then the bound can be simplified into +RegretT (u1:T ) ≤ ET + ˜O(UT ). +Note that the big-Oh bound is meant for the regime where for all n ∈ [1 : N], either (i) u1:T,n = 0; or +(ii) �T +t=1 ⟨gt, ut,n⟩2 ≫ ˆε2 �T +t=1 ⟨gt, ht,n⟩2 and �T +t=1 ⟨gt, ht,n⟩2 ≫ G2. Compared to Theorem 1, we improve +G �N +n=1 +��T +t=1 ∥ut,n∥2 to �N +n=1 +��T +t=1 ⟨gt, ut,n⟩2, which adapts to the complexity of both the gradient sequence +and the comparator. +32 + +Proof of Lemma B.8. By summing Lemma B.7, +RegretT (u1:T ) ≤ +N +� +n=1 +� +� +�ˆεn +� +� +� +� +� +� +T +� +t=1 +⟨gt, ht,n⟩2 + 6G +� +� + +� +� +� +� +T +� +t=1 +⟨gt, ut,n⟩2O +� +�log +� +�ˆε−1 +n +��T +t=1 ⟨gt, ut,n⟩2 +��T +t=1 ⟨gt, ht,n⟩2 + 6G +� +� +� +� +� +� +� += ET + UT · O +� N +� +n=1 +qn log qnUT +πnET +� +≤ ET + UT · O +� +log UT +ET ++ +N +� +n=1 +qn log qn +πn +� += ET + UT · O +� +log UT +ET ++ KL(q||π) +� +. +33 + diff --git a/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/load_file.txt b/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e5bb195c6527dd6923a119e9feab751dfd0c0a0d --- /dev/null +++ b/cdFQT4oBgHgl3EQfiTZ3/content/tmp_files/load_file.txt @@ -0,0 +1,1264 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf,len=1263 +page_content='Unconstrained Dynamic Regret via Sparse Coding Zhiyu Zhang Boston University zhiyuz@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='edu Ashok Cutkosky Boston University ashok@cutkosky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='com Ioannis Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Paschalidis Boston University yannisp@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='edu Abstract Motivated by time series forecasting, we study Online Linear Optimization (OLO) under the coupling of two problem structures: the domain is unbounded, and the performance of an algorithm is measured by its dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Handling either of them requires the regret bound to depend on certain complexity measure of the comparator sequence – specifically, the comparator norm in unconstrained OLO, and the path length in dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In contrast to a recent work [JC22] that adapts to the combination of these two complexity measures, we propose an alternative complexity measure by recasting the problem into sparse coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Adaptivity can be achieved by a simple modular framework, which naturally exploits more intricate prior knowledge of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Along the way, we also present a new gradient adaptive algorithm for static unconstrained OLO, designed using novel continuous time machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This could be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1 Introduction Time series forecasting is a fundamental problem in science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To design forecasting strategies, a classical procedure is to model the time series based on batched data, either statistically or empirically, and then deploy such models online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The effectiveness of this procedure critically relies on certain stationarity of the environment, thus may fail under distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The present work addresses this issue from the perspective of online learning – we design an online fine-tuning framework such that given any oracle forecaster, the fine-tuned predictions are equipped with robustness guarantees that do not rely on statistical assumptions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, we study the following variant of Online Convex Optimization (OCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In the t-th round, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We query an oracle forecaster A for its prediction wt ∈ Rd, determine a fine-tuning adjustment xt ∈ Rd, and then predict their sum xt + wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The environment reveals a true value zt ∈ Rd and a convex G-Lipschitz1 loss function lt : Rd → R, minimized at zt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We suffer the loss lt(xt + wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our goal is to achieve low regret against any alternative sequence of predictions y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , yT ∈ Rd selected in hindsight, where T is the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The y1:T sequence does not have to be the true time series z1:T , which will be clear shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Since for any subgradient gt ∈ ∂lt(xt + wt) we have lt(xt + wt) − lt(yt) ≤ ⟨gt, xt + wt − yt⟩, for the rest of the paper we will assume only observing gt (instead of lt and zt), and define the regret in the formulation of Online Linear Optimization (OLO) [Haz16, Ora19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let ut = yt − wt be the “ideal” fine-tuning adjustment had we known the comparing sequence y1:T beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, RegretT (u1:T ) := T � t=1 ⟨gt, xt − ut⟩ = T � t=1 ⟨gt, xt + wt − yt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (1) As an example, if y1:T = z1:T , then bounding the regret subsumes bounding the forecasting error �T t=1 lt(xt +wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Through the lens of OLO, we call Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (1) the unconstrained dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to the most standard setting of OLO, the challenge here is due to the coupling of two problem structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1With respect to ∥·∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='13349v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='LG] 31 Jan 2023 The domain Rd is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The comparator sequence u1:T is time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Even under only one of these conditions, it appears that the environment is given too much power: no matter how we predict, there always exist some g1:T and u1:T sequences inducing large regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Circumventing this issue relies on comparator adaptivity – instead of only depending on the time horizon T, the regret bound also depends on certain complexity measures of u1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For the time series application, this allows us to write2 T � t=1 lt(xt + wt) ≤ inf y1:T � T � t=1 lt(yt) + RegretT (y1:T − w1:T ) � , (2) where the minimizing argument y1:T trades off its cumulative loss and the complexity of u1:T = y1:T − w1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Choosing the complexity measure introduces inductive bias into the associated algorithm, which is ubiquitous in high dimensional statistics and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The rationale is that no algorithm works well universally for all problem instances, therefore a meaningful goal is to find the suitable inductive bias for the considered application, and design the corresponding optimal algorithm for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Specifically for our setting, prior works mostly studied the two problem structures separately, as reviewed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For static regret (ut = u) on unconstrained domains, the standard complexity measure is the comparator norm ∥u∥ [MO14, OP16, CO18], whereas for dynamic regret on bounded domains, one typically considers the path length �T −1 t=1 ∥ut − ut+1∥ [Zin03, ZLZ18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A recent work [JC22] studied unconstrained dynamic regret by combining these two complexity measures, resulting in the regret bound (simplified) RegretT (u1:T ) = ˜O � �G � � � � � T � t=1 ∥ut∥2 � �T −1 � t=1 ∥ut+1 − ut∥2 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, by closely examining its inductive bias, such a bound may not be the most natural one for “non- converging” environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In order to achieve low regret (hence low cumulative loss via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (2)), it is implicitly assumed that the residual sequence u1:T = y1:T −w1:T is small and almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The latter poses a somewhat stringent requirement on the oracle forecaster A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For example, if u1:T is periodic, as often encountered in time series with seasonality, then the regret bound is in general linear in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, the regret bound is achieved by a heavily customized mirror descent algorithm, which deviates from classical frameworks and relies on rather sophisticated algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In this paper, we will take a conceptually different sparse coding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The obtained regret bound adapts to a new complexity measure of u1:T , which naturally exploits more intricate prior knowledge of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 Contribution The contributions of this paper are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our first contribution is a simple framework that achieves a new type of unconstrained dynamic regret bounds (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In a broad sense, it is based on two ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We consider the sequence space RdT that contains x1:T , g1:T and u1:T , rather than the default domain Rd that contains per-round quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is a fundamental view for the batch analysis of sequential data, such as in signal processing [Mal08, VKG14] and time series modeling [BD16, SS17], but (in our opinion) under-explored in the online learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 Static online learning can be considered as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We use advances in static unconstrained OLO to aggregate dynamic base algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In contrast to expert-based model selection approaches, this enables learning linear combinations of the base algorithms, rather than their convex combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2Such bounds are called oracle inequalities in statistical learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3Possibly due to the emphasis on static regret by the community: the sequence u1:T collapses into a time-invariant u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2 Combining the two ideas converts our problem into Online Linear Regression (OLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If the comparator u1:T can be linearly represented by a certain collection of feature vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', a dictionary) in RdT , then our regret bound adapts to (i) the energy of u1:T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) the sparsity of its representation, without knowing either conditions beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This brings two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our approach is built upon close connections to signal processing, thus can benefit from prior works there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For example, a major research topic4 in signal processing is finding the appropriate (typically redundant) dictionary for specific applications, such that the considered signal admits a sparse representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We allow taking such a dictionary as prior knowledge and adapting to its quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Instead of requiring heavy customization like [JC22], many static unconstrained OLO algorithm, given the dictionary, can be used as a black box to solve OLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, our approach automatically inherits a wide range of favorable properties from the static regret setting, such as Lipschitz constant adaptivity [Cut19a], scale-freeness [MK20] and generalized loss-regret tradeoffs [ZCP22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Overall, the proposed approach is not a replacement, but a complement to [JC22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' They represent different inductive bias, thus should be selected based on the specific application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Nonetheless, simply adding them can always theoretically guarantee the best of both worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our second contribution is a new static unconstrained OLO algorithm, which can be used as a subroutine of the sparse coding framework (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To explain what it does, let us consider again the time series application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Intuitively, given an oracle forecaster A, we have to determine how much we trust it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is essentially a tradeoff: if we want low regret on comparators y1:T that are close to w1:T , we have to sacrifice the regret with respect to far-away comparators, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In the setting of static regret, our prior work [ZCP22a] proposed a continuous-time-inspired algorithm with the optimal tradeoff,5 but the bound is not simultaneously adaptive to the gradient variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Such gradient adaptivity has been a hallmark of practical algorithms, as popularized by AdaGrad [DHS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In this paper, we propose an algorithm that closes this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key technique is a new discretization argument that quantifies the deviation of the discrete time algorithm from its ideal, continuous time counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging it into the sparse coding framework, we obtain a dynamic regret bound that adapts to not only the sparsity of the comparator (on the transform domain), but also the sparsity of the observed gradients (on the time domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 Related work Our paper addresses the connection between unconstrained online learning and dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Although they both embody the idea of comparator adaptivity, unified studies have been scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Unconstrained OLO To obtain static regret bounds in OLO, Online Gradient Descent (OGD) [Zin03] is often the default approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With learning rate η, it guarantees O(η−1 ∥u∥2 2 +ηT) regret with respect to any static comparator u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Without the prior knowledge of ∥u∥2, it is impossible to tune η optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To address this issue, a series of works (also called parameter-free online learning) [SM12, MO14, OP16, CO18, FRS18, MK20, ZCP22a] developed vastly different strategies to achieve the oracle optimal rate O(∥u∥ √ T) up to logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Most recent works are based on a dual space analysis and an elegant loss-regret duality [MO14], with the model selection approach from [FKMS17, CLW21, JC22] being a notable exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In these regret bounds, the complexity of u is measured by the comparator norm ∥u∥, or more generally, ∥u − w∥ given a prior w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' L1 and L2 norm bounds were presented in [SM12], while general Banach norm bounds were developed by [FRS18, CO18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Historically, the L1 norm has renowned connections to sparsity, as suggested by LASSO [Tib96], compressed sensing [CRT06], and several works in online learning [KW95, SM12, Ger13, vdH19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, we are not aware of any prior use of such regret bounds in characterizing the structural simplicity of nonstationary environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4As the title of [Mal08] suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Often framed as representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5Defined as achieving O( √ T) regret without the doubling trick, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3 Our second contribution is dedicated to static unconstrained OLO itself, thus requires a more detailed review of existing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is deferred to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 for cleaner exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Dynamic regret Although the field of online learning primarily focused on the static regret, comparing against dynamic sequences has been studied by several lines of works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The closest topic to ours is the universal dynamic regret, where the regret bound adapts to the complexity of the comparator u1:T on a bounded domain with diameter D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Typically, the complexity measure is the path length PT,p = �T −1 t=1 ∥ut − ut+1∥p [HW01] or its generalization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', norm squared [KMBAY15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The optimal bound for OLO is O(G � DTPT,2) [Zin03, HW15, JRSS15, ZLZ18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With curved losses, the accelerated rate ˜O(T 1/3P 2/3 T,1 ) is achievable [BW21, BW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As expected, one cannot go beyond linear dynamic regret in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The hope is that for “converging” environments where reasonable comparators have short path lengths, the overall regret bound can be sublinear in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Except [JC22, LZZZ22], a shared limitation is the requirement of a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A practical solution is to estimate the range of the problem offline, but since the diameter D is used to select the hyperparameter, wrong estimates will deteriorate the empirical performance of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Besides the universal dynamic regret, there are other notions of dynamic regret that do not induce oracle inequalities like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (2), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', (i) the restricted dynamic regret [YZJY16, ZYY+17, BW19, BW20, BZW21], which depends on the complexity of certain offline optimal comparator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) regret bounds that depend on the functional variation �T −1 t=1 maxx |lt(x) − lt+1(x)| [BGZ15, CWW19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' They are both incompatible with OLO on unbounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Notably, we emphasize the difference between our work and a dynamic model approach from [HW15, ZLZ18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' On a bounded domain X, their algorithms can take N dynamic models Φt,n : X → X, n ∈ [1 : N] as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The regret bound has a similar form as path length bounds [Zin03], but replaces the path length with the error of the best dynamic model on the comparator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', minn �T −1 t=1 ∥ut+1 − Φt,n(ut)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our dictionary also represents certain dynamic prior knowledge, but a key difference is that instead of using the best dictionary element to model the comparator, we use the best linear combination of the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This allows handling unconstrained domain through subspace modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Online regression Our framework builds on online regression, which, in its nonparametric form, has been connected to the path length characterization of dynamic regret [RS14, GG15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Prior works are mostly restricted to the square loss, and efficient computation can be a challenge [BW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For the special case of Online Linear Regression (OLR) with square loss, the celebrated VAW forecaster [AW01, Vov01] guarantees O(N log T) regret against any unbounded coefficient vector ˆu ∈ RN, where N is the dimension of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Such a fast rate becomes vacuous when N > T [GY14], therefore [Ger13] proposed a sparsity regret bound ˜O(∥ˆu∥0) and an accompanying inefficient algorithm as its high dimensional generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Efficient computation was addressed by [GW18], but the obtained result only applies to bounded ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In some sense, such sparsity regret bounds are the square loss analogue of the L1-norm parameter-free bounds in OLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' They are also closely related to sparsity oracle inequalities in statistics, as reviewed by [Ger13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Parametric time series models Besides the dynamic regret approach to time series forecasting, significant research effort has been devoted to parametric strategies with stronger inductive bias, such as the ARMA model, state space models, and more recent deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Online learning has been applied to such models as well [AHMS13, AHZ15, AM16, KM16, HLS+18], leading to forecasting guarantees under mild statistical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Taking the autoregressive (AR) model for example, we will show that learning it can be converted to an instance of the sparse coding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Other sparsity topics in OL Finally, we review other sparsity-related topics in online learning, which do not fit into the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [LLZ09, Xia09, DSSST10, SST11] considered using online learning to solve batch L1 regularized problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The goal is to achieve sparse predictions instead of sparsity adaptive regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [Kal14, FKK16, KKLP17] studied online sparse regression, where only a subset of features are available in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The challenge is to handle bandit feedback in OLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 Notation For two integers a ≤ b, [a : b] is the set of all integers c such that a ≤ c ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Treating all vectors as column vectors, span(A) denotes the column space of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For a function Φ : R × R → R, assuming differentiability, let ∂1Φ and ∂2Φ be its first order partial derivatives with respect to the two arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Similarly, ∂11Φ, ∂12Φ and ∂22Φ denote second order partial derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' f ∗ is the Fenchel conjugate of a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' log represents natural logarithm when the base is omitted, and log+(·) := 0 ∨ log(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' KL(·||·) is the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ΠX(x) is the Euclidean projection from x to a closed convex set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We define the imaginary error function as erfi(x) = � x 0 exp(u2)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that it is scaled by √π/2 from the usual definition, thus can also be queried from standard software packages like SciPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let erfi−1 be its inverse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Specialized notations for the sparse coding framework are detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2 Sparsity adaptive dynamic regret In this section we present our sparse coding framework for unconstrained dynamic regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The basic setting is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Focusing on the sequence space RdT , Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 presents our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 discusses our framework from a generalized primal-dual perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 Setting We start by formally introducing our setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For sequences x1:T , g1:T and u1:T , we will flatten everything and treat them as dT dimensional vectors, concatenating per-round quantities in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' They are called signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our framework requires online access to a dictionary matrix H ∈ RdT ×N, whose columns are N nonzero feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We write H in an equivalent block form as [ht,n]1≤t≤T,1≤n≤N, where each block ht,n ∈ Rd×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The accompanied linear transform u = Hˆu relates a signal u ∈ RdT to a coefficient vector ˆu ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Adopting the convention in signal processing, we will call RdT the time domain, and RN the transform domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In general, symbols without hat refer to time domain quantities, while their transform domain counterparts are denoted with hat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With such notations, we consider the following interaction protocol, which could be termed multivariate OLR with linear loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In the t-th round, our algorithm observes a d-by-N feature matrix Ht := [ht,n]1≤n≤N, makes a prediction xt ∈ Rd, receives a loss gradient gt ∈ Rd satisfying ∥gt∥2 ≤ G, and then suffers the loss ⟨gt, xt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The performance metric is the dynamic regret defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (1), where the comparator u1:T is unconstrained in RdT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 Main result Overall, our strategy is to apply a static unconstrained OLO algorithm on the direction of each feature vector, and then aggregate their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, let us start with a single feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Size 1 dictionary Consider an index n ∈ [1 : N], which is associated to the feature h1:T,n := [h1,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , hT,n] ∈ RdT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We suppress the index n and write it as h1:T = [h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , hT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For any comparator u1:T ∈ span(h1:T ), there exists ˆu ∈ R such that u1:T = h1:T ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The cumulative loss of u1:T can be rewritten as ⟨g1:T , u1:T ⟩ = ⟨g1:T , h1:T ⟩ ˆu = T � t=1 ⟨gt, ht⟩ ˆu, which is the loss of the coefficient ˆu on surrogate losses ⟨gt, ht⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To compete with u1:T ∈ span(h1:T ), it suffices to run a 1D static regret algorithm that competes with ˆu ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Formally, we present this procedure as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By further assuming bounded ∥ht∥2, Algorithm 1 could take any static unconstrained OLO algorithm as a black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, since the feature ht is revealed before picking xt, we can use a better black box that adapts to the scale of ht, even if ∥ht∥2 is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is crucial for our purpose, as it allows the dynamic regret bound to adapt to the energy of the comparator, E(u1:T ) := ∥u1:T ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As an example, we present such a black box as Algorithm 5 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2, which generalizes a recent result [ZCP22a] to the setting with time-varying but known Lipschitz constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5 Algorithm 1 Sparse coding with size 1 dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Require: An algorithm A for static 1D unconstrained OLO, and a nonzero feature vector h1:T ⊂ RdT revealed online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , T do 2: Receive ht ∈ Rd, and pass G ∥ht∥2 to A as the Lipschitz constant of its next (t-th) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3: Query A for its t-th output, and assign it to ˆxt ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4: Predict xt = ˆxtht ∈ Rd, and receive a loss gradient gt ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5: Compute ˆgt = ⟨gt, ht⟩, and send it to A as its t-th surrogate loss gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6: end for Although not simultaneously adaptive to the magnitude of gt, Algorithm 5 enjoys other appealing properties in the static setting, such as the optimal loss-regret tradeoff (reviewed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1) and the optimal leading constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Its analysis goes through a non-gradient-adaptive discretization argument (the Discrete Itˆo formula [HLPR20]), which sets the stage for our improved technique later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging it into Algorithm 1 yields Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proofs for this subsection are deferred to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let ˆε > 0 be an arbitrary hyperparameter for Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Applying it as a subroutine, for all T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees RegretT (u1:T ) ≤ GεT + √ 2 ∥u1:T ∥2 G �� log � 1 + ∥u1:T ∥2 √ 2εT � + 1 � , where εT = ˆε ∥h1:T ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The subscript emphasizes that εT depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' General dictionary With the single direction learner above, let us turn to the general setting with N features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We run N copies of Algorithm 1 in parallel, aggregate their predictions, and the regret bound sums Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, similar to [Cut19b] in the static setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' An extra twist is that each feature is associated with a different hyperparameter: it introduces a prior on the transform domain, which is essential for the overparameterized regime with N ≫ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In summary, the pseudocode is presented as Algorithm 2, and the regret bound is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Algorithm 2 Sparse coding with general dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Require: A dictionary H = [ht,n], where ht,n ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Constants ˆε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , ˆεN > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1: For all n ∈ [1 : N], initialize a copy of Algorithm 1 as An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It runs Algorithm 5 as a subroutine, with hyperparameter ˆεn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , T do 3: Receive Ht = [ht,n]1≤n≤N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all n, send ht,n to An, and query its prediction wt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4: Predict xt = �N n=1 wt,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5: Receive loss gradient gt, and send it to A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , AN as loss gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6: end for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all T ∈ N+ and u1:T ∈ RdT , Algorithm 2 guarantees RegretT (u1:T ) ≤ 2GET + √ 2GUT �� log+ UT √ 2ET + � KL(q||π) + 2 � + G T � t=1 ∥ut,0∥2 , where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N n=1 u1:T,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ET = �N n=1 ˆεn ∥h1:T,n∥2 and UT = �N n=1 ∥u1:T,n∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' π and q are N dimensional probability vectors defined by πn = ˆεn ∥h1:T,n∥2 /ET , and qn = ∥u1:T,n∥2 /UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6 To interpret this result, we start from the simplest case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If the size N = d, the dictionary Ht = Id (the d dimensional identity matrix), and the hyperparameters satisfy �N n=1 ˆεn = ε, then against any static comparator (ut = u ∈ Rd), Theorem 1 guarantees RegretT (u1:T ) ≤ 2εG √ T + √ 2 ∥u∥1 G √ T �� log+ ∥u∥1 √ 2ε + � KL(q||π) + 2 � , (3) where πn = ˆεn/ε, and q = u/∥u∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Intuitively, since ε can be tuned arbitrarily low, the first term on the RHS is typically negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Within the rest, the KL term gives the bound a Bayesian flavor:6 we use a prior π to guess the posterior distribution q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', how the “strength” of the comparator is spread across different feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Simply picking π as the uniform distribution results in KL(q||π) ≤ log N, and the bound recovers the standard ˜O(∥u∥1 √ T) bound in static unconstrained OLO [Ora19, Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next, we enter the dynamic realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Assume feature vectors are orthogonal, and the comparator u1:T ∈ span(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Within Theorem 1, we are free to set u1:T,0 = 0, and let u1:T,n be the projection of u1:T onto span(h1:T,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to orthogonality, the projection preserves the energy of the comparator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e, E(u1:T ) = T � t=1 ∥ut∥2 2 = N � n=1 ∥u1:T,n∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By further defining SH(u1:T ) := (�N n=1 ∥u1:T,n∥2)2/�N n=1 ∥u1:T,n∥2 2, we have UT = � SH(u1:T )E(u1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that SH(u1:T ) is a classical sparsity measure of {u1:T,n}1≤n≤N [HR09]: if there are only N0 ≤ N nonzero vectors within this collection, then SH(u1:T ) ≤ N0 due to the Cauchy-Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, through the complexity measure UT , Theorem 1 adapts to (i) the energy of u1:T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) the sparsity of its rep- resentation, without knowing either condition beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With low enough ET , the bound has the order ˜O( � SH(u1:T )E(u1:T )) ≤ ˜O( √ NT): the easier the comparator is (low energy, and sparse on H), the lower the bound becomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' So far we have only considered the underparameterized regime (N ≤ dT) where feature vectors can be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, recent trends in signal processing have emphasized overparameterization (N ≫ dT) as a key to obtain sparser representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Theorem 1 can be nicely interpreted in this context as well: since it applies to any decomposition of u1:T , as long as u1:T can be represented by a subset ˜H of orthogonal features within H, the regret bound adapts to S ˜ H(u1:T ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', the sparsity of u1:T measured on ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In other words, Theorem 1 adapts to the quality of the optimal (comparator-dependent) sub-dictionary ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that: Algorithm 2 runs N base algorithms in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For efficient computation with large N, the dictionary itself has to be sparse, which is called the local property in signal processing [Mal08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 for a comparison between Fourier and wavelet dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Theorem 1 suffers a large-N penalty through the KL term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In practice, one may pick a good prior π, instead of the uniform distribution, to reduce this root-logarithmic overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Power law phenomenon To further demonstrate the quantitative benefit, let us consider an empirically justified setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In signal processing, the study of sparsity has been partially motivated by the power law [Pri21]: for many real world signals, even with a standard Fourier or wavelet dictionary, the n-th largest transform domain coefficient has magnitude roughly proportional to n−α, where α ∈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Suppose d = 1, and the comparator u1:T exhibits the power law through an orthogonal transformation of RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, when T is large, SH(u1:T ) = (�T n=1 n−α)2 �T n=1 n−2α ≈ 2α − 1 (1 − α)2 (T)2−2α = O � T 2−2α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With E(u1:T ) = O(T), we obtain a sublinear ˜O(T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5−α) dynamic regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6Analogous to comparator adaptive bounds in the expert problem [LS15, KVE15, CLW21, NBC+21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 7 Example Finally, we note that the strength of our framework lies in the incorporation of domain knowledge through the dictionary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4, we discuss several concrete examples, including classical Fourier and wavelet dictionaries, the autoregressive dictionary defined by time series, and dictionaries learned by online learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As an added bonus, different unconstrained dynamic regret bounds, such as [JC22] and the different instances of Theorem 1, can be combined by simply summing their corresponding predictions (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 Primal-dual interpretation Concluding this section, we discuss our framework from a primal-dual perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In static OLO [Ora19], the primal space refers to the domain Rd, while the dual space refers to the space of linear maps on Rd, or intuitively, where we store a sufficient statistic of the observed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The same algorithm can have different but equivalent analysis on the primal space and the dual space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='g, the Follow the Regularized Leader (FTRL) versus the potential method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our framework generalizes the static setting, thus can be understood in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Specifically, we consider an analogous primal-dual relation between the time domain RdT and the transform domain RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' From this angle, Algorithm 2 runs as follows, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Environment Transform domain ℝ𝑁 𝑆𝑡 = 𝑆𝑡−1 − ො𝑔𝑡 ො𝑔𝑡 = ℋ𝑡 𝑇𝑔𝑡 Potential on ℝ𝑁 ො𝑥𝑡+1 = 𝜕Φ(𝑆𝑡) 𝑥𝑡+1 = ℋ𝑡+1 ො𝑥𝑡+1 Figure 1: Algorithm 2 as potential method with general sufficient statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' At the end of the t-th round, we multiply gt ∈ Rd by the d-by-N feature matrix Ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The sufficient statistic in RN is updated as St = St−1 − HT t gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By evaluating the gradient of a potential function Φ at St, we obtain a transform domain prediction ˆxt+1 ∈ RN (analogous to Line 3 of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' After the next feature matrix Ht+1 is revealed, we define the t + 1-th prediction as xt+1 = Ht+1ˆxt+1 ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Crucially, instead of storing the sum of loss gradients (as in static OLO), we store a N dimensional filtered version of the gradient sequence g1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Roughly speaking, N captures the complexity of the comparator class, against which our algorithm guarantees sublinear regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As for the proof strategy, we have focused on aggregating regret bounds on the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Alternatively, one could use a transform domain analysis to obtain the same result, generalizing the standard workflow in static unconstrained OLO [MO14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key is a loss-regret duality for sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If there exists a function fT : RdT → R such that the prediction sequence x1:T guarantees a loss upper bound ⟨g1:T , x1:T ⟩ ≤ −fT (g1:T ), then for all u1:T ∈ RdT , RegretT (u1:T ) ≤ f ∗ T (−u1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In general, the function fT can be fully nonlinear, so we consider a more structured function class where the nonlinearity acts on a linear sketch of the input, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', fT (x) = Φ(HT x) for some nonlinear potential function Φ : RN → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By picking xt = Htˆxt, constructing the loss upper bound is converted into a coin-betting problem with decision ˆxt ∈ RN, where existing theoretical results are available [Cov66, OP16, ZCP22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 7We present two other versions in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6, which are more closely tied to path-length-based dynamic regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 8 3 A better static algorithm As shown in Figure 1, the sparse coding framework consists of two components: (i) choosing a dictionary H that captures the dynamics of the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) designing a good potential function (or static unconstrained OLO algorithm) with low quantitative regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We now present our second contribution, which addresses the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 surveys the background of this topic, while our new algorithm, including its implication for the sparse coding framework, is presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For static comparators ut = u ∈ Rd, we will write the regret as RegretT (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 Loss-regret tradeoff An important topic in static unconstrained OLO is the loss-regret tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to a celebrated no free lunch theorem [Cov66], all such algorithms are required to trade off their cumulative loss RegretT (0) with their leading regret term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', RegretT (u) for large ∥u∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Roughly speaking, such a tradeoff represents how much we trust the initialization of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Most prior works [MO14, OP16] are natively designed with O(1) loss and O(∥u∥ � T log(∥u∥ T)) regret, while in principle, the optimal tradeoff corresponds to O( √ T) loss and O(∥u∥ � T log(∥u∥)) regret, which matches the minimax optimal O( √ T) rate on bounded domains (with respect to T alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Although different loss-regret tradeoffs are mutually convertible through the doubling trick [SS11], doing so significantly downgrades the empirical performance of the algorithm,8 thus should (ideally) be avoided in theory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A recent work of ours [ZCP22a] achieved the optimal tradeoff in an “anytime” manner without doubling tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, compared to other frontiers in this field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', [MK20]), the regret bound does not simultaneously adapt to the observed gradient variance VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The importance of such gradient adaptivity has been demonstrated in practice [DHS11], but from a technical perspective, it is challenging to add this property to non-gradient- adaptive unconstrained algorithms, as both the algorithm and the analysis need to be modified with considerable sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Existing techniques [CO18, MK20, JC22] are closely tied to the suboptimal loss-regret tradeoff,9 and their extensions to our objective are unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' At the center of the optimal tradeoff [ZCP22a] is a nonstandard erfi potential function, which solves a Partial Differential Equation (PDE) that characterizes the continuous time (CT) limit of the learning game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In a broader context, the interplay between discrete time (DT) online learning and its CT limit has received growing attention [KS10, Zhu14, BEZ20, DK20, HLPR20, KKW20, PLH22, ZCP22b], as the latter is often easier to analyze and gain intuition from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, a bottleneck here is the discretization of CT-derived algorithms – the standard technique is the Discrete Itˆo formula [HLPR20], which by construction is not gradient adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, although gradient adaptivity has been studied in CT before, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', [Fre09] and [HLPR20, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4], the obtained benefits have not been extended to the DT online learning problem we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In this section, we improve [ZCP22a] by simultaneously achieving gradient adaptivity and the optimal loss-regret tradeoff, without doubling tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key technique is a new discretization argument that further induces gradient adaptivity, which could be of separate interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 Main result Concretely, we consider OLO with domain Rd and a known Lipschitz constant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to the setting of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, we now focus on the static regret RegretT (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our approach requires two potential functions defined as follows, where the parameters satisfy ε > 0, α > 0 and z > k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' φ(x, y) = ε√αx � 2 � y √ 4αx 0 erfi(u)du − 1 � , Φ(V, S) = φ(V + z + kS, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (4) φ is the “basic” potential function from [ZCP22a], which solves the Backward Heat Equation (BHE) ∂1φ+α∂22φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that φ can be evaluated efficiently using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Φ is the potential function we actually apply, which is constructed from φ with a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 8And incurs a multiplicative constant in the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 9With O(√VT log VT ) dependence on VT alone, worse than the optimal O(√VT ) rate achieved by adaptive OGD on bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 9 Overall, our algorithm has a hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key component is the 1D base algorithm (Algorithm 3), where for clarity, all the algorithmic quantities are denoted with tilde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We enforce the requirement − �t i=1 ˜li ≥ −1 to make sure ˜St ≥ −1, thus the gradient computation in Line 3 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, the meta-algorithm (Algorithm 4) applies two standard techniques [CO18, Cut20] on top of the base algorithm: the first reduces the domain of the base algorithm from R to R+, while the second extends it from R+ to Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Algorithm 3 1D base algorithm Require: The potential function Φ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Constants ε > 0, α > 0 and z > k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Surrogate loss gradients ˜l1:T satisfying ˜lt ∈ [−1, 1] and − �t i=1 ˜li ≥ −1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1: Initialize ˜V0 = 0, ˜S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , T do 3: Predict ˜zt = ∂2Φ( ˜Vt−1, ˜St−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4: Receive the surrogate loss gradient ˜lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5: Let ˜Vt = ˜Vt−1 + ˜l2 t , and ˜St = ˜St−1 − ˜lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6: end for Algorithm 4 Meta algorithm on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1: Define A1d as a copy of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Define AB as OGD on the d-dimensional unit L2 norm ball, with adaptive learning rate ηt = � 2/�t i=1 ∥gi∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Initialization of AB is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' do 3: Query A1d for its prediction ˜zt ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let zt = ΠR+(zt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4: Query AB for its prediction wt ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5: Predict xt = ztwt, receive the loss gradient gt ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6: Send gtG−1 as the surrogate loss to AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 7: Define lt = � gtG−1, wt � , and ˜lt = � lt, lt˜zt ≥ ltzt, 0, else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 8: Send ˜lt as the surrogate loss to A1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 9: end for Before analyzing its performance, Proposition 3 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 shows that the surrogate loss ˜lt defined in the meta-algorithm indeed satisfies − �t i=1 ˜li ≥ −1, therefore the entire hierarchical procedure is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, with the gradient variance defined as VT = �T t=1 ∥gt∥2 2, we present the regret bound as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With ε > 0, α = 1, k = 2 and z = 16, Algorithm 4 guarantees for all T ∈ N+ and u ∈ Rd, RegretT (u) ≤ ε � VT + 2G ¯S + ∥u∥2 � ¯S + 2 � 2VT � , where ¯S = 8G � 1 + � log(2 ∥u∥2 ε−1 + 1) �2 + 2 � VT + 16G2 � 1 + � log(2 ∥u∥2 ε−1 + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us make this bound a bit more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Using asymptotic orders, we can simplify it into (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 for the derivation) RegretT (u) ≤ ε �� VT + 6G � + ∥u∥2 O �� VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) � , which is simultaneously valid in two regimes: (i) ∥u∥2 ≫ ε and VT ≫ G2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) u = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', RegretT (0) ≤ ε �√VT + 6G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that the logarithmic residual term (outside the root) is standard in gradient adaptive uncon- strained OLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, with a O(√VT ) maximum loss bound, Algorithm 4 guarantees a O(∥u∥2 √VT log ∥u∥2) 10 regret bound10 – this matches the minimax optimal rate O(√VT ) on bounded domains, achieved by adaptive OGD [DHS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to prior works, we improve [ZCP22a] by achieving second order gradient adaptivity, and [CO18, MK20, JC22] by a better asymptotic rate on VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Sketch of the analysis We now sketch our analysis of the base algorithm (Algorithm 3), including the key idea of discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' At one point we consider two-case cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Alternate expressions for the second case are provided in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Overall, the analysis has a similar procedure as typical potential methods: we first upper-bound the cumulative loss �T t=1 ˜lt˜zt, and then obtain the regret bound through a loss-regret duality [MO14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The loss upper bound follows from a telescopic sum on the one-step bound: ˜lt˜zt = ˜lt∂2Φ( ˜Vt−1, ˜St−1) ≤ Φ( ˜Vt−1, ˜St−1) − Φ( ˜Vt, ˜St).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proving it is the main technical challenge of our analysis (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3), as in most prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To this end, we aim to show that for all V ≥ 0, S ≥ −1 and c ∈ [−1, 1] satisfying S + c ≥ −1, fV,S(c) := Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It is clear that fV,S(0) = f ′ V,S(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, to prove fV,S(c) ≤ 0, a sufficient condition is the concavity of fV,S on the considered input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By calculating the Hessian and using |c| ≤ 1, f ′′ V,S(c) ≤ 2∂1Φ(V + c2, S + c) + 4∂11Φ(V + c2, S + c) + 4 ��∂12Φ(V + c2, S + c) �� + ∂22Φ(V + c2, S + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Furthermore, due to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (4), the derivatives of Φ are concisely related to the derivatives of φ: if ∂12Φ(V + c2, S + c) ≤(≥) 0, then f ′′ V,S(c) ≤ 2∂1φ + [k −(+) 2]2∂11φ + 2[k −(+) 2]∂12φ � �� � :=∆ +∂22φ, (5) where the derivatives on the RHS are evaluated at the input pair (V + c2 + z + k(S + c), S + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now, the key observation is that the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (5) has a striking similarity to the Backward Heat Equation ∂1φ + α∂22φ = 0, which the basic potential function φ satisfies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This motivates us to view ∆ as the discretization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Ideally, if ∆ ≤ 0, then f ′′ V,S(c) ≤ 0 by simply picking α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The reality is only slightly more complicated: We pick k = 2 to eliminate the harder case within the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As for the other case, it only occurs when S + c is at most constant-away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Picking a large enough constant offset z, we have ∆ ≤ ∂22φ, therefore f ′′ V,S(c) ≤ 0 follows from α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In summary, through a change of variable, we show how to utilize the CT property (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', the BHE) of potential functions in the verification of DT algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The discretization error is more finely characterized compared to the Discrete Itˆo formula (surveyed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2), which results in additional gradient adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, since the BHE succinctly captures a family of adaptive potentials [ZCP22a], the argument above could be applicable to other loss-regret tradeoffs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Such generality and simplicity may provide benefits over existing techniques without CT connections [CO18, MK20, JC22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Application to dynamic regret Finally, we apply this static algorithm to bound the dynamic regret, through our sparse coding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Slightly different from Section 2, we impose an additional assumption, ∥ht,n∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As shown in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='8, with asymptotic simplification, the dynamic bound against any u1:T ∈ span(H) becomes RegretT (u1:T ) ≤ N � n=1 ˆεn � � � � � � T � t=1 ⟨gt, ht,n⟩2 + 6G � � + ˜O � � N � n=1 � � � � T � t=1 ⟨gt, ut,n⟩2 � � , where the first sum is a cumulative loss term tuned to be small, and {ut,n} is an arbitrary decomposition of the comparator satisfying �N n=1 ut,n = u1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to Theorem 1 which guarantees a similar form RegretT (u1:T ) ≤ 2G N � n=1 ˆεn � � � � T � t=1 ∥ht,n∥2 2 + ˜O � �G N � n=1 � � � � T � t=1 ∥ut,n∥2 2 � � , 10Loosely assimilating the residual term for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 11 our improved approach further achieves gradient adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In other words, the obtained algorithm adapts to not only the sparsity of the comparator (on the transform domain), but also the sparsity of the observed gradients (on the time domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4 Conclusion In this paper, we presented two complementary results for unconstrained OLO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Through a sparse coding framework, one can convert static unconstrained OLO algorithms to the dynamic setting, and the regret bound adapts to both the energy and the sparsity of the comparator sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is closely connected to representation learning, thus may lead to deeper integration of the two research areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We propose an algorithm that simultaneously achieves the gradient variance adaptivity and the optimal loss- regret tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key technique is a new discretization argument, which could facilitate the continuous time analysis of online learning in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' References [AHMS13] Oren Anava, Elad Hazan, Shie Mannor, and Ohad Shamir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Online learning for time series prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In Conference on learning theory, pages 172–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PMLR, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [AHZ15] Oren Anava, Elad Hazan, and Assaf Zeevi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Online time series prediction with missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In International conference on machine learning, pages 2191–2199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PMLR, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [AM16] Oren Anava and Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Heteroscedastic sequences: beyond gaussianity.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [vdH19] Dirk van der Hoeven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' User-specified local differential privacy in unconstrained adaptive online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [VKG14] Martin Vetterli, Jelena Kovaˇcevi´c, and Vivek K Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Foundations of signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Cambridge University Press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [Vov01] Volodya Vovk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Competitive on-line statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' International Statistical Review, 69(2):213–248, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [Xia09] Lin Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Dual averaging method for regularized stochastic learning and online optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 22, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [YZJY16] Tianbao Yang, Lijun Zhang, Rong Jin, and Jinfeng Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Tracking slowly moving clairvoyant: Optimal dynamic regret of online learning with true and noisy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 449–457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PMLR, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [ZCP22a] Zhiyu Zhang, Ashok Cutkosky, and Ioannis Paschalidis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PDE-based optimal strategy for uncon- strained online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 26085–26115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PMLR, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [ZCP22b] Zhiyu Zhang, Ashok Cutkosky, and Ioannis Ch Paschalidis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Optimal parameter-free online learning with switching cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='06846, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [Zhu14] Kangping Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Two problems in applications of PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' PhD thesis, New York University, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 15 [Zin03] Martin Zinkevich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Online convex programming and generalized infinitesimal gradient ascent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In Proceedings of the 20th International Conference on Machine Learning, pages 928–936, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [ZLZ18] Lijun Zhang, Shiyin Lu, and Zhi-Hua Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Adaptive online learning in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Advances in neural information processing systems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' [ZYY+17] Lijun Zhang, Tianbao Yang, Jinfeng Yi, Rong Jin, and Zhi-Hua Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Improved dynamic regret for non-degenerate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 16 Appendix Organization Appendix A presents details on our sparse coding framework (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix B presents details on the improved static unconstrained OLO algorithm (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A Detail on sparse coding The sparse coding framework requires running a static OLO algorithm as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We present some basic facts on the erfi function in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, and a non-gradient-adaptive static OLO subroutine in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 contains the proof of our sparsity adaptive regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 discusses several concrete choices of the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 shows how to combine algorithms with different unconstrained dynamic regret guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6 complements the primal-dual interpretation of our framework from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 Fact on the erfi function First of all, we will use the following facts on the erfi function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that in this paper, we scale it from its usual definition, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all x ∈ R, � x 0 erfi(u)du = xerfi(x) − 1 2 exp(x2) + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The proof follows from a simple integration by parts, therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all x ≥ 1, erfi(x) ≥ exp(x2)/2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let f(x) = erfi(x) − exp(x2)/2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' f(1) = erfi(1) − e/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all x ≥ 1, f ′(x) = 1 2x2 exp(x2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 (From Theorem 4 of [ZCP22a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all x ≥ 0, erfi−1(x) ≤ 1 + � log(x + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 Unconstrained OL with varying Lipschitzness We present a non-gradient-adaptive, static 1D unconstrained OLO algorithm as Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It is designed to exploit time-varying, but known Lipschitz constants on the loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The regret bound is Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Algorithm 5 1D Static unconstrained OLO with time-varying Lipschitzness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Require: A hyperparameter ˆε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A sequence of Lipschitz constants G1:T such that each loss gradient ˆgt ∈ R satisfies |ˆgt| ≤ Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 1: Initialize V0 = S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Define a potential function as Φ(V, S) = ˆε √ V � 2 � S √ 2V 0 erfi(x)dx − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (6) Note that � erfi(x)dx can be evaluated using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2: for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' do 3: Receive the t-th Lipschitz constant Gt, and let Vt = Vt−1 + G2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 4: If Gt = 0, predict ˆxt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Otherwise, predict ˆxt = 1 2Gt [Φ (Vt, St−1 + Gt) − Φ (Vt, St−1 − Gt)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 5: Observe the loss gradient ˆgt ∈ R, and let St = St−1 − ˆgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 6: end for 17 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all T ∈ N+ and ˆu ∈ R, Algorithm 5 guarantees T � t=1 ⟨ˆgt, ˆxt − ˆu⟩ ≤ ˆε � � � � T � t=1 G2 t + √ 2 |ˆu| � � � � T � t=1 G2 t �� log � 1 + |ˆu| √ 2ˆε � + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 generalizes the argument of [HLPR20, ZCP22a] by allowing arbitrary time-varying gap parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It also demonstrates the existing technique (the Discrete Itˆo formula) for discretizing continuous- time-derived algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is in contrast to our improved technique in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' First, consider a function Φ : R × R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In light of standard partial derivatives ∂1Φ, ∂2Φ and ∂22Φ, given a gap parameter δ > 0, we define discrete derivatives (denoted with bars) as ¯∂δ 1Φ(V, S) = 1 δ2 � Φ(V, S) − Φ(V − δ2, S) � , ¯∂δ 2Φ(V, S) = 1 2δ [Φ(V, S + δ) − Φ(V, S − δ)] , ¯∂δ 22Φ(V, S) = 1 δ2 [Φ(V, S + δ) + Φ(V, S − δ) − 2Φ(V, S)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If δ = 0, define ¯∂δ 1Φ(V, S) = ¯∂δ 2Φ(V, S) = ¯∂δ 22Φ(V, S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The Discrete Itˆo formula [HLPR20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='13 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='14] has been shown useful in connecting discrete time online learning algorithms with their continuous time counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We generalize it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 (Discrete Itˆo formula with general gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Consider any function Φ : R≥0 × R → R, convex in its second argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all V ≥ 0, S, c ∈ R and δ ≥ 0 satisfying |c| ≤ δ, we have Φ(V + δ2, S + c) − Φ(V, S) ≤ c¯∂δ 2Φ(V + δ2, S) + δ2 � ¯∂δ 1Φ(V + δ2, S) + 1 2 ¯∂δ 22Φ(V + δ2, S) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (7) Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The case of δ = 0 trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As for δ > 0, applying the discrete derivatives, LHS = Φ(V + δ2, S + c) − 1 2 � Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) � + 1 2 � Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) � − Φ(V, S) = Φ(V + δ2, S + c) − 1 2 � Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) � + δ2 � ¯∂δ 1Φ(V + δ2, S) + 1 2 ¯∂δ 22Φ(V + δ2, S) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Comparing it with our objective, it remains to show Φ(V + δ2, S + c) − 1 2 � Φ(V + δ2, S + δ) + Φ(V + δ2, S − δ) � ≤ c¯∂δ 2Φ(V + δ2, S) = c 2δ � Φ(V + δ2, S + δ) − Φ(V + δ2, S − δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Regrouping the terms, it suffices to show Φ(V + δ2, S + c) ≤ δ + c 2δ Φ(V + δ2, S + δ) + δ − c 2δ Φ(V + δ2, S − δ), which follows from the convexity of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To apply the Discrete Itˆo formula to Algorithm 5, we need to fix a minor issue: the function Φ(V, S) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (6) is not well-defined for V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Without loss of generality, we will impose Φ(0, S) = 0 for all S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Notice 18 that Φ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (6) is convex in S, and the prediction in Algorithm 5 is precisely the discrete derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', ˆxt = ¯∂Gt 2 Φ(Vt, St−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging in V ← Vt−1, S ← St−1, δ ← Gt and c ← −ˆgt into Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5, we have Φ(Vt, St) − Φ(Vt−1, St−1) ≤ −ˆgtˆxt + G2 t � ¯∂Gt 1 Φ(Vt, St−1) + 1 2 ¯∂Gt 22 Φ(Vt, St−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The second term on the RHS can be seen as a perturbation on an otherwise clean recursive inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The form of this perturbation term also closely resembles the Backward Heat Equation (BHE) ∂1Φ + 1 2∂22Φ = 0, which, as shown in [ZCP22a], is satisfied by existing potential functions in unconstrained OLO, including Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This explains why the Discrete Itˆo formula is useful: to convert continuous-time-derived algorithms to discrete time, it suffices to characterize the discretization error on the BHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As long as the discretization error (the perturbation term above) is upper-bounded, we can still control the cumulative loss of the algorithm by a telescopic sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', T � t=1 ˆgtˆxt ≤ Φ(0, 0) − Φ(VT , ST ) + T � t=1 G2 t � ¯∂Gt 1 Φ(Vt, St−1) + 1 2 ¯∂Gt 22 Φ(Vt, St−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (8) Specifically for Algorithm 5, we bound the perturbation term as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It uses a key result from [HLPR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all t ∈ N+, Algorithm 5 guarantees G2 t � ¯∂Gt 1 Φ(Vt, St−1) + 1 2 ¯∂Gt 22 Φ(Vt, St−1) � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us define f(x) = 2xerfi(x) − exp(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to the definition of Φ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (6) and the simplification of � erfi(x)dx in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, for all V > 0 and S ∈ R, Φ(V, S) = ˆε √ V f � S √ 2V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' When V = 0, we have defined Φ(0, S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now, consider the quantities in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To proceed, there are two cases: (i) Vt−1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (ii) Vt−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If Vt−1 > 0, plugging in the discrete derivatives, G2 t � ¯∂Gt 1 Φ(Vt, St−1) + 1 2 ¯∂Gt 22 Φ(Vt, St−1) � = 1 2Φ(Vt, St−1 + Gt) + 1 2Φ(Vt, St−1 − Gt) − Φ(Vt−1, St−1) = 1 2 ˆε � Vt � f �St−1 + Gt √2Vt � + f �St−1 − Gt √2Vt � − 2 � Vt−1 Vt f � St−1 � 2Vt−1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to [HLPR20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='10], for all x ∈ R and z ∈ [0, 1), f �x + z √ 2 � + f �x − z √ 2 � ≤ 2 � 1 − z2f � x � 2(1 − z2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Taking x = St−1/√Vt and z = Gt/√Vt proves the first case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As for the second case (Vt−1 = 0), note that St−1 = 0 and Vt = G2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, G2 t � ¯∂Gt 1 Φ(Vt, St−1) + 1 2 ¯∂Gt 22 Φ(Vt, St−1) � = 1 2Φ(G2 t, Gt) + 1 2Φ(G2 t, −Gt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If Gt = 0, then it holds trivially that RHS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Otherwise, RHS = ˆε 2Gt � f � 1 √ 2 � + f � − 1 √ 2 �� ≤ 0, due to straightforward evaluation of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This completes the proof of the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 19 With Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6, it becomes fairly standard to prove the guarantee of Algorithm 5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' See, for example, [Ora19, Chapter 9] for the overall proof strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (8), we obtain a cumulative loss bound T � t=1 ˆgtˆxt ≤ −Φ(VT , ST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to a standard loss-regret duality [Ora19, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6], the regret can be bounded by T � t=1 ⟨ˆgt, ˆxt − ˆu⟩ ≤ Φ∗ VT (ˆu), where Φ∗ VT denotes the Fenchel conjugate of the function ΦVT (·) := Φ(VT , ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Finally, due to the proof of [ZCP22a, Theorem 4], Φ∗ VT (ˆu) ≤ ˆε � VT + |ˆu| � 2VT �� log � 1 + |ˆu| √ 2ˆε � + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 Proof of main results This subsection presents the omitted proofs for Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let ˆε > 0 be an arbitrary hyperparameter for Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Applying it as a subroutine, for all T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees RegretT (u1:T ) ≤ GεT + √ 2 ∥u1:T ∥2 G �� log � 1 + ∥u1:T ∥2 √ 2εT � + 1 � , where εT = ˆε ∥h1:T ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The subscript emphasizes that εT depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We start by rewriting the dynamic regret as RegretT (u1:T ) = T � t=1 ⟨gt, xt − ut⟩ = T � t=1 ⟨gt, htˆxt − htˆut⟩ = T � t=1 ⟨ˆgt, ˆxt − ˆut⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The static regret on the RHS can be bounded from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4, with the Lipschitz constant Gt = G ∥ht∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, RegretT (u1:T ) ≤ Gˆε � � � � T � t=1 ∥ht∥2 2 + √ 2G |ˆu| � � � � T � t=1 ∥ht∥2 2 �� log � 1 + |ˆu| √ 2ˆε � + 1 � = Gˆε ∥h1:T ∥2 + √ 2G |ˆu| ∥h1:T ∥2 � � � � � �log � 1 + |ˆu| ∥h1:T ∥2 √ 2ˆε ∥h1:T ∥2 � + 1 � � = GεT + √ 2 ∥u1:T ∥2 G �� log � 1 + ∥u1:T ∥2 √ 2εT � + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all T ∈ N+ and u1:T ∈ RdT , Algorithm 2 guarantees RegretT (u1:T ) ≤ 2GET + √ 2GUT �� log+ UT √ 2ET + � KL(q||π) + 2 � + G T � t=1 ∥ut,0∥2 , where 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N n=1 u1:T,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ET = �N n=1 ˆεn ∥h1:T,n∥2 and UT = �N n=1 ∥u1:T,n∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' π and q are N dimensional probability vectors defined by πn = ˆεn ∥h1:T,n∥2 /ET , and qn = ∥u1:T,n∥2 /UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To begin with, we apply a dynamic analogue of [Cut19b] to sum the regret bound of single direction learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Any comparator u1:T can be decomposed into the directions of feature vectors plus an unconstrained residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, for all decomposition u1:T = �N n=0 u1:T,n such that u1:T,n ∈ span(h1:T,n) for all n ∈ [1 : T], we have RegretT (u1:T ) = ⟨g1:T , x1:T − u1:T ⟩ = ⟨−g1:T , u1:T,0⟩ + N � n=1 ⟨g1:T , w1:T,n − u1:T,n⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' On each direction we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, ⟨−g1:T , u1:T,0⟩ ≤ G �T t=1 ∥ut,0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It leads to RegretT (u1:T ) ≤ G N � n=1 � � �ˆεn ∥h1:T,n∥2 + √ 2 ∥u1:T,n∥2 � � � � � �log � 1 + ∥u1:T,n∥2 √ 2ˆεn ∥h1:T,n∥2 � + 1 � � � � � + G T � t=1 ∥ut,0∥2 = GET + √ 2GUT + √ 2GUT N � n=1 qn � log � 1 + qnUT √ 2πnET � + G T � t=1 ∥ut,0∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now consider the third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Without loss of generality, assume qn > 0 for all n, and UT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Applying log(1 + x) ≤ log x + x−1, N � n=1 qn � log � 1 + qnUT √ 2πnET � ≤ N � n=1 qn �√ 2πnET qnUT + log UT √ 2ET + log qn πn = N � n=1 √qn �√ 2πnET UT + qn log UT √ 2ET + qn log qn πn ≤ �√ 2ET UT + log UT √ 2ET + KL(q||π) (Cauchy-Schwarz) ≤ �√ 2ET UT + � log+ UT √ 2ET + � KL(q||π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Finally, note that √UT ET ≤ (UT + ET )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Combining everything completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 Example The idea of the sparse coding framework is closely related to signal processing and representation learning, where a fundamental objective is to find a dictionary that sparsely represents the signal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Through a few examples, we show that it ties several distinct applications together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Fourier dictionary Many prediction tasks exhibit natural periodicity, such as the daily temperature, the seasonal sale of a product, and the load on a power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Here, trigonometric feature vectors are a reasonable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Taking d = 1 for example, with a known base frequency ω and an order K ∈ N, one can define a size 2K dictionary from (for all k ∈ [1 : K]) ht,2k−1 = cos(kωt), ht,2k = sin(kωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It is also optional to add an all-one feature to track the constant offset of u1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Alternatively, if T is fixed, we may set N = T and define H as the Discrete Fourier Transform (DFT) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Since we only consider real inputs, the complex DFT dictionary can be simplified into the real form above with ω = 2π/T, which is intuitively suitable for tasks with unknown periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 21 Wavelet dictionary Wavelets are powerful tools to handle multi-scale signal structures, and specifically in our framework, “shifting” environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With d = 1 and N = T, we consider the simplest Haar wavelet, where the dictionary H is set as the transpose of the (un-normalized) Haar matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The precise definition is standard, but out of our scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, the idea can be clearly illustrated in the special case with T = 8: H = � ����������� 1 1 1 0 1 0 0 0 1 1 1 0 −1 0 0 0 1 1 −1 0 0 1 0 0 1 1 −1 0 0 −1 0 0 1 −1 0 1 0 0 1 0 1 −1 0 1 0 0 −1 0 1 −1 0 −1 0 0 0 1 1 −1 0 −1 0 0 0 −1 � ����������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It is an orthogonal basis of RdT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Projecting a signal onto features on the left is equivalent to downsampling, while the removed local details are captured by features on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to using the dense DFT matrix, such local property simplifies the computation in our framework, as the base algorithm An in Algorithm 2 trivially outputs wt,0 = 0 when the input feature ht,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, in each round, the Haar-wavelet-based algorithm only maintains O(log T) black-box 1D algorithms, as opposed to O(T) in the Fourier-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Dictionary from time series Specifically for time series forecasting, we can learn classical parametric strategies, such as the autoregressive (AR) model, by choosing H properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As shown in [AHMS13], learning it is a fundamental task for learning the more general ARMA models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If a time series z1:T is generated by a (noiseless) AR(k) model, then with parameters α1:k, it satisfies zt = �k i=1 αizt−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We consider the time series setup from the beginning of the paper, with d = 1 and w1:T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Setting N = p and ht,n = zt−n, Theorem 1 translates to a forecasting regret bound against any prediction sequence y1:T generated by an AR(k) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In particular, the bound adapts to the magnitude and sparsity of the comparator model parameter, which induces an oracle inequality similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This improves the non-adaptive approach from [AHMS13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Learned dictionary Since H is only queried online, we may generate H itself using an online learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If the base learner guarantees a regret bound against certain normalized comparators, then our approach can enhance it by adapting to the actual scale of the comparator, which is unbounded a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, consider N = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For any (unknown) ˆu ∈ R, RegretT (u1:T ) = ˆu T � t=1 � gt, ht − ut ˆu � + T � t=1 ⟨gt, ht⟩ (ˆxt − ˆu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The first sum can be bounded by the guarantee of h1:T – this is the ideal adaptive bound we aim for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In this regard, the second sum is the overhead of such adaptivity, similar to the objective in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Specific applications of this technique can be found in [CO18] and [JC22, Appendix I], while here we show that in general, it can be viewed as an instance of the sparse coding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 Model selection by summation For unconstrained dynamic regret, an appealing property is that different regret bounds can be simply aggregated by summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is essentially the idea of Theorem 1 itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' From this angle, our sparse coding framework and the path length bound from [JC22] are mutually complementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let A1 and A2 be two algorithms, each guaranteeing an unconstrained dynamic regret bound fi(u1:T ) for all u1:T ∈ RdT , i = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Consider a master algorithm A that simply predicts the sum of their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, for any decomposition of the comparator u1:T = u(1) 1:T + u(2) 1:T , RegretT (u1:T ) ≤ f1(u(1) 1:T ) + f2(u(2) 1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 22 For example, take A1 as the sparse coding algorithm, and A2 as the algorithm from [JC22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, within Theorem 1, we can replace the trivial characterization of ut,0 by a path length bound on ut,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The sacrifice is only a slightly larger cumulative loss term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', 2GET in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The result can also be plugged into the oracle inequality Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6 Detail on the primal-dual interpretation Supplementing the primal-dual discussion in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3, we present several versions of the loss-regret duality on the sequence space RdT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The first one generalizes a classical argument in static unconstrained OLO [MO14, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The other two are to our knowledge new, and are more closely tied to the path length characterization of the comparator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Define st = �t i=1 gi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', the sum of past gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If there exists a function fT : RdT → R such that the prediction sequence x1:T guarantees a loss upper bound ⟨g1:T , x1:T ⟩ ≤ −fT (g1:T ), then for all u1:T ∈ RdT , RegretT (u1:T ) ≤ f ∗ T (−u1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This follows from a standard Fenchel duality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' RegretT (u1:T ) = ⟨g1:T , x1:T − u1:T ⟩ ≤ ⟨g1:T , −u1:T ⟩ − fT (g1:T ) ≤ sup x∈RdT ⟨x, −u1:T ⟩ − fT (x) = f ∗ T (−u1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Recall that we defined st = �t i=1 gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If there exists fT : RdT → R such that x1:T guarantees ⟨g1:T , x1:T ⟩ ≤ −fT (s1:T ), then for all u1:T ∈ RdT , RegretT (u1:T ) ≤ f ∗ T (u2 − u1, u3 − u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , uT − uT −1, −uT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We start by rewriting the comparator loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ⟨g1:T , u1:T ⟩ = ⟨sT , uT ⟩ + T � t=1 ⟨gt, ut − uT ⟩ = ⟨sT , uT ⟩ + T � t=1 T −1 � i=t ⟨gt, ui − ui+1⟩ = ⟨sT , uT ⟩ + T −1 � i=1 i � t=1 ⟨gt, ui − ui+1⟩ = ⟨sT , uT ⟩ + T −1 � t=1 ⟨st, ut − ut+1⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Given the loss upper bound, RegretT (u1:T ) ≤ T −1 � t=1 ⟨st, ut+1 − ut⟩ + ⟨sT , −uT ⟩ − fT (s1:T ) ≤ sup x1:T ∈RdT �T −1 � t=1 ⟨xt, ut+1 − ut⟩ + ⟨xT , −uT ⟩ − fT (x1:T ) � = f ∗ T (u2 − u1, u3 − u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , uT − uT −1, −uT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By reversing the index, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If there exists fT : RdT → R such that x1:T guarantees ⟨g1:T , x1:T ⟩ ≤ −fT (sT , sT − s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , sT − sT −1), then for all u1:T ∈ RdT , RegretT (u1:T ) ≤ f ∗ T (−u1, u1 − u2, u2 − u3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , uT −1 − uT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Similar to the proof above, ⟨g1:T , u1:T ⟩ = ⟨sT , u1⟩ + T � t=1 ⟨gt, ut − u1⟩ = ⟨sT , u1⟩ + T � t=1 t−1 � i=1 ⟨gt, ui+1 − ui⟩ = ⟨sT , u1⟩ + T −1 � i=1 T � t=i+1 ⟨gt, ui+1 − ui⟩ = ⟨sT , u1⟩ + T −1 � t=1 ⟨sT − st, ut+1 − ut⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 23 RegretT (u1:T ) ≤ ⟨sT , −u1⟩ + T −1 � t=1 ⟨sT − st, ut − ut+1⟩ − fT (sT , sT − s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , sT − sT −1) ≤ sup x1:T ∈RdT � ⟨x1, −u1⟩ + T � t=2 ⟨xt, ut−1 − ut⟩ − fT (x1:T ) � = f ∗ T (−u1, u1 − u2, u2 − u3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' , uT −1 − uT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' B Detail on the improved static algorithm This section presents the second contribution of the paper, an improved static unconstrained OLO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 contains the derivatives of our potential functions, which will be useful in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 analyzes the base algorithm (Algorithm 3), with its regret bound presented as Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 presents the analysis of the meta algorithm, resulting in Theorem 2, the main theorem of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 discusses the application of this static algorithm to the dynamic regret problem, through the sparse coding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1 Facts of the potential function For the two potential functions defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2, we compute their derivatives as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This will be useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ∂1φ(x, y) = −ε√α 2√x exp � y2 4αx � , ∂2φ(x, y) = εerfi � y √ 4αx � , ∂11φ(x, y) = ε√α 4x3/2 � y2 2αx + 1 � exp � y2 4αx � , ∂12φ(x, y) = − εy 4√αx3/2 exp � y2 4αx � , ∂22φ(x, y) = ε 2√αx exp � y2 4αx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to the change of variable, the derivatives of Φ can be concisely represented as ∂1Φ(V, S) = ∂1φ(V + z + kS, S), ∂2Φ(V, S) = k∂1φ(V + z + kS, S) + ∂2φ(V + z + kS, S), ∂11Φ(V, S) = ∂11φ(V + z + kS, S), ∂12Φ(V, S) = k∂11φ(V + z + kS, S) + ∂12φ(V + z + kS, S), ∂22Φ(V, S) = k2∂11φ(V + z + kS, S) + 2k∂12φ(V + z + kS, S) + ∂22φ(V + z + kS, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2 Analysis of the base algorithm The key component of our approach is the base algorithm (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Within its analysis, the most crucial part is the characterization of the one step change of the potential (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This subsection is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We first present two simple lemmas on the property of our potential function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, we prove the key lemma (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3), which leads to a cumulative loss upper bound (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As in the standard analysis of potential methods, converting the loss upper bound to the regret bound relies on computing the Fenchel conjugate of Φ – this is the focus of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Finally, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6 combines everything into the regret bound of the base algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To begin with, we first show that Φ(V, S) is convex in S, just like more standard potential functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 24 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If ε > 0, α > 0 and z > k > 0, the potential function Φ(V, S) satisfies ∂22Φ(V, S) ≥ 0 for all V ≥ 0 and S ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Define the shorthands x = V + z + kS and y = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all V ≥ 0 and S ≥ −1, we have x > 0, therefore ∂22Φ(V, S) = k2∂11φ(x, y) + 2k∂12φ(x, y) + ∂22φ(x, y) = ε√α 4x3/2 exp � y2 4αx � �k2y2 2αx + k2 − 2ky α + 2x α � = ε√α 4x3/2 exp � y2 4αx � �k2y2 2αx + k2 + 2(V + z) α � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next, we show that the base algorithm makes strictly negative prediction ˜zt when St−1 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This will be exploited by the meta-algorithm to ensure that the surrogate losses received by the base algorithm satisfy the St ≥ −1 constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If ε > 0, α > 0 and z > k > 0, the potential function Φ(V, S) satisfies ∂2Φ(V, S) < 0 for all V ≥ 0 and −1 ≤ S ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us check ∂2Φ(V, 0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Indeed, ∂2Φ(V, 0) = k∂1φ(V + z, 0) + ∂2φ(V + z, 0) = − εk√α 2 √ V + z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, ∂22Φ(V, S) ≥ 0 due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, we have ∂2Φ(V, S) < 0 for all V ≥ 0 and −1 ≤ S ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The key lemma in our analysis is the following, which says a suitable combination of parameters yields a one-step loss bound on the potential function Φ, as long as the second argument of Φ is always larger than −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Such a lemma is typically the central component in the classical potential analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 (Key lemma: one step potential bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With α = 1, k = 2, z = 16 and an arbitrary ε > 0, the potential function Φ(V, S) satisfies Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S) ≤ 0, for all V ≥ 0, S ≥ −1 and c ∈ [−1, 1] ∩ [−1 − S, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us view our objective as a function of c, fV,S(c) := Φ(V + c2, S + c) − Φ(V, S) − c∂2Φ(V, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Taking the derivatives, f ′ V,S(c) = 2c∂1Φ(V + c2, S + c) + ∂2Φ(V + c2, S + c) − ∂2Φ(V, S), f ′′ V,S(c) = 2∂1Φ(V + c2, S + c) + 4c2∂11Φ(V + c2, S + c) + 4c∂12Φ(V + c2, S + c) + ∂22Φ(V + c2, S + c) ≤ 2∂1Φ(V + c2, S + c) + 4∂11Φ(V + c2, S + c) + 4 ��∂12Φ(V + c2, S + c) �� + ∂22Φ(V + c2, S + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (9) Note that fV,S(0) = f ′ V,S(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, to prove fV,S(c) ≤ 0, it suffices to show f ′′ V,S(c) ≤ 0 for all considered values of V , S and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (9) has a striking similarity to the Backward Heat Equation – in fact, after a change of variable, the resulting expressions, namely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (11) below, will resemble a BHE on φ (∂1φ + α∂22φ = 0) plus a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The main goal of this proof is to control such perturbations by properly choosing α, k and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, due to the absolute value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (9), we will analyze two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Technically, the first case is harder, therefore we pick k to simplify its analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The second case requires S + c to be around zero – this is an “edge” case and relatively easier to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 25 Case 1: ∂12Φ(V + c2, S + c) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Substituting the derivatives of Φ by the derivatives of φ, we have f ′′ V,S(c) ≤ 2∂1φ + (k − 2)2∂11φ + 2(k − 2)∂12φ + ∂22φ ��� (V +c2+z+k(S+c),S+c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (10) The RHS means we evaluate all the derivative functions at (V + c2 + z + k(S + c), S + c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging in our specific choice of k and α, f ′′ V,S(c) ≤ 2∂1φ + ∂22φ ��� (V +c2+z+2(S+c),S+c) (k = 2) ≤ 2∂1φ + 2∂22φ ��� (V +c2+z+2(S+c),S+c) (∂22φ ≥ 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (φ satisfies the BHE with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=') Case 2: ∂12Φ(V + c2, S + c) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Similar to the first case, f ′′ V,S(c) ≤ 2∂1φ + (k + 2)2∂11φ + 2(k + 2)∂12φ + ∂22φ ��� (V +c2+z+k(S+c),S+c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (11) Consider the k-dependent “perturbation” terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (11), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', (k + 2)2∂11φ + 2(k + 2)∂12φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our goal is to upper bound it by ∂22φ, such that an upper bound of f ′′ V,S(c) follows from the BHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging in the derivatives of φ from Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, for all inputs (x, y), (k + 2)2∂11φ + 2(k + 2)∂12φ − ∂22φ ��� (x,y) = ε 4√αx3/2 exp � y2 4αx � � (k + 2)2 � y2 2x + α � − 2(k + 2)y − 2x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We aim to show the bracket on the RHS is negative at x = V + c2 + z + k(S + c) and y = S + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Also plugging in our choice of α = 1 and k = 2, this amounts to showing ♦ := 2(S + c)2 V + c2 + z + 2S + 2c + 4 − 3(S + c) − 1 2(V + c2 + z) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The idea is that we can pick a large enough z to make it hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Concretely, If S + c > 0, then ♦ ≤ 2(S + c)2 2S + 2c + 4 − 3(S + c) − 1 2z ≤ 4 − 1 2z, and it suffices to pick z ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If S + c ≤ 0, then since c ∈ [−1 − S, ∞), we have S + c ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As long as z > 2, ♦ ≤ 2 z − 2 + 7 − 1 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It suffices to pick z ≥ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In summary, z = 16 ensures ♦ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to the BHE on φ, f ′′ V,S(c) ≤ 2∂1φ + 2∂22φ ��� (V +c2+z+2(S+c),S+c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Combining the two cases completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Based on Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3, we immediately obtain a cumulative loss bound of the base algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The proof is a straightforward telescopic sum, therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 (Cumulative loss bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With α = 1, k = 2 and z = 16, Algorithm 3 guarantees for all T ∈ N+, T � t=1 ˜lt˜zt ≤ Φ(0, 0) − Φ( ˜VT , ˜ST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 26 As for the regret bound, similar to the standard duality argument [Ora19, Chapter 9], we need the Fenchel conjugate of the potential function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With any V ≥ 0, define Φ∗ V (u) := sup S∈[−1,∞) uS − Φ(V, S) as the conjugate of Φ(V, S) with respect to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Slightly different from the standard definition where the supremum is over R, here the supremum is over [−1, ∞), since the surrogate losses in the base algorithm satisfy ˜St ≥ −1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To proceed, we will only consider the dual variable satisfying u ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 (Conjugate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With ε > 0, α > 0 and z > k > 0, for all u ≥ 0, Φ∗ V (u) := sup S∈[−1,∞) uS − Φ(V, S) ≤ u ¯S + ε � α(V + z + k ¯S), where ¯S = 4αk � 1 + � log(2uε−1 + 1) �2 + � 4α(V + z) � 1 + � log(2uε−1 + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We first show that the supremum over S in the Fenchel conjugate is attainable by some S∗ ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' To this end, define a function f(S) = uS − Φ(V, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' f is continuous, with f ′(S) = u − ∂2Φ(V, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, f is concave on [−1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The existence of S∗ then follows from analyzing the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all S ∈ [−1, 0], we have f ′(S) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The reason is u ≥ 0, and ∂2Φ(V, S) ≤ 0 due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For sufficiently large S, we aim to show f ′(S) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us begin by writing down ∂2Φ(V, S), from Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ∂2Φ(V, S) = εerfi � S � 4α(V + z + kS) � − εk√α 2 √ V + z + kS exp � S2 4α(V + z + kS) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now consider large S that satisfies S ≥ � 4α(V + z + kS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to an estimate of the erfi function (Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2), erfi � S � 4α(V + z + kS) � ≥ � α(V + z + kS) S exp � S2 4α(V + z + kS) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, � α(V + z + kS) S − k√α √ V + z + kS = √α(V + z) S √ V + z + kS ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, ∂2Φ(V, S) = � ε 2erfi � S � 4α(V + z + kS) � − εk√α 2 √ V + z + kS exp � S2 4α(V + z + kS) �� + ε 2erfi � S � 4α(V + z + kS) � ≥ ε 2erfi � S � 4α(V + z + kS) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (12) For sufficiently large S, we have RHS > u, hence f ′(S) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Summarizing the above, we have shown that there exists S∗ ∈ [0, ∞) such that Φ∗ V (u) := sup S∈[−1,∞) uS − Φ(V, S) = uS∗ − Φ(V, S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Moreover, S∗ should satisfy the first order condition f ′(S∗) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', u = ∂2Φ(V, S∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Our goal next is to upper bound S∗ by a function of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Again, we analyze two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 27 Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If S∗ satisfies S∗ < � 4α(V + z + kS∗), then by regrouping the terms, we have (S∗)2 − 4αkS∗ − 4α(V + z) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Solving this quadratic inequality, S∗ ≤ 1 2 � 4αk + � (4αk)2 + 16α(V + z) � = 2αk + � 4α2k2 + 4α(V + z) ≤ 4αk + � 4α(V + z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If S∗ satisfies S∗ ≥ � 4α(V + z + kS∗), then same as the earlier analysis in the present proof, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (12), we have u ≥ ε 2erfi � S∗ � 4α(V + z + kS∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For conciseness, define the notation p = erfi−1(2uε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, (S∗)2 − 4αkp2S∗ − 4αp2(V + z) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Solving the quadratic inequality, S∗ ≤ 1 2 � 4αkp2 + � (4αkp2)2 + 16αp2(V + z) � ≤ 2αkp2 + � 4α2k2p4 + 4αp2(V + z) ≤ 4αkp2 + � 4α(V + z)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now we can combine the above two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Specifically, p ≤ 1+ � log(2uε−1 + 1) due to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, S∗ ≤ 4αk � 1 + � log(2uε−1 + 1) �2 + � 4α(V + z) � 1 + � log(2uε−1 + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Define the RHS as ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, from the definition of the Fenchel conjugate, Φ∗ V (u) = uS∗ − Φ(V, S∗) = uS∗ − ε � α(V + z + kS∗) � 2 � S∗ √ 4α(V +z+kS∗) 0 erfi(u)du − 1 � ≤ u ¯S + ε � α(V + z + k ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging in ¯S completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Finally, we assemble the cumulative loss bound (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4) and the conjugate of the potential (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5) into the regret bound of the base algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6 (Regret of the base algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With ε > 0, α = 1, k = 2 and z = 16, Algorithm 3 guarantees for all T ∈ N+ and ˜u ≥ 0, T � t=1 ˜lt(˜zt − ˜u) ≤ ε � ˜VT + 2 ¯S + ˜u ¯S, where ¯S = 8 � 1 + � log(2˜uε−1 + 1) �2 + 2 � ˜VT + 16 � 1 + � log(2˜uε−1 + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Due to the standard loss-regret duality [Ora19, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6], starting from the cumulative 28 loss bound (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4), the regret can be bounded by T � t=1 ˜lt(˜zt − ˜u) ≤ ST ˜u + Φ(0, 0) − Φ( ˜VT , ˜ST ) ≤ Φ(0, 0) + sup S∈[−1,∞) � S˜u − Φ( ˜VT , S) � = Φ(0, 0) + Φ∗ ˜VT (˜u) ≤ −4ε + ε � ˜VT + 16 + 2 ¯S + ˜u ¯S ≤ ε � ˜VT + 2 ¯S + ˜u ¯S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging in ¯S from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='5 completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='3 Proof of the main result This subsection presents the theoretical guarantees of the meta algorithm (Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We first show that when combined with the base algorithm (Algorithm 3), the whole procedure is well-posed, in the sense that the surrogate loss ˜lt satisfies − �t i=1 ˜li ≥ −1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proposition 3 (Well-posedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The surrogate loss ˜lt defined in Algorithm 4 satisfies − �t i=1 ˜li ≥ −1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' First, notice that |˜lt|≤ |lt| = ��� gtG−1, wt ��� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next, we prove by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Consider − �t−1 i=1 ˜li, which is defined as ˜St−1 in the base algorithm (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Suppose ˜St−1 ≥ −1, which trivially holds at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let us analyze two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If ˜St−1 ≥ 0, then − �t i=1 ˜li = ˜St−1 − ˜lt ≥ ˜St−1 − |˜lt|≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If −1 ≤ ˜St−1 < 0, then due to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2, the prediction ˜zt of the base algorithm satisfies ˜zt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The meta algorithm projects it to zt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, due to our definition of ˜lt in the meta algorithm, ˜lt = � lt, lt ≤ 0, 0, else, which is non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, − �t i=1 ˜li = ˜St−1 − ˜lt ≥ ˜St−1 ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' An induction completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next we present the main result, the static regret bound of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Here we define the gradient variance VT = �T t=1 ∥gt∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With ε > 0, α = 1, k = 2 and z = 16, Algorithm 4 guarantees for all T ∈ N+ and u ∈ Rd, RegretT (u) ≤ ε � VT + 2G ¯S + ∥u∥2 � ¯S + 2 � 2VT � , where ¯S = 8G � 1 + � log(2 ∥u∥2 ε−1 + 1) �2 + 2 � VT + 16G2 � 1 + � log(2 ∥u∥2 ε−1 + 1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Since the meta algorithm simply applies two existing black-box reductions [CO18, Cut20], the proof is straightforward given Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' First, due to a polar decomposition theorem [CO18, Theorem 2], the regret can be decomposed into the regret of AB with respect to u/∥u∥2, plus the regret of zt with respect to ∥u∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, the latter is upper-bounded by the regret of ˜zt – this is because our definition of zt and ˜lt follows 29 the procedure of [Cut20, Theorem 2], where a convex constraint can be added to an unconstrained algorithm without changing its regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In summary, we have RegretT (u) ≤ G T � t=1 lt(zt − ∥u∥2) + ∥u∥2 T � t=1 ⟨gt, wt − u/∥u∥2⟩ ≤ G T � t=1 ˜lt(˜zt − ∥u∥2) + ∥u∥2 T � t=1 ⟨gt, wt − u/∥u∥2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The two regret terms on the RHS represent the regret bound of A1d and AB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Now, the first term follows from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='6, where ˜VT = �T t=1 ˜l2 t ≤ �T t=1 l2 t = �T t=1 � gtG−1, wt �2 ≤ VT /G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' As for the regret of AB, due to [Ora19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='14], T � t=1 ⟨gt, wt − u/∥u∥2⟩ ≤ 2 � 2VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Combining these two components completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Finally, let us use asymptotic orders to make this bound a bit more interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Consider the regime of large ∥u∥2 and VT , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', ∥u∥2 ≫ ε and VT ≫ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' We preserve the dependence of ε, as it is an arbitrary hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In contrast, α, z and k are absolute constants, therefore subsumed by the big-Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Using log(1 + x) ≤ x, we can crudely bound ¯S by ¯S ≤ 8G � 1 + � 2 ∥u∥2 ε−1 �2 + 2 � VT + 16G2 � 1 + � 2 ∥u∥2 ε−1 � = 8G + 2 � VT + 16G2 + o � ∥u∥2 ε−1� VT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Plugging this crude bound of ¯S into the first term of the regret bound, we have RegretT (u) ≤ ε � VT + 16G2 + 4G � VT + 16G2 + ε � o � G ∥u∥2 ε−1� VT � + ∥u∥2 � ¯S + 2 � 2VT � ≤ ε( � VT + 16G2 + 2G) + o �√ G ∥u∥2 V 1/4 T � + ∥u∥2 � ¯S + 2 � 2VT � ≤ ε( � VT + 6G) + o �√ G ∥u∥2 V 1/4 T � + ∥u∥2 � ¯S + 2 � 2VT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next, notice that ¯S = O �� VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Using it to replace the remaining ¯S above, RegretT (u) ≤ ε �� VT + 6G � + o �√ G ∥u∥2 V 1/4 T � + ∥u∥2 O �� VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The second term can be assimilated into the third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The result becomes RegretT (u) ≤ ε �� VT + 6G � + ∥u∥2 O �� VT log(∥u∥2 ε−1) ∨ G log(∥u∥2 ε−1) � , (13) where O(·) subsumes absolute constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that this bound is not only valid for large ∥u∥2, but also valid when u = 0 (this can be directly verified from Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Therefore, we can use it to characterize the loss-regret tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' It is the same loss-regret tradeoff as [ZCP22a], but with time T replaced by the gradient variance VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Towards the optimal leading constant Without considering gradient adaptivity, [ZCP22a] showed that the optimal leading term in the regret bound (including the multiplicative constant) is ∥u∥2 G � 2T log(∥u∥2 ε−1), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In Theorem 2, if we ignore the logarithmic residue log(∥u∥2 ε−1) outside the square root,11 then the 11Which does not depend on VT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 30 leading term is 2 ∥u∥2 � VT log(∥u∥ ε−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In the worst case with VT = G2T, the constant of the latter has a √ 2 gap with respect to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' This is essentially due to our analysis, where α is picked as 1 instead of 1/2 to handle the second case in the proof sketch (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' One can use a smaller α (corresponding to smaller leading constant in the regret) in exchange for a larger z (the additive constant on VT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' However, achieving the lower bound √ 2 without blowing up the additive term remains to be studied in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 Application to dynamic regret Given the improved static algorithm, we now apply its 1D version to the sparse coding framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For clarity, we will adopt the asymptotic regret bound, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (13), and loosely assimilate the residual term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Since it is applied on the transform domain, we will denote transform domain quantities with hat, according to our convention in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, analogous to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4 applied in the main paper, our improved static algorithm, given an arbitrary hyperparameter ˆε > 0, guarantees for all T ∈ N+ and ˆu ∈ R, T � t=1 ⟨ˆgt, ˆxt − ˆu⟩ ≤ ˆε � � � � � � T � t=1 ˆg2 t + 6 ˆG � � + |ˆu| O � � � � � � T � t=1 ˆg2 t log(|ˆu| ε−1) � � , (14) where ˆG is the Lipschitz constant for the surrogate loss ˆgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In the sparse coding framework, we will assume the dictionary satisfies ∥ht∥2 ≤ 1, which holds for Fourier and (un-normalized) wavelet dictionaries (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Then, let us apply the improved static algorithm to a single feature vector (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that instead of setting the surrogate Lipschitz constant as G ∥ht∥2 (Line 2 of Algorithm 1), we now set it as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The resulting dynamic regret bound is the following, which is analogous to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Let ˆε > 0 be an arbitrary hyperparameter for our improved static algorithm (the 1D version of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Applying it as a subroutine, for all T ∈ N+ and u1:T ∈ span(h1:T ), Algorithm 1 guarantees RegretT (u1:T ) ≤ εT + � � � � T � t=1 ⟨gt, ut⟩2O � �log � �ε−1 T � � � � T � t=1 ⟨gt, ut⟩2 � � � � , where εT = ˆε � � � � � � T � t=1 ⟨gt, ht⟩2 + 6G � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In particular, the big-Oh bound holds in two regimes: (i) �T t=1 ⟨gt, ut⟩2 ≫ ˆε2 �T t=1 ⟨gt, ht⟩2 and �T t=1 ⟨gt, ht⟩2 ≫ G2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' and (ii) u1:T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, the better underlying static algorithm essentially improves G2 ∥h1:T ∥2 2 in the dynamic regret bound with �T t=1 ⟨gt, ht⟩2, and G2 ∥u1:T ∥2 2 with �T t=1 ⟨gt, ut⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Similar to the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='1, the dynamic regret of Algorithm 1 equals Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (14) on the transform domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' In particular, ˆu satisfies u1:T = ˆuh1:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' The surrogate Lipschitz constant ˆG in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' (14) 31 equals G, the actual Lipschitz constant for the dynamic regret problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' With �T t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ht⟩2 ≫ G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' RegretT (u1:T ) ≤ ˆε � � � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ht⟩2 + 6G � � + |ˆu| O � � � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ht⟩2 log(|ˆu| ˆε−1) � � = εT + |ˆu| � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ht⟩2O � �log � �ε−1 T |ˆu| � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ht⟩2 � � � � = εT + � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' |ˆu| |ht|⟩2O � �log � �ε−1 T � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' |ˆu| |ht|⟩2 � � � � = εT + � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ut⟩2O � �log � �ε−1 T � � � � T � t=1 ⟨gt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ut⟩2 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Next, let us consider general size N dictionaries, analogous to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Still, we assume that for all t and n, ∥ht,n∥2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Consider Algorithm 2, with its static subroutine replaced by the 1D version of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all T ∈ N+ and u1:T ∈ RdT , it guarantees RegretT (u1:T ) ≤ ET + UT · O � log UT ET + KL(q||π) � + G T � t=1 ∥ut,0∥2 , where 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' For all n ∈ [1 : N], u1:T,n is any vector in span(h1:T,n), and u1:T,0 = u1:T − �N n=1 u1:T,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' ET and UT are non-negative numbers defined by ET = N � n=1 ˆεn � � � � � � T � t=1 ⟨gt, ht,n⟩2 + 6G � � , UT = N � n=1 � � � � T � t=1 ⟨gt, ut,n⟩2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' π and q are N dimensional probability vectors defined by πn = ˆεn ET � � � � � � T � t=1 ⟨gt, ht,n⟩2 + 6G � � , qn = 1 UT � � � � T � t=1 ⟨gt, ut,n⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' If π is the uniform distribution, and u1:T ∈ span(H), then the bound can be simplified into RegretT (u1:T ) ≤ ET + ˜O(UT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Note that the big-Oh bound is meant for the regime where for all n ∈ [1 : N], either (i) u1:T,n = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' or (ii) �T t=1 ⟨gt, ut,n⟩2 ≫ ˆε2 �T t=1 ⟨gt, ht,n⟩2 and �T t=1 ⟨gt, ht,n⟩2 ≫ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' Compared to Theorem 1, we improve G �N n=1 ��T t=1 ∥ut,n∥2 to �N n=1 ��T t=1 ⟨gt, ut,n⟩2, which adapts to the complexity of both the gradient sequence and the comparator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 32 Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' By summing Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content='7, RegretT (u1:T ) ≤ N � n=1 � � �ˆεn � � � � � � T � t=1 ⟨gt, ht,n⟩2 + 6G � � + � � � � T � t=1 ⟨gt, ut,n⟩2O � �log � �ˆε−1 n ��T t=1 ⟨gt, ut,n⟩2 ��T t=1 ⟨gt, ht,n⟩2 + 6G � � � � � � � = ET + UT · O � N � n=1 qn log qnUT πnET � ≤ ET + UT · O � log UT ET + N � n=1 qn log qn πn � = ET + UT · O � log UT ET + KL(q||π) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} +page_content=' 33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFQT4oBgHgl3EQfiTZ3/content/2301.13349v1.pdf'} diff --git a/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf b/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4662448b04c0c347753377cd517eac2c895e79b5 Binary files /dev/null and b/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf differ diff --git a/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/2301.00655v1.pdf.txt b/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/2301.00655v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ece1f97d0ca90fce7c0aeea5a3917ec60e6ae9de --- /dev/null +++ b/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/2301.00655v1.pdf.txt @@ -0,0 +1,220 @@ +arXiv:2301.00655v1 [math.OC] 2 Jan 2023 +On Some Characterization of GS-exponential kind +of Convex Functions +Ehtesham Akhtera and Musavvir Alib,∗ +a, b Department of Mathematics, +Aligarh Muslim University, Aligarh-202002, India +E-mail addresses: b∗musavvir.alig@gmail.com (Corresponding author), +ehteshamakhter111@gmail.com. +Abstract +This manuscript introduces the idea of GS-exponential kind of convex functions and some +of their algebraic features, and we introduce a new class GS-exponential kind of convex +sets. In addition, we describe certain fundamental GS-exponential kind of convex func- +tion with characteristics in both the general and the differentiable cases. We establish +the sufficient conditions of optimality and offer the proof for unconstrained as well as +inequality-constrained programming while considering the assumption of GS-exponential +kind of convexity. +MSC: 26A51, 26B25, 90C26. +Keywords: GS-exponential kind of convex functions and sets, Inequalities, Opti- +mality conditions, Optimization. +1 +Introduction +Due to the importance of convexity and its generalisations in the study of optimality +to resolve mathematical issues, researchers have concentrated a lot of their efforts +on generalised convex functions for this purpose. As an illustration, Hudzik and +Maligranda (1994) [7], investigated at two distinct forms of s-convexity and found +that s-convexity in the next meaning is basically more significant than in the first +sense whenever (0 < s < 1). Youness (1999) [20] expanded the definitions of convex +sets and functions to create a new class of sets and functions known as E-convex +sets and E-convex functions. Yang (2001) [19] enhanced Youness’s paper [20] by +incorporating certain illustrations. +In recent years, academic experts have given these generalized convex functions +in additional consideration. The semi-preinvex functions were studied by X.J. Long +and J.W. Peng in 2006 [12] as a generalization of the semi-preinvex functions and +1 + +the b-vex functions. Y. Syau et al. (2009)[17] developed the E-b-vex function fam- +ily, a novel class of functions which are the generalizations of b-vex functions and +E-vex functions. In 2011, T. Emam investigated a novel class of functions known as +approximately b-invex functions. He also discussed some of its properties and dis- +covered the necessary optimality conditions for nonlinear programming using these +functions. In their investigation of a novel class of generalized sub-b-convex func- +tions and sub-b-convex sets, M.T. Chao et al. (2012) [13] showed the conditions for +the existence of optimal solutions for both unconstrained and inequality-constrained +sub-b-convex programming. +The study in our paper aims to introduce a new class of generalized exponential +kind of convex functions termed as GS-exponential kind of convex functions and +explores certain characteristics of the same class. This paper draws inspiration from +a number of research papers [2, 5, 6, 8, 10, 14, 15, 16, 18, 21]. Additionally, we offer +the adequate GS-exponential kind of convexity-derived criteria of optimality for pro- +gramming with variables which are both unconstrained and inequality-constrained. +2 +Preliminaries +We will go through the definitions of sub-b-s-convexity, exponential kind of convexity, +and s-convexity of functions in this section of the manuscript. For the remainder of +this work, let V stand for any non-empty convex subset in Rn. +Definition 2.1. [11] The function Q : V → R is known as sub-b-s-convex in the +second sense associated with the map G : V × V × (0, 1] → R, if +Q(am1 + (1 − a)m2) ≤ asQ(m1) + (1 − a)sQ(m2) + G(m1, m2, s) +holds for all m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1]. +Definition 2.2. [7] The function Q : V → R is known as s-convex in the second +sense, if for all m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1], we have +Q(am1 + (1 − a)m2) ≤ asQ(m1) + (1 − a)sQ(m2) +Definition 2.3. [9] A positive function Q : V → R is known as exponential kind of +convex function, if +Q(am1 + (1 − a)m2) ≤ (ea − 1)Q(m1) + (e1−a − 1)Q(m2) +holds for all m1, m2 ∈ V, a ∈ [0, 1]. +The concepts defined as in 2.1, 2.2 and 2.3, motivates us to explore a new idea +known as GS-exponential kind of convex function. +2 + +3 +Main Results +Definition 3.1. The function Q : V → R is known as GS-exponential kind of +convex function on V associated with the map G : V × V × (0, 1] → R, if +Q(am1 + (1 − a)m2) ≤ (ea − 1)sQ(m1) + (e1−a − 1)sQ(m2) + aG(m1, m2, s) (3.1) +holds for each m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1]. +Remark 3.1. If we take s = 1, Q(m1) is non-negative and G(m1, m2, s) = 0, the +GS-exponential kind of convex function reduces to be exponential kind of convex +function. +Theorem 3.2. If Q1, Q2 : V → R are GS-exponential kind of convex function +associated with the map G1, G2 respectively, then Q1 +Q2 and βQ1, (β ≥ 0) are also +a GS-exponential kind of convex function. +Corollary 3.2.1. If Qi : V → R, (i = 1, 2, ....., n) are GS-exponential kind of +convex function associated with the map Gi : V × V × (0, 1] → R, (i = 1, 2, ...., n), +respectively, then Q = �n +i=1 βiQi, β ≥ 0, (i = 1, 2, ..., n) is GS-exponential kind of +convex function associated with the map G = �n +i=1 βiGi. +Lemma 3.3. For all a ∈ [0, 1] and s ∈ (0, 1], the inequalities (ea − 1)s ≥ a and +(e1−a − 1)s ≥ 1 − a hold. +Proposition 3.4. Every convex function is GS-exponential kind of convex function +if it has a map G associated with it that is non-negative. +Theorem 3.5. If Q : V → R is the GS-exponential kind of convex function associ- +ated with the map G and S : R → R is a non-negative function in addition to being +linear, then S ◦ Q is a GS-exponential kind of convex function associated with the +map S ◦ G. +Definition 3.6. Assume that U be a non-empty subset of Rn+1. Then, U is known +as GS-exponential kind of convex set associated with the map G : Rn×Rn×(0, 1] → +R if for all (m1, α1), (m2, α2) ∈ U, m1, m2 ∈ Rn, a ∈ [0, 1] and some fixed s ∈ (0, 1], +we have +(am1 + (1 − a)m2, (ea − 1)sα1 + (e1−a − 1)sα2 + aG(m1, m2, s)) ∈ U. +Now, we provide a characterization of GS-exponential kind of convex function +Q : V → R based on their respective epigraphs, given by +E(Q) = {(m, α) : m ∈ V, α ∈ R, Q(m) ≤ α}. +3 + +Theorem 3.7. A function Q : V → R is a GS-exponential kind of convex function +associated with the map G : V × V × (0, 1] → R, if and only if E(Q) is a GS- +exponential kind of convex set associated with the map G. +Theorem 3.8. Assume that m2 > 0 and Qβ : [m1, m2] → R is a family of numer- +ical functions associated with the map Gβ and each Gβ is a GS-exponential kind +of convex functions and each Gβ is bounded function, also assume that Q(m) = +supβ Qβ(m) and G(m1, m2, s) = supβ Gβ(m1, m2, s). If the set (non-empty) K = +{r ∈ [m1, m2]|Q(r) < ∞}, then K is an interval and Q is GS-exponential kind of +convex function on K. +Theorem 3.9. Let Q : [m1, m2] → R be a GS-exponential kind of convex function +associated with the map G : [m1, m2]×[m1, m2]×(0, 1] → R and also let G(m1, m2, s) +is bounded, then Q is also bounded on [m1, m2]. +In this section, Q is considered to be a differentiable function and s, a ∈ (0, 1]. +Theorem 3.10. Let Q : V → R be a non-negative differentiable GS-exponential +kind of convex function associated with the map G. Then +(i)∇Q(m2)T(m1 − m2) < (ea − 1)s +a +Q(m1) + e(1−a)s +a +Q(m2) + G(m1, m2, s) − o(a) +a , +(ii)∇Q(m2)T(m1−m2) < (es − 1)s(Q(m1) − Q(m2)) + 3Q(m2) − o(a) +a ++G(m1, m2, s) +Theorem 3.11. Let Q : V → R be a non-positive differentiable GS-exponential +kind of convex function associated with the map G. Then +∇Q(m2)T(m1 − m2) ≤ (ea − 1)s +a +[Q(m1) − Q(m2)] + G(m1, m2, s) − o(a) +a . +Corollary 3.11.1. Assume that Q : V → R is a positive differentiable GS-exponential +kind of convex function, then +∇[Q(m2) − Q(m1)]T(m1 − m2) +< +(ea − 1)s +a +[Q(m1) + Q(m2)] + e(1−a)s +a +[Q(m1) + Q(m2)] ++G(m1, m2, s) + G(m2, m1, s) − 2o(a) +a . +In case if Q is a negative valued, then +∇[Q(m2) − Q(m1)]T(m1 − m2) ≤ G(m1, m2, s) + G(m2, m1, s) − 2o(a) +a . +4 + +The following methods are then utilized to apply the above outcomes to nonlinear +programming. So, we take the unconstrained problem (S). +(S) : min{Q(m), m ∈ V } +(3.2) +Theorem 3.12. Let Q : V → R be a positive differentiable GS-exponential kind +of convex function associated with the map G. Also, suppose that m ∈ V and the +inequality +∇Q(m)T(n − m) > G(n, m, s) + 3Q(m) − o(a) +a +(3.3) +holds for each n ∈ V, a ∈ (0, 1), and for any particular s ∈ (0, 1],, then n is the +solution optimal to the problem (3.2) related to Q on V . +The following example of unconstrained programming is taken into consideration +References +[1] Alomari, M., Darus, M., Dragomir, S.S.:New inequalities of Simpson’s type for +s-convex functions with applications. Research report collection, 12(4) (2009). +[2] Butt, S.I., Kashuri, A., Nasir, J.:Hermite-Hadamard type inequalities via new +exponential type convexity and their applications. Filomat, 35(6), pp.1803-1822 +(2021). +[3] Du, T., Wang, H., Khan, M.A., Zhang, Y.:Certain integral inequalities consid- +ering generalized m-convexity on fractal sets and their applications. Fractals, +27(07), p.1950117 (2019). +[4] Emam, T.:Roughly B-invex programming problems. 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Filomat (2022). +[17] Syau, Y.R., Jia, L., Lee, E.S.:Generalizations of E-convex and B-vex functions. +Computers & Mathematics with Applications, 58(4), pp.711-716 (2009). +[18] Wang, G., He, Y.:Generalized convexity of the inverse hyperbolic cosine func- +tion. Miskolc Mathematical Notes, 19(2), pp.873-881 (2018). +[19] Yang, X.M.:On E-convex sets, E-convex functions, and E-convex programming. +Journal of Optimization Theory and Applications, 109(3), p.699 (2001). +[20] Youness, E.A.:E-convex sets, E-convex functions, and E-convex programming. +Journal of Optimization Theory and Applications, 102(2), pp.439-450 (1999). +[21] Zhao, Y.X., Wang, S.Y., Coladas Uria, L.:Characterizations of r-convex func- +tions. Journal of optimization theory and applications, 145(1), pp.186-195 +(2010). +6 + diff --git a/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/load_file.txt b/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3100fdc29fba54fe89997879d663aa11b675909f --- /dev/null +++ b/d9AyT4oBgHgl3EQfwvm8/content/tmp_files/load_file.txt @@ -0,0 +1,243 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf,len=242 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='00655v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='OC] 2 Jan 2023 On Some Characterization of GS-exponential kind of Convex Functions Ehtesham Akhtera and Musavvir Alib,∗ a, b Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India E-mail addresses: b∗musavvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='alig@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='com (Corresponding author), ehteshamakhter111@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Abstract This manuscript introduces the idea of GS-exponential kind of convex functions and some of their algebraic features, and we introduce a new class GS-exponential kind of convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In addition, we describe certain fundamental GS-exponential kind of convex func- tion with characteristics in both the general and the differentiable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' We establish the sufficient conditions of optimality and offer the proof for unconstrained as well as inequality-constrained programming while considering the assumption of GS-exponential kind of convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' MSC: 26A51, 26B25, 90C26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Keywords: GS-exponential kind of convex functions and sets, Inequalities, Opti- mality conditions, Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 1 Introduction Due to the importance of convexity and its generalisations in the study of optimality to resolve mathematical issues, researchers have concentrated a lot of their efforts on generalised convex functions for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' As an illustration, Hudzik and Maligranda (1994) [7], investigated at two distinct forms of s-convexity and found that s-convexity in the next meaning is basically more significant than in the first sense whenever (0 < s < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Youness (1999) [20] expanded the definitions of convex sets and functions to create a new class of sets and functions known as E-convex sets and E-convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Yang (2001) [19] enhanced Youness’s paper [20] by incorporating certain illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In recent years, academic experts have given these generalized convex functions in additional consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' The semi-preinvex functions were studied by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Long and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Peng in 2006 [12] as a generalization of the semi-preinvex functions and 1 the b-vex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Syau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' (2009)[17] developed the E-b-vex function fam- ily, a novel class of functions which are the generalizations of b-vex functions and E-vex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In 2011, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Emam investigated a novel class of functions known as approximately b-invex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' He also discussed some of its properties and dis- covered the necessary optimality conditions for nonlinear programming using these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In their investigation of a novel class of generalized sub-b-convex func- tions and sub-b-convex sets, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' (2012) [13] showed the conditions for the existence of optimal solutions for both unconstrained and inequality-constrained sub-b-convex programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' The study in our paper aims to introduce a new class of generalized exponential kind of convex functions termed as GS-exponential kind of convex functions and explores certain characteristics of the same class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' This paper draws inspiration from a number of research papers [2, 5, 6, 8, 10, 14, 15, 16, 18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Additionally, we offer the adequate GS-exponential kind of convexity-derived criteria of optimality for pro- gramming with variables which are both unconstrained and inequality-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 2 Preliminaries We will go through the definitions of sub-b-s-convexity, exponential kind of convexity, and s-convexity of functions in this section of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' For the remainder of this work, let V stand for any non-empty convex subset in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' [11] The function Q : V → R is known as sub-b-s-convex in the second sense associated with the map G : V × V × (0, 1] → R, if Q(am1 + (1 − a)m2) ≤ asQ(m1) + (1 − a)sQ(m2) + G(m1, m2, s) holds for all m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' [7] The function Q : V → R is known as s-convex in the second sense, if for all m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1], we have Q(am1 + (1 − a)m2) ≤ asQ(m1) + (1 − a)sQ(m2) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' [9] A positive function Q : V → R is known as exponential kind of convex function, if Q(am1 + (1 − a)m2) ≤ (ea − 1)Q(m1) + (e1−a − 1)Q(m2) holds for all m1, m2 ∈ V, a ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' The concepts defined as in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='3, motivates us to explore a new idea known as GS-exponential kind of convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 2 3 Main Results Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' The function Q : V → R is known as GS-exponential kind of convex function on V associated with the map G : V × V × (0, 1] → R, if Q(am1 + (1 − a)m2) ≤ (ea − 1)sQ(m1) + (e1−a − 1)sQ(m2) + aG(m1, m2, s) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1) holds for each m1, m2 ∈ V, a ∈ [0, 1] and for any fixed s ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' If we take s = 1, Q(m1) is non-negative and G(m1, m2, s) = 0, the GS-exponential kind of convex function reduces to be exponential kind of convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' If Q1, Q2 : V → R are GS-exponential kind of convex function associated with the map G1, G2 respectively, then Q1 +Q2 and βQ1, (β ≥ 0) are also a GS-exponential kind of convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' If Qi : V → R, (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', n) are GS-exponential kind of convex function associated with the map Gi : V × V × (0, 1] → R, (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='., n), respectively, then Q = �n i=1 βiQi, β ≥ 0, (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', n) is GS-exponential kind of convex function associated with the map G = �n i=1 βiGi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' For all a ∈ [0, 1] and s ∈ (0, 1], the inequalities (ea − 1)s ≥ a and (e1−a − 1)s ≥ 1 − a hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Every convex function is GS-exponential kind of convex function if it has a map G associated with it that is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' If Q : V → R is the GS-exponential kind of convex function associ- ated with the map G and S : R → R is a non-negative function in addition to being linear, then S ◦ Q is a GS-exponential kind of convex function associated with the map S ◦ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Assume that U be a non-empty subset of Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Then, U is known as GS-exponential kind of convex set associated with the map G : Rn×Rn×(0, 1] → R if for all (m1, α1), (m2, α2) ∈ U, m1, m2 ∈ Rn, a ∈ [0, 1] and some fixed s ∈ (0, 1], we have (am1 + (1 − a)m2, (ea − 1)sα1 + (e1−a − 1)sα2 + aG(m1, m2, s)) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Now, we provide a characterization of GS-exponential kind of convex function Q : V → R based on their respective epigraphs, given by E(Q) = {(m, α) : m ∈ V, α ∈ R, Q(m) ≤ α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' A function Q : V → R is a GS-exponential kind of convex function associated with the map G : V × V × (0, 1] → R, if and only if E(Q) is a GS- exponential kind of convex set associated with the map G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Assume that m2 > 0 and Qβ : [m1, m2] → R is a family of numer- ical functions associated with the map Gβ and each Gβ is a GS-exponential kind of convex functions and each Gβ is bounded function, also assume that Q(m) = supβ Qβ(m) and G(m1, m2, s) = supβ Gβ(m1, m2, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' If the set (non-empty) K = {r ∈ [m1, m2]|Q(r) < ∞}, then K is an interval and Q is GS-exponential kind of convex function on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Let Q : [m1, m2] → R be a GS-exponential kind of convex function associated with the map G : [m1, m2]×[m1, m2]×(0, 1] → R and also let G(m1, m2, s) is bounded, then Q is also bounded on [m1, m2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In this section, Q is considered to be a differentiable function and s, a ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Let Q : V → R be a non-negative differentiable GS-exponential kind of convex function associated with the map G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Then (i)∇Q(m2)T(m1 − m2) < (ea − 1)s a Q(m1) + e(1−a)s a Q(m2) + G(m1, m2, s) − o(a) a , (ii)∇Q(m2)T(m1−m2) < (es − 1)s(Q(m1) − Q(m2)) + 3Q(m2) − o(a) a +G(m1, m2, s) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Let Q : V → R be a non-positive differentiable GS-exponential kind of convex function associated with the map G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Then ∇Q(m2)T(m1 − m2) ≤ (ea − 1)s a [Q(m1) − Q(m2)] + G(m1, m2, s) − o(a) a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Assume that Q : V → R is a positive differentiable GS-exponential kind of convex function, then ∇[Q(m2) − Q(m1)]T(m1 − m2) < (ea − 1)s a [Q(m1) + Q(m2)] + e(1−a)s a [Q(m1) + Q(m2)] +G(m1, m2, s) + G(m2, m1, s) − 2o(a) a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' In case if Q is a negative valued, then ∇[Q(m2) − Q(m1)]T(m1 − m2) ≤ G(m1, m2, s) + G(m2, m1, s) − 2o(a) a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 4 The following methods are then utilized to apply the above outcomes to nonlinear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' So, we take the unconstrained problem (S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' (S) : min{Q(m), m ∈ V } (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Let Q : V → R be a positive differentiable GS-exponential kind of convex function associated with the map G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Also, suppose that m ∈ V and the inequality ∇Q(m)T(n − m) > G(n, m, s) + 3Q(m) − o(a) a (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='3) holds for each n ∈ V, a ∈ (0, 1), and for any particular s ∈ (0, 1],, then n is the solution optimal to the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='2) related to Q on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' The following example of unconstrained programming is taken into consideration References [1] Alomari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Darus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Dragomir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=':New inequalities of Simpson’s type for s-convex functions with applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Research report collection, 12(4) (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' [2] Butt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Kashuri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Nasir, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=':Hermite-Hadamard type inequalities via new exponential type convexity and their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Filomat, 35(6), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='1803-1822 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' [3] Du, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Khan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=', Coladas Uria, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=':Characterizations of r-convex func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' Journal of optimization theory and applications, 145(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content='186-195 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AyT4oBgHgl3EQfwvm8/content/2301.00655v1.pdf'} +page_content=' 6' metadata={'source': 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sha256:2b49e755a4d1eb832dd379d8666839b5f81fcf70a31ea3832924049a1691df22 +size 132857 diff --git a/iNE3T4oBgHgl3EQfggp5/content/tmp_files/2301.04562v1.pdf.txt b/iNE3T4oBgHgl3EQfggp5/content/tmp_files/2301.04562v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c9327ec1afe58b394560ccdb13072d05f337581f --- /dev/null +++ b/iNE3T4oBgHgl3EQfggp5/content/tmp_files/2301.04562v1.pdf.txt @@ -0,0 +1,1756 @@ +arXiv:2301.04562v1 [math.DG] 11 Jan 2023 +Morse actions of discrete groups on symmetric spaces: +Local-to-global principle +Michael Kapovich, Bernhard Leeb, Joan Porti +January 12, 2023 +Abstract +Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. +As an application of our techniques we show algorithmic recognizability of Morse actions +and construct Morse “Schottky subgroups” of higher rank semisimple Lie groups via +arguments not based on Tits’ ping-pong. Our argument is purely geometric and proceeds +by constructing equivariant Morse quasiisometric embeddings of trees into higher rank +symmetric spaces. +Contents +1 +Introduction +2 +2 +Preliminaries +4 +2.1 +Basic notions of geometry of symmetric spaces . . . . . . . . . . . . . . . . . . . +4 +2.2 +Standing notation and conventions +. . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.3 +ζ-angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2.4 +Distances to parallel sets versus angles +. . . . . . . . . . . . . . . . . . . . . . . +6 +3 +Morse maps +8 +3.1 +Morse quasigeodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Morse maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3.3 +Continuity at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +3.4 +A Morse Lemma for straight sequences . . . . . . . . . . . . . . . . . . . . . . . +11 +3.5 +Lipschitz retractions to straight paths . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.6 +Local Morse quasigeodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +3.7 +Local-to-global principle for Morse maps . . . . . . . . . . . . . . . . . . . . . . +18 +1 + +4 +Group-theoretic applications +19 +4.1 +Stability of Morse actions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +4.2 +Schottky actions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +4.3 +Algorithmic recognition of Morse actions . . . . . . . . . . . . . . . . . . . . . . +25 +5 +Appendix: Further properties of Morse quasigeodesics +28 +5.1 +Finsler geometry of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . +28 +5.2 +Stability of diamonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +5.3 +Finsler approximation of Morse quasigeodesics . . . . . . . . . . . . . . . . . . . +31 +5.4 +Altering Morse quasigeodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +1 +Introduction +This is a sequel to our paper [KLP5] and mostly consists of the material of section 7 of our ear- +lier paper [KLP1] (the only additional material appears in Theorem 4.8 and the appendix +to the paper). +We recall that quasigeodesics in Gromov hyperbolic spaces can be recog- +nized locally by looking at sufficiently large finite pieces, see [CDP]. +In our earlier papers +[KLP4, KLP5, KLP2, KL1, KL2], for higher rank symmetric spaces X (of noncompact type) +we introduced an analogue of hyperbolic quasigeodesics, which we call Morse quasigeodesics. +Morse quasigeodesics are defined relatively to a certain face τmod of the model spherical face +σmod of X. In addition to the quasiisometry constants L, A, τmod-Morse quasigeodesics come +equipped with two other parameters, a positive number D and a Weyl-convex subset Θ of the +open star of τmod in the modal spherical chamber σmod. In [KLP1, KLP5, KLP2] we also defined +τmod-Morse maps Y Ñ X from Gromov-hyperbolic spaces to symmetric spaces. These maps +are defined by the property that they send geodesics to uniformly τmod-Morse quasigeodesics, +i.e. τmod-Morse quasigeodesics with a fixed set of parameters, pΘ, D, L, Aq. +The main result of this paper is a local characterization of Morse quasigeodesics in X: +Theorem 1.1 (Local-to-global principle for Morse quasigeodesics). For L, A, Θ, Θ1, D +there exist S, L1, A1, D1 such that every S-local pΘ, D, L, Aq-local Morse quasigeodesic in X is a +pΘ1, D1, L1, A1q–Morse quasigeodesic. +Here S-locality of a certain property of a map means that this property is satisfied for +restrictions of this map to subintervals of length S. We refer to Definition 3.34 and Theorem +3.34 for the details. Based on this principle, we prove in Section 3.7 a local-to-global principle +for Morse maps from hyperbolic metric spaces to symmetric spaces. +We prove several consequences of these local-to-global principles: +1. The structural stability of Morse subgroups of G, generalizing Sullivan’s Structural Sta- +bility Theorem in rank one [Su] (see also [KKL] for a detailed proof); see Theorems 4.4 and 4.6. +2 + +While structural stability for Anosov subgroups was known earlier (Labourie and Guichard– +Wienhard), our method is more general and applies to a wider class of discrete subgroups, see +[KL4]. +Theorem 1.2 (Openness of the space of Morse actions). For a word hyperbolic group +Γ, the subset of τmod-Morse actions is open in HompΓ, Gq. +Theorem 1.3 (Structural stability). Let Γ be word hyperbolic. Then for τmod-Morse actions +ρ : Γ ñ X, the boundary embedding αρ : B8Γ Ñ Flagpτmodq depends continuously on the action +ρ. +In particular, actions sufficiently close to a faithful Morse action are again discrete and +faithful. We supplement this structural stability theorem with a stability theorem on domains +of proper discontinuity, Theorem 4.8. +2. The locality of the Morse property implies that Morse subgroups are algorithmically +recognizable; Section 4.3: +Theorem 1.4 (Semidecidability of Morse property of group actions). Let Γ be word +hyperbolic. Then there exists an algorithm whose inputs are homomorphisms ρ : Γ Ñ G (defined +on generators of Γ) and which terminates if and only if ρ defines a τmod-Morse action Γ ñ X. +If the action is not Morse, the algorithm runs forever. Note that in view of [K2], there are +no algorithms (in the sense of BSS computability) which would recognize if a representation +Γ Ñ IsompH3q is not geometrically finite. +3. We illustrate our techniques by constructing Morse-Schottky actions of free groups on +higher rank symmetric spaces; Section 4.2. Unlike all previously known constructions, our proof +does not rely on ping-pong arguments, but is purely geometric and proceeds by constructing +equivariant quasi-isometric embeddings of trees. The key step is the observation that a certain +local straightness property for sufficiently spaced sequences of points in the symmetric space +implies the global Morse property. This observation is also at the heart of the proof of the +local-to-global principle for Morse actions. +Since [KLP1] was originally posted in 2014, several improvements on the material of section +7 of [KLP1] and, hence, of the present paper were made: +(a) Different forms of Combination Theorems for Anosov subgroups were proven in [DKL, +DK1, DK2] written in collaboration with Subhadip Dey by the 1st and the 2nd author and, +subsequently, by the 1st author. The first one was a generalization of the technique in section +4.2 the present paper, but the other two generalizations are based on a form of the ping-pong +argument. +(b) Explicit estimates in the local-to-global principle for Morse quasigeodesics and, hence, +Morse embeddings, were obtained by Max Riestenberg in [1]. Riestenberg’s estimates are based +on replacing certain limiting arguments used in the present paper with differential-geometric +and Lie-theoretic arguments. +3 + +Organization of the paper. +The notions of Morse quasigeodesics and actions are discussed in detail in section 3. In that +section, among other things, we establish local-to-global principles for Morse quasigeodesics. +In section 4 we apply local-to-global principles to discrete subgroups of Lie groups: We show +that Morse actions are structurally stable and algorithmically recognizable. We also construct +Morse-Schottky actions of free groups on symmetric spaces. In section 5 (the appendix to the +paper) we prove further properties of Morse quasigeodesics that we found to be useful in our +work. +Acknowledgements. The first author was supported by NSF grants DMS-12-05312 and +DMS-16-04241, by KIAS (the Korea Institute for Advanced Study) through the KIAS scholar +program, and by a Simons Foundation Fellowship, grant number 391602. The last author was +supported by grants Mineco MTM2012-34834 and AGAUR SGR2009-1207. The three authors +are also grateful to the GEAR grant which partially supported the IHP trimester in Winter +of 2012 (DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures and representation +varieties” (the GEAR Network), and to the Max Planck Institute for Mathematics in Bonn, +where some of this work was done. +2 +Preliminaries +2.1 +Basic notions of geometry of symmetric spaces +Throughout the paper we will be using definitions, notations and results of our earlier work. +We refer the reader to our earlier papers, e.g. [KLP4, KLP5, KLP2, KL1, KL2] for the vari- +ous notions related to symmetric spaces, such as polyhedral Finsler metrics on symmetric spaces +([KL1]), the opposition involution ι of σmod, model faces τmod of σmod and the associated τmod-flag +manifolds Flagpτmodq (sections 2.2.2 and 2.2.3 of [KLP5]), type map θ : B8X Ñ σmod, open Schu- +bert cells Cpτq Ă Flagpτmodq (section 2.4 of [KLP5]), ∆-valued distances d∆ on X (section 2.6 +of [KLP5]), Θ-regular geodesic segments (see §2.5.3 of [KLP5]), parallel sets, stars, open stars +and Θ-stars, stpτq, ostpτq, and stΘpτq, Weyl sectors V px, τq (section 2.4 of [KLP5]), Weyl cones +V px, stpτqq and Θ-cones V px, stΘpτqq, diamonds ♦τmodpx, yq and Θ-diamonds ♦Θpx, yq (section +2.5 of [KLP5]), τmod-regular sequences and groups (section 4.2 of [KLP5]), τmod-convergence +subgroups, flag-convergence, the Finsler interpretation of flag-convergence (see [KL1, §4.5 and +5.2] and [KLP5]), τmod-limit sets ΛτmodpΓq Ă Flagpτmodq (section 4.5 of [KLP5]), visual limit set +(page 4 of [KLP5]), uniformly τmod-regular sequences and subgroups (section 4.6 of [KLP5]), +Morse subgroups (section 5.4 of [KLP5]) and, more generally, Morse quasigeodesics and Morse +maps (Definitions 5.31, 5.33 of [KLP2]), antipodal limit sets (Definition. 5.1 of [KLP5]) and +antipodal maps to flag-manifolds (Definition 6.11 of [KLP2]). +In the paper we will be frequently using convexity of Θ-cones in X: +Proposition 2.1 (Proposition 2.10 in [KLP5]). For every Weyl-convex subset Θ Ă stpτmodq, +4 + +for every x P X and τ P Flagpτmodq, the cone V px, stΘpτqq Ă X is convex. +2.2 +Standing notation and conventions +• We will use the notation X for a symmetric space of noncompact type, G for a semisimple +Lie group acting isometrically and transitively on X, and K is a maximal compact sub- +group of G, so that X is diffeomorphic to G{K. We will assume that G is commensurable +with the isometry group IsompXq in the sense that we allow finite kernel and cokernel for +the natural map G Ñ IsompXq. In particular, the image of G in IsompXq contains the +identity component IsompXqo. +• We let τmod Ď σmod be a fixed ι-invariant face type. +• We will use the notation xn +f +ÝÑ τ P Flagpτmodq for the flag-convergence of a τmod-regular +sequence xn P X to a simplex τ P Flagpτmodq. +• We will be using the notation Θ, Θ1 for an ι-invariant, compact, Weyl-convex (see Defini- +tion 2.7 in [KLP5]) subset of the open star ostpτmodq Ă σmod. +• We will always assume that Θ ă Θ1, meaning that Θ Ă intpΘ1q. +• Constants L, A, D, ǫ, δ, l, a, s, S are meant to be always strictly positive and L ě 1. +2.3 +ζ-angles +We fix as auxiliary datum a ι-invariant type ζ “ ζmod P intpτmodq. (We will omit the subscript +in ζmod in order to avoid cumbersome notation for ζ-angles.) For a simplex τ Ă B8X of type +τmod, i.e. τ P Flagpτmodq, we define ζpτq P τ as the ideal point of type ζmod. Given two such +simplices τ˘ P Flagpτmodq and a point x P X, define the ζ-angles +=ζ +xpτ´, τ`q “ =ζ +xpτ´, ξ`q :“ =xpξ´, ξ`q, +(2.2) +where ξ˘ “ ζpτ˘q. +Similarly, define the ζ-Tits angle +=ζ +Titspτ´, τ`q “ =ζ +Titspτ´, ξ`q :“ =xpξ´, ξ`q, +(2.3) +where x belongs to a flat F Ă X such that τ´, τ` Ă BTitsF. Then simplices τ˘ (of the same +type) are antipodal iff +=ζ +Titspτ´, τ`q “ π +for some, equivalently, every, choice of ζ as above. +Remark 2.4. We observe that the ideal points ζ˘ are opposite, =Titspζ´, ζ`q “ π, if and only +if they can be seen under angle » π (i.e., close to π) from some point in X. More precisely, +there exists ǫpζmodq such that: +5 + +If =xpζ´, ζ`q ą π ´ ǫpζmodq for some point x then ζ˘ are opposite. +This follows from the angle comparison =xpζ´, ζ`q ď =Titspζ´, ζ`q and the fact that the Tits +distance between ideal points of the fixed type ζmod takes only finitely many values. +For a τmod-regular unit tangent vector v P TX we denote by τpvq Ă B8X the unique simplex +of type τmod such that ray ρv with the initial direction v represents an ideal point in ostpτpvqq. +We put ζpvq “ ζpτpvqq. Note that ζpvq depends continuously on v. +For a τmod-regular segment xy in X we let τpxyq “ τpvq, where v is the unit vector tangent +to xy. +Then, for a τmod-regular segments xy, xz and τ P Flagpτmodq, we define the ζ-angles +=ζ +xpy, τq “ =ζ +xpτpxyq, τq, +=ζ +xpy, zq “ =ζ +xpτpxyq, τpxzqq +Thus, the ζ-angle depends not on y, z but rather on the simplices τpxyq, τpxzq. These ζ- +angles will play the role of angles the between diamonds ♦τmodpx, yq and ♦τmodpx, zq, meeting +at x. Note that if X has rank 1, then the ζ-angles are just the ordinary Riemannian angles. +2.4 +Distances to parallel sets versus angles +In this section we collect some geometric facts regarding parallel sets in symmetric spaces, +primarily dealing with estimation of distances from points in X to parallel sets. +Remark 2.5. The constants and functions in this section are not explicit and their existence +is proven by compactness arguments. For explicit computations here and in Theorem 3.18, we +refer the reader to the PhD thesis of ... +We first prove a lemma (Lemma 2.6) which strengthens Corollary 2.46 of [KLP5]. +Lemma 2.6. Suppose that τ˘ are antipodal simplices in BTitsX. Then every geodesic ray γ +asymptotic to a point ξ P ostpτ`q, is strongly asymptotic to a geodesic ray in Ppτ´, τ`q. +Proof. If ξ belongs to the interior of the simplex τ`, then the assertion follows from Corollary +2.46 of [KLP5]: +Weyl sectors V px1, τq and V px2, τq are strongly asymptotic if and only if x1 and x2 lie in +the same horocycle at τ. +We now consider the general case. Suppose, that ξ belongs to an open simplex intpτ 1q, such +that τ is a face of τ 1. Then there exists an apartment a Ă BTitsX containing both ξ (and, +hence, τ 1 as well as τ) and the simplex τ´. Let F Ă X be the maximal flat with B8F “ a. +Then F contains a geodesic asymptotic to points in τ´ and τ`. Therefore, F is contained in +Ppτ´, τ`q. On the other hand, by the same Corollary 2.46 of [KLP5], applied to the simplex +τ 1, we conclude that γ is strongly asymptotic to a geodesic ray in F. +The following lemma provides a quantitative strengthening of the conclusion of Lemma 2.6: +6 + +Lemma 2.7. Let Θ be a compact subset of ostpτ`q. Then those rays xξ with θpξq P Θ are uni- +formly strongly asymptotic to Ppτ´, τ`q, i.e. dp¨, Ppτ´, τ`qq decays to zero along them uniformly +in terms of dpx, Ppτ´, τ`qq and Θ. +Proof. Suppose that the assertion of lemma is false, i.e., there exists ǫ ą 0, a sequence Ti P R` +diverging to infinity, and a sequence of rays ρi “ xiξi with ξi P Θ and dpxi, Ppτ´, τ`qq ď d, so +that +dpy, Ppτ´, τ`qq ě ǫ, @y P ρpr0, Tisq. +(2.8) +Using the action of the stabilizer of Ppτ´, τ`q, we can assume that the points xi belong to a +certain compact subset of X. Therefore, the sequence of rays xiξi subconverges to a ray xξ with +dpx, Ppτ´, τ`qq ď d and ξ P Θ. The inequality (2.8) then implies that the entire limit ray xξ is +contained outside of the open ǫ-neighborhood of the parallel set Ppτ´, τ`q. However, in view +of Lemma 2.6, the ray xξ is strongly asymptotic to a geodesic in Ppτ´, τ`q. Contradiction. +We next relate distances from points x P X to parallel sets and the ζ-angles at x. Suppose +that the simplices τ˘, equivalently, the ideal points ζ˘ “ ζpτ˘q (see section 2.3), are opposite. +Then +=ζ +xpτ´, τ`q “ =xpζ´, ζ`q “ π +if and only if x lies in the parallel set Ppτ´, τ`q. Furthermore, =ζ +xpτ´, τ`q » π if and only if x is +close to Ppτ´, τ`q, and both quantities control each other near the parallel set. More precisely: +Lemma 2.9. (i) If dpx, Ppτ´, τ`qq ď d, then =ζ +xpτ´, τ`q ě π ´ ǫpdq with ǫpdq Ñ 0 as d Ñ 0. +(ii) For sufficiently small ǫ, ǫ ď ǫ1pζmodq, we have: The inequality =ζ +xpτ´, τ`q ě π´ǫ implies +that dpx, Ppτ´, τ`qq ď dpǫq for some function dpǫq which converges to 0 as ǫ Ñ 0. +Proof. The intersection of parabolic subgroups Pτ´ X Pτ` preserves the parallel set Ppτ´, τ`q +and acts transitively on it. Compactness and the continuity of =¨pζ´, ζ`q therefore imply that +π ´ =¨pζ´, ζ`q attains on the boundary of the tubular r-neighborhood of Ppτ´, τ`q a strictly +positive maximum and minimum, which we denote by φ1prq and φ2prq. Furthermore, φiprq Ñ 0 +as r Ñ 0. We have the estimate: +π ´ φ1pdpx, Ppτ´, τ`qqq ď =xpζ´, ζ`q ď π ´ φ2pdpx, Ppτ´, τ`qqq +The functions φiprq are (weakly) monotonically increasing. This follows from the fact that, +along rays asymptotic to ζ´ or ζ`, the angle =¨pζ´, ζ`q is monotonically increasing and the +distance dp¨, Ppτ´, τ`qq is monotonically decreasing. The estimate implies the assertions. +The control of dp¨, Ppτ´, τ`qq and =¨pζ´, ζ`q “spreads” along the Weyl cone V px, stpτ`qq, +since the latter is asymptotic to the parallel set Ppτ´, τ`q. Moreover, the control improves, if +one enters the cone far into a τmod-regular direction. More precisely: +Lemma 2.10. Let y P V px, stΘpτ`qq be a point with dpx, yq ě l. +(i) If dpx, Ppτ´, τ`qq ď d, then +dpy, Ppτ´, τ`qq ď D1pd, Θ, lq ď d +7 + +with D1pd, Θ, lq Ñ 0 as l Ñ `8. +(ii) For sufficiently small ǫ, ǫ ď ǫ1pζmodq, we have: If =xpζ´, ζ`q ě π ´ ǫ, then +=ypζ´, ζ`q ě π ´ ǫ1pǫ, Θ, lq ě π ´ ǫpdpǫqq +with ǫ1pǫ, Θ, lq Ñ 0 as l Ñ `8. +Proof. The distance from Ppτ´, τ`q takes its maximum at the tip x of the cone V px, stpτ`qq, +because it is monotonically decreasing along the rays xξ for ξ P stpτ`q. This yields the right- +hand bounds d and, applying Lemma 2.9 twice, ǫpdpǫqq. +Those rays xξ with uniformly τmod-regular type θpξq P Θ are uniformly strongly asymptotic +to Ppτ´, τ`q, i.e. dp¨, Ppτ´, τ`qq decays to zero along them uniformly in terms of d and Θ, see +Lemma 2.7. This yields the decay D1pd, Θ, lq Ñ 0 as l Ñ `8. The decay of ǫ1 follows by +applying Lemma 2.9 again. +3 +Morse maps +In this section we investigate the Morse property of sequences and maps. The main aim of +this section is to establish a local criterion for being Morse. To do so we introduce a local +notion of straightness for sequences of points in X. Morse sequences are in general not straight, +but they become straight after suitable modification, namely by sufficiently coarsifying them +and then passing to the sequence of successive midpoints. Conversely, the key result is that +sufficiently spaced straight sequences are Morse. We conclude that there is a local-to-global +characterization of the Morse property. +3.1 +Morse quasigeodesics +Definition 3.1 (Morse quasigeodesic). A pΘ, D, L, Aq-Morse quasigeodesic in X is an +pL, Aq-quasigeodesic p : I Ñ X (defined on an interval I Ă R) such that for all t1, t2 P I +the subpath p|rt1,t2s is D-close to a Θ-diamond ♦Θpx1, x2q with dpxi, pptiqq ď D. +We will refer to a quadruple pΘ, D, L, Aq as a Morse datum and abbreviate M “ pΘ, D, L, Aq. +Set M `D1 “ pΘ, D`D1, L, A`2D1q. We say that M contains Θ if M has the form pΘ, D, L, Aq +for some D ě 0, L ě 1, A ě 0. +The following lemma is immediate from the definiton of a M-Morse quasigeodesic. +Lemma 3.2 (Perturbation lemma). If p, p1 are paths in X such that p is M-Morse and +dpp, p1q ď D1 then p1 is M ` D1-Morse. +A Morse quasigeodesic p is called a Morse ray if its domain is a half-line. If I “ R then a +Morse quasigeodesic is called a Morse quasiline. +8 + +Morse quasirays do in general not converge at infinity (in the visual compactification of X), +but they τmod-converge at infinity. This is a consequence of: +Lemma 3.3 (Conicality). Every Morse quasiray p : r0, 8q Ñ X is uniformly Hausdorff close +to a subset of a cone V ppp0q, stΘpτqq for a unique simplex τ of type τmod. +Proof. The subpaths p|r0,t0s are uniformly Hausdorff close to Θ-diamonds. These subconverge +to a cone V px, stΘpτqq x uniformly close to pp0q and τ a simplex of type τmod. This establishes +the existence. Since ppnq +f +ÝÑ τ, the uniqueness of τ follows from the uniqueness of τmod-limits, +see [KLP5, Lemma 4.23]. +Definition 3.4 (End of Morse quasiray). We call the unique simplex given by the previous +lemma the end of the Morse quasiray p : r0, 8q Ñ X and denote it by +pp`8q P Flagpτmodq. +Hausdorff close Morse quasirays have the same end by Lemma 3.3. In section 3.3 we will +prove uniform continuity of ends of Morse quasirays with respect to the topology of coarse +convergence of quasirays. +3.2 +Morse maps +We now turn to Morse maps with more general domains (than just intervals). +Definition 3.5. Let Y be a Gromov-hyperbolic geodesic metric space. A map f : Y Ñ X is +called M-Morse if it sends geodesics in Y to M-Morse quasigeodesics. +Thus, every Morse map is a quasiisometric embedding. While this definition makes sense +for general metric spaces, in [KLP2] we proved that the domain of a Morse map is necessarily +hyperbolic. +More generally, one can define Morse maps on quasigeodesic metric spaces: +Definition 3.6 (Quasigeodesic metric space). A metric space Z is called pl, aq-quasigeodesic +if all pairs of points in Y can be connected by pl, aq-quasigeodesics. A space is called quasi- +geodesic if it is pl, aq-quasigeodesic for some pair of parameters l, a. +Every quasigeodesic space is quasiisometric to a geodesic metric space. Namely, if Z is pλ, αq- +quasigeodesic space then it is quasiisometric to its pλ ` αq-Rips complex. The quasigeodesic +spaces considered in this paper are discrete groups equipped with word metrics. +Definition 3.7 (Morse embedding). Let pΘ, D, L, Aq be a Morse datum. +An pΘ, D, L, A, l, aq-Morse embedding (or a map) from an pl, aq-quasigeodesic space Z into X is +a map f : Z Ñ X which sends pl, aq-quasigeodesics in Z to pΘ, D, L, Aq-Morse quasigeodesics +in X. +9 + +Of course, every pl, aq-quasigeodesic metric space is also pl1, a1q-quasigeodesic space for any +l1 ě l, a1 ě a. The next lemma shows that this choice of quasigeodesic constants is essentially +irrelevant. +Lemma 3.8. Let f : Z Ñ X be a map from a Gromov-hyperbolic pl, aq-quasigeodesic space Z. +If f is M “ pΘ, D, L, A, l, aq-Morse then for any pl1, a1q, it sends pl1, a1q-quasigeodesics in Z to +M1 “ pΘ, D1, L1, A1q-Morse quasigeodesics in X. Here the datum M1 depends only on M, l1, a1 +and the hyperbolicity constant δ of Z. +Proof. This is a consequence of the definition of Morse quasigeodesics, and the Morse Lemma +applied to Z. +Notice that the parameter Θ in the Morse datum M1 is the same as in M. Hence, we arrive +to +Definition 3.9. A map f : Z Ñ X of a quasigeodesic hyperbolic space Z is called Θ-Morse if +it sends uniform quasigeodesics in Z to Θ-Morse uniform quasigeodesics in X. +This notion depends only on the quasi-isometry class of Z, i.e. the precomposition of a +Θ-Morse embedding with a quasi-isometry is again Θ-Morse. For this to be true we have to +require control on the images of quasigeodesics of arbitrarily bad (but uniform) quality. +Let Γ be a hyperbolic group with fixed a finite generating set S, and let Y be the Cayley +graph of Γ with respect to S. For x P X, an isometric action Γ ñ X determines the orbit +map ox : Γ Ñ Γx Ă X. Every such map extends to the Cayley graph Y of Γ, sending edges to +geodesics in X. +Definition 3.10. An isometric action Γ ñ X or a representation ρ : Γ Ñ G, is called M-Morse +(with respect to a base-point x P X) if the (extended) orbit map ox : Y Ñ X is M-Morse. +Similarly, a subgroup Γ ă G is Morse if the inclusion homomorphism Γ ãÑ G is Morse. +The Morse property of an action and the parameter Θ, of course, does not depend on +the choice of a generating set of Γ and a base-point x, but the triple pD, L, Aq does. Thus, +it makes sense to talk about a Θ-Morse and τmod-Morse actions of hyperbolic groups, where +Θ Ă ostpτmodq. In [KLP5, KLP2, KL1] we gave many alternative definitions of Morse actions, +including the equivalence of this definition to the notion of Anosov subgroups. +3.3 +Continuity at infinity +Let X, Y be proper metric spaces. We fix a base point y P Y . +Definition 3.11. A sequence of maps fn : Y Ñ X is said to coarsely converge to a map +f : Y Ñ X if there exists C ă 8 such that for every R there exists N “ NpC, Rq for which +dpfn|B, f|Bq ď C, +where B “ Bpy, Rq. +10 + +Note the difference of this definition with the notion of uniform convergence on compacts: +Since we are working in the coarse setting, requiring the distance between maps to be less than +ǫ close to zero is pointless. +In view of the Arzela–Ascoli theorem, the space of pL, Aq-coarse Lipschitz maps Y Ñ X +sending y to a fixed bounded subset of X, is coarsely sequentially compact: Every sequence +contains a coarsely converging subsequence. +In the next lemma we assume that Y is a geodesic δ-hyperbolic space and X is a symmetric +space of noncompact type. The lemma itself is an immediate consequence of the perturbation +lemma, Lemma 3.2. +Lemma 3.12. Suppose that pn : R` Ñ X is a sequence of M-Morse rays which coarsely +converges to a map p : R` Ñ X. Then p is M1-Morse, where M1 “ M ` C and the constant C +is the one appearing in the definition of coarse convergence. +In particular, a coarse limit of a sequence of (uniformly) Morse quasigeodesics is again +Morse. +For the next lemma, we equip the flag manifold F “ Flagpτmodq with some background +metric dF. +Lemma 3.13. Suppose that pn : R` Ñ X is a sequence of M-Morse rays coarsely converging +to a M-Morse ray p : R` Ñ X. Then the sequence τn :“ pnp8q of ends of the quasirays pn +converges to τ “ pp8q. Moreover, the latter convergence is uniform in the following sense. For +every ǫ ą 0 there exists n0 depending only on M and C and NpR, Cq (appearing in Definition +3.11) such that for all n ě n0, dFpτn, τq ď ǫ. +Proof. Suppose that the claim is false. Then in view of coarse compactness of the space of +M-Morse maps sending y to a fixed compact subset of X, there exists a sequence ppnq as +in the lemma, coarsely converging to p, such that the sequence pnp8q “ τn converges to +τ 1 ‰ pp8q “ τ. +By the coarse convergence pn Ñ p, there exists C ă 8 and a sequence +tn Ñ 8 such that dppnptnq, pptnqq ď C. By the definition of Morse quasigeodesics, there exists +a sequence of cones V pxn, stpτnqq (with xn in a bounded subset B Ă X) such that the image +of pn is contained in the D-neighborhood of V pxn, stpτnqq. Thus, the sequence ppnptnqq flag- +converges to τ 1, while ppptnqq flag-converges to τ. According to [KLP5, Lemma 4.23], altering +a sequence by a uniformly bounded amount, does not change the flag-limit. Therefore, the +sequence ppptnqq also flag-converges to τ 1. Hence, τ “ τ 1. A contradiction. +3.4 +A Morse Lemma for straight sequences +In order to motivate the results of this section we recall the following sufficient condition for a +piecewise-geodesic path in a Hadamard manifold Y of curvature ď ´1 to be quasigeodesic (see +e.g. [KaLi]): +11 + +Proposition 3.14. Suppose that c is a piecewise-geodesic path in Y whose angles at the vertices +are ě α ą 0 and whose edges are longer than L, where α and L satisfy +coshpL{2q sinpα{2q ě ν ą 1. +(3.15) +Then c is an pLpνq, Apνqq-quasigeodesic. +By considering c with vertices on a horocycle in the hyperbolic plane, one see that the +inequality in this proposition is sharp. +Corollary 3.16. If L is sufficiently large and α is sufficiently close to π then c is (uniformly) +quasigeodesic. +In higher rank, we do not have an analogue of the inequality (3.15), instead, we will be +generalizing the corollary. However, angles in the corollary will be replaced with ζ-angles. We +will show (in a String of Diamonds Theorem, theorem 3.30) that if a piecewise-geodesic path +c in X has sufficiently long edges and ζ-angles between consecutive segments sufficiently close +to π, then c is M-Morse for a suitable Morse datum. +In the following, we consider finite or infinite sequences pxnq of points in X. +Definition 3.17 (Straight and spaced sequence). We call a sequence pxnq pΘ, ǫq-straight +if the segments xnxn`1 are Θ-regular and +=ζ +xnpxn´1, xn`1q ě π ´ ǫ +for all n. We call it l-spaced if the segments xnxn`1 have length ě l. +Note that every straight sequence can be extended to a biinfinite straight sequence. +Straightness is a local condition. The goal of this section is to prove the following local- +to-global result asserting that sufficiently straight and spaced sequences satisfy a higher rank +version of the Morse Lemma (for quasigeodesics in hyperbolic space). +Theorem 3.18 (Morse Lemma for straight spaced sequences). For Θ, Θ1, δ there exist +l, ǫ such that: +Every pΘ, ǫq-straight l-spaced sequence pxnq is δ-close to a parallel set Ppτ´, τ`q with sim- +plices τ˘ of type τmod, and it moves from τ´ to τ` in the sense that its nearest point projection +¯xn to Ppτ´, τ`q satisfies +¯xn˘m P V p¯xn, stΘ1pτ˘qq +(3.19) +for all n and m ě 1. +Remark 3.20 (Global spacing). 1. As a corollary of this theorem, we will show that straight +spaced sequences are quasigeodesic: +dpxn, xn`mq ě clm ´ 2δ +with a constant c “ cpΘ1q ą 0. See Corollary 3.29. In particular, by interpolating the sequence +pxnq via geodesic segments we obtain a Morse quasigeodesic in X. +12 + +2. +Theorem 3.18 is a higher-rank generalization of two familiar facts from geometry of +Gromov-hyperbolic geodesic metric spaces: The fact that local quasigeodesics (with suitable +parameters) are global quasigeodesics and the Morse lemma stating that quasigeodesics stay +uniformly close to geodesics. In the higher rank, quasigeodesics, of course, need not be close +to geodesics, but, instead (under the straightness assumption), are close to diamonds/Weyl +cones/parallel sets. +3. One can obviously strengthen the Corollary 3.16 by stating that for each ǫ ă π there +exists L0pǫq such that if α ě π ´ ǫ and L ě L0pǫq then c is a uniform quasigeodesic in X. A +similar strengthening is false for symmetric spaces of rank ě 2. For instance, when W – S3 and +ǫ “ 2π{3, then no matter what Θ, Θ1 and l are, the conclusion of Theorem 3.18 fails already +for sequences contained in a single flat. +In order to prove the theorem, we start by considering half-infinite sequences and prove that +they keep moving away from an ideal simplex of type τmod if they do so initially. +Definition 3.21 (Moving away from an ideal simplex). Given a face τ Ă BTitsX of type +τmod and distinct points x, y P X, define the angle +=ζ +xpτ, yq :“ =xpz, yq +where z is a point (distinct from x) on the geodesic ray xξ, where ξ P τ is the point of type ζ. +We say that a sequence pxnq moves ǫ-away from a simplex τ of type τmod if +=ζ +xnpτ, xn`1q ě π ´ ǫ +for all n. +Lemma 3.22 (Moving away from ideal simplices). For small ǫ and large l, ǫ ď ǫ0 and +l ě lpǫ, Θq, the following holds: +If the sequence pxnqně0 is pΘ, ǫq-straight l-spaced and if +=ζ +x0pτ, x1q ě π ´ 2ǫ, +then pxnq moves ǫ-away from τ. +Proof. By Lemma 2.10(ii), the unit speed geodesic segment c : r0, t1s Ñ X from pp0q to pp1q +moves ǫpdp2ǫqq-away from τ at all times, and ǫ1p2ǫ, Θ, lq-away at times ě l, which includes the +final time t1. For lpǫ, Θq sufficiently large, we have ǫ1p2ǫ, Θ, lq ď ǫ. Then c moves ǫ-away from +τ at time t1, which means that =ζ +x1pτ, x0q ď ǫ. Straightness at x1 and the triangle inequality +yield that again =ζ +x1pτ, x2q ě π ´ 2ǫ. One proceeds by induction. +Note that there do exist simplices τ satisfying the hypothesis of the previous lemma. For +instance, one can extend the initial segment x0x1 backwards to infinity and choose τ “ τpx1x0q. +Now we look at biinfinite sequences. +13 + +We assume in the following that pxnqnPZ is pΘ, ǫq-straight l-spaced for small ǫ and large l. As +a first step, we study the asymptotics of such sequences and use the argument for Lemma 3.22 +to find a pair of opposite ideal simplices τ˘ such that pxnq moves from τ´ towards τ`. +Lemma 3.23 (Moving towards ideal simplices). For small ǫ and large l, ǫ ď ǫ0 and +l ě lpǫ, Θq, the following holds: +There exists a pair of opposite simplices τ˘ of type τmod such that the inequality +=ζ +xnpτ¯, xn˘1q ě π ´ 2ǫ +(3.24) +holds for all n. +Proof. 1. For every n define a compact set C¯ +n Ă Flagpτmodq +C˘ +n “ tτ˘ : =ζ +xnpτ˘, xn¯1q ě π ´ 2ǫu. +As in the proof of Lemma 3.22, straightness at xn`1 implies that C´ +n Ă C´ +n`1. Hence the family +tC´ +n unPZ form a nested sequence of nonempty compact subsets and therefore have nonempty +intersection containing a simplex τ´. Analogously, there exists a simplex τ` which belongs to +C` +n for all n. +2. It remains to show that the simplices τ´, τ` are antipodal. Using straightness and the +triangle inequality, we see that +=ζ +xnpτ´, τ`q ě π ´ 5ǫ +for all n. Hence, if 5ǫ ă ǫpζq, then the simplices τ´, τ` are antipodal in view of Remark 2.4. +The pair of opposite simplices pτ´, τ`q which we found determines a parallel set in X. The +second step is to show that pxnq is uniformly close to it. +Lemma 3.25 (Close to parallel set). For small ǫ and large l, ǫ ď ǫpδq and l ě lpΘ, δq, the +sequence pxnq is δ-close to Ppτ´, τ`q. +Proof. The statement follows from the combination of the inequality (3.4) (in the second part +of the proof of Lemma 3.23) and Lemma 2.9. +The third and final step of the proof is to show that the nearest point projection p¯xnq of +pxnq to Ppτ´, τ`q moves from τ´ towards τ`. +Lemma 3.26 (Projection moves towards ideal simplices). For small ǫ and large l, ǫ ď ǫ0 +and l ě lpǫ, Θ, Θ1q, the segments ¯xn¯xn`1 are Θ1-regular and +=ζ +¯xnpτ´, ¯xn`1q “ π +for all n. +Proof. By the previous lemma, pxnq is δ0-close to Ppτ´, τ`q if ǫ0 is sufficiently small and l is +sufficiently large. Since xnxn`1 is Θ-regular, the triangle inequality for ∆-lengths yields that +the segment ¯xn¯xn`1 is Θ1-regular, again if l is sufficiently large. +14 + +Let ξ` denote the ideal endpoint of the ray extending this segment, i.e. ¯xn`1 P ¯xnξ`. Then +xn`1 is 2δ0-close to the ray xnξ`. We obtain that +=ζ +Titspτ´, ξ`q ě =ζ +xnpτ´, ξ`q » =ζ +xnpτ´, xn`1q » π +where the last step follows from inequality (3.24). The discreteness of Tits distances between +ideal points of fixed type ζ implies that in fact +=ζ +Titspτ´, ξ`q “ π, +i.e. the ideal points ζpτ´q and ζpξ`q are antipodal. But the only simplex opposite to τ´ in +B8Ppτ´, τ`q is τ`, so τpξ`q “ τ` and +=ζ +¯xnpτ´, ¯xn`1q “ =ζ +¯xnpτ´, ξ`q “ π, +as claimed. +Proof of Theorem 3.18. It suffices to consider biinfinite sequences. +The conclusion of Lemma 3.26 is equivalent to ¯xn`1 P V p¯xn, stΘ1pτ`qq. Combining Lem- +mas 3.25 and 3.26, we thus obtain the theorem for m “ 1. +The convexity of Θ1-cones, cf. Proposition 2.1, implies that +V p¯xn`1, stΘ1pτ`qq Ă V p¯xn, stΘ1pτ`qq, +and the assertion follows for all m ě 1 by induction. +Remark 3.27. The conclusion of the theorem implies flag-convergence x˘n Ñ τ˘ as n Ñ `8. +However, the sequences pxnqnP˘N do in general not converge at infinity, but accumulate at +compact subsets of stΘ1pτ˘q. +3.5 +Lipschitz retractions to straight paths +Consider a (possibly infinite) closed interval J in R; we will assume that J has integer or infinite +bounds. Suppose that p : J X Z Ñ P “ Ppτ´, τ`q Ă X is an l-separated, λ-Lipschitz, pΘ, 0q- +straight coarse sequence pointing away from τ´ and towards τ`. We extend p to a piecewise- +geodesic map p : J Ñ P by sending intervals rn, n ` 1s to geodesic segments ppnqppn ` 1q via +affine maps. We retain the name p for the extension. +Lemma 3.28. There exists L “ Lpl, λ, Θq and an L-Lipschitz retraction of X to p, i.e., an +L-Lipschitz map r : X Ñ J so that r ˝ p “ Id. In particular, p : J X Z Ñ X is a p¯L, ¯Aq- +quasigeodesic, where ¯L, ¯A depend only on l, λ, Θ. +Proof. It suffices to prove existence of a retraction. +Since P is convex in X, it suffices to +construct a map P Ñ J. Pick a generic point ξ “ ξ` P τ` and let bξ : P Ñ R denote the +Busemann function normalized so that bξpppzqq “ 0 for some z P J X Z. Then the Θ-regularity +15 + +assumption on p implies that the slope of the piecewise-linear function bξ ˝ p : J Ñ R is strictly +positive, bounded away from 0. The assumption that p is l-separated λ-Lipschitz implies that +l ď |p1ptq| ď λ +for each t (where the derivative exists). The straightness assumption on p implies that the +function h :“ bξ ˝ p : J Ñ R is strictly increasing. +By combining these observations, we +conclude that h is an L-biLipschitz homeomorphism for some L “ Lpl, λ, Θq. Lastly, we define +r : P Ñ J, +r “ h´1 ˝ bξ. +Since bξ is 1-Lipschitz, the map r is L-Lipschitz. By the construction, r ˝ p “ Id. +Corollary 3.29. Suppose that p : JXZ Ñ X is a l-spaced, λ-Lipschitz, pΘ, ǫq-straight sequence. +Pick some Θ1 such that Θ Ă intpΘ1q and let δ “ δpl, Θ, Θ1, ǫq be the constant as in Theorem +3.18. Then for L “ Lpl ´ 2δ, λ ` 2δ, Θ1q we have: +1. There exists an pL, 2δq-coarse Lipschitz retraction X Ñ J. +2. The map p is a pΘ1, D1, L1, A1q-quasigeodesic with D1, L1, A1 depending only on l, λ, Θ, Θ1, ǫ. +Proof. The statement immediately follows the above lemma combined with Theorem 3.18. +Reformulating in terms of piecewise-geodesic paths, we obtain +Theorem 3.30 (String of diamonds theorem). For any pair of Weyl convex subsets Θ ă Θ1 +and a number D ě 0 there exist positive numbers ǫ, S, L, A depending on the datum pΘ, Θ1, Dq +such that the following holds. +Suppose that c is an arc-length parameterized piecewise-geodesic path (finite or infinite) in +X obtained by concatenating geodesic segments xixi`1 satisfying for all i: +1. Each segment xixi`1 is Θ-regular and has length ě S. +2. +=ζ +xipxi´1, xi`1q ě π ´ ǫ. +Then the path c is pΘ1, D, L, Aq-Morse. +3.6 +Local Morse quasigeodesics +According to Theorem 3.30, sufficiently straight and spaced straight piecewise-geodesic paths +are Morse. +In this section we will now prove that, conversely, the Morse property implies +straightness in a suitable sense, namely that for sufficiently spaced quadruples the associated +midpoint triples are arbitrarily straight. (For the quadruples themselves this is in general not +true.) +Definition 3.31 (Quadruple condition). For points x, y P X we let midpx, yq denote the +midpoint of the geodesic segment xy. A map p : I Ñ X satisfies the pΘ, ǫ, l, sq-quadruple +condition if for all t1, t2, t3, t4 P I with t2 ´ t1, t3 ´ t2, t4 ´ t3 ě s the triple of midpoints +pmidpt1, t2q, midpt2, t3q, midpt3, t4qq +16 + +is pΘ, ǫq-straight and l-spaced. +Proposition 3.32 (Morse implies quadruple condition). For L, A, Θ, Θ1, D, ǫ, l exists a +scale s “ spL, A, Θ, Θ1, D, ǫ, lq such that every pΘ, D, L, Aq-Morse quasigeodesic satisfies the +pΘ1, ǫ, l, s1q-quadruple condition for every s1 ě s. +Proof. Let p : I Ñ X be an pL, A, Θ, Dq-Morse quasigeodesic, and let t1, . . . , t4 P I such that +t2 ´ t1, t3 ´ t2, t4 ´ t3 ě s. We abbreviate pi :“ pptiq and mi “ midppi, pi`1q. +Regarding straightness, it suffices to show that the segment m2m1 is Θ1-regular and that +=ζ +m2pp2, m1q ď ǫ +2 provided that s is sufficiently large in terms of the given data. +By the Morse property, there exists a diamond ♦Θpx1, x3q such that dpx1, p1q, dpx3, p3q ď D +and p2 P NDp♦Θpx1, x3qq. The diamond spans a unique parallel set Ppτ´, τ`q. (Necessarily, +x3 P V px1, stΘpτ`qq and x1 P V px3, stΘpτ´qq.) +We denote by ¯pi and ¯mi the projections of pi and mi to the parallel set. +We first observe that m2 (and m3) is arbitrarily close to the parallel set if s is large enough. +If this were not true, a limiting argument would produce a geodesic line at strictly positive +finite Hausdorff distance P p0, Ds from Ppτ´, τ`q and asymptotic to ideal points in stΘpτ˘q. +However, all lines asymptotic to ideal points in stΘpτ˘q are contained in Ppτ´, τ`q. +Next, we look at the directions of the segments ¯m2 ¯m1 and ¯m2¯p2 and show that they +have the same τ-direction. +Since ¯p2 is 2D-close to V p¯p1, stΘpτ`qq, we have that the point +¯p1 is 2D-close to V p¯p2, stΘpτ´qq, and hence also ¯m1 is 2D-close to V p¯p2, stΘpτ´qq. +There- +fore, ¯p1, ¯m1 P V p¯p2, stΘ1pτ´qq if s is large enough. Similarly, ¯m2 P V p¯p2, stΘ1pτ`qq and hence +¯p2 P V p ¯m2, stΘ1pτ´qq. The convexity of Θ1-cones, see Proposition 2.1, implies that also ¯m1 P +V p ¯m2, stΘ1pτ´qq. In particular, =ζ +¯m2p¯p2, ¯m1q “ 0 if s is sufficiently large. +Since m2 is arbitrarily close to the parallel set if s is sufficiently large, it follows by another +limiting argument that =ζ +m2pp2, m1q ď ǫ +2 if s is sufficiently large. +Regarding the spacing, we use that ¯m1 P V p¯p2, stΘ1pτ´qq and ¯m2 P V p¯p2, stΘ1pτ`qq. It follows +that +dp ¯m1, ¯m2q ě c ¨ pdp ¯m1, ¯p2q ` dp¯p2, ¯m2qq +with a constant c “ cpΘ1q ą 0, and hence that dpm1, m2q ě l if s is sufficiently large. +Theorem 3.18 and Proposition 3.32 tell that the Morse property for quasigeodesics is equiv- +alent to straightness (of associated spaced sequences of points). Since straightness is a local +condition, this leads to a local to global result for Morse quasigeodesics, namely that the Morse +property holds globally if it holds locally up to a sufficiently large scale. +Definition 3.33 (Local Morse quasigeodesic). An S-local pΘ, D, L, Aq-Morse quasigeode- +sic in X is a map p : I Ñ X such that for all t0 the subpath p|rt0,t0`Ss is a pΘ, D, L, Aq-Morse +quasigeodesic. +Note that local Morse quasigeodesics are uniformly coarse Lipschitz. +17 + +Theorem 3.34 (Local-to-global principle for Morse quasigeodesics). For L, A, Θ, Θ1, D +exist S, L1, A1, D1 such that every S-local pΘ, D, L, Aq-local Morse quasigeodesic in X is an +pΘ1, D1, L1, A1q-Morse quasigeodesic. +Proof. We choose an auxiliary Weyl convex subset Θ2 such that Θ ă Θ2 ă Θ1. +Let p : I Ñ X be an S-local pΘ, D, L, Aq-local Morse quasigeodesic. We consider its coarsi- +fication on a (large) scale s and the associated midpoint sequence, i.e. we put ps +n “ ppnsq and +ms +n “ midpps +n, ps +n`1q. Whereas the coarsification itself does in general not become arbitrarily +straight as the scale s increases, this is true for its midpoint sequence due to Proposition 3.32. +We want it to be sufficiently straight and spaced so that we can apply to it the Morse Lemma +from Theorem 3.18. Therefore we first fix an auxiliary constant δ, and further auxiliary con- +stants l, ǫ as determined by Theorem 3.18 in terms of Θ1, Θ2 and δ. Then Proposition 3.32 +applied to the pΘ, D, L, Aq-Morse quasigeodesics p|rt0,t0`Ss yields that pms +nq is pΘ2, ǫq-straight +and l-spaced if S ě 3s and the scale s is large enough depending on L, A, Θ, Θ2, D, ǫ, l. +Now we can apply Theorem 3.18 to pms +nq. It yields a nearby sequence p ¯ms +nq, dp ¯ms +n, ms +nq ď δ, +with the following property: For all n1 ă n2 ă n3 the segments ¯ms +n1 ¯ms +n3 are uniformly regular +and the points ms +n2 are δ-close to the diamonds ♦Θ1p ¯ms +n1, ¯ms +n3q. +Since the subpaths p|rns,pn`1qss filling in pps +nq are pL, Aq-quasigeodesics (because S ě s), +and it follows that for all t1, t2 P I the subpaths p|rt1,t2s are D1-close to Θ1-diamonds with D1 +depending on L, A, s. +The conclusion of Theorem 3.18 also implies a global spacing for the sequence pms +nq, compare +Remark 3.20, i.e. dpms +n, ms +n1q ě c ¨ |n ´ n1| with a positive constant c depending on Θ1, l. Hence +p is a global pL1, A1q-quasigeodesic with L1, A1 depending on L, A, s, c. +Combining this information, we obtain that p is an pΘ1, D1, L1, A1q-Morse quasigeodesic for +certain constants L1, A1 and D1 depending on L, A, Θ, Θ1 and D, provided that the scale S is +sufficiently large in terms of the same data. +3.7 +Local-to-global principle for Morse maps +We now deduce from our local-to-global result for Morse quasigeodesics, Theorem 3.34, a local- +to-global result for Morse embeddings. +We restrict to the setting of maps of Gromov-hyperbolic pl, aq-quasigeodesic metric spaces +Z to symmetric spaces X. +Definition 3.35 (Local Morse embedding). We call a map f : Z Ñ X an S-local +pΘ, D, L, Aq-Morse map if for any pl, aq-quasigeodesic q : I Ñ Z defined on an interval I +of length ď S the image path f ˝ q is a pΘ, D, L, Aq-Morse quasigeodesic in X. +Theorem 3.36 (Local-to-global principle for Morse embeddings of Gromov hyper- +bolic spaces). For l, a, L, A, Θ, Θ1, D exists a scale S and a datum pD1, L1, A1q such that every +S-local pΘ, D, L, Aq-Morse embedding from an pl, aq-quasigeodesic Gromov hyperbolic space into +X is a pΘ1, D1, L1, A1q-Morse embedding. +18 + +Proof. Let f : Z Ñ X denote the local Morse embedding. It sends every pl, aq-quasigeodesic +q : I Ñ Z to an S-local pΘ, D, L, Aq-Morse quasigeodesic p “ f ˝ q in X. By Theorem 3.34, +p is pL1, A1, Θ1, D1q-Morse if S ě Spl, a, L, A, Θ, Θ1, Dq, where L1, A1, D1 depend on the given +data. +Below is a reformulation of this theorem in the case of geodesic Gromov-hyperbolic spaces. +Let Z be a δ-hyperbolic geodesic space. An R-ball Bpz, Rq in Z need not be convex, but +it is δ-quasiconvex. In particular, the restriction of the metric from Z to Bpz, Rq results in a +p1, δq-quasigeodesic metric space. +Theorem 3.37 (Local-to-global principle for Morse embeddings of geodesic spaces). +For L, A, Θ, Θ1, D, δ exists a scale R and a datum pD1, L1, A1q such that if Z is a δ-hyperbolic +geodesic metric space and the restriction of f to any R-ball is pΘ, D, L, A, 1, δq-Morse, then +f : Z Ñ X is pΘ1, D1, L1, A1q-Morse. +4 +Group-theoretic applications +As a consequence of the local-to-global criterion for Morse maps, in this section we establish +that the Morse property for isometric group actions is an open condition. Furthermore, for +two nearby Morse actions, the actions on their τmod-limit sets are also close, i.e. conjugate by +an equivariant homeomorphism close to identity. In view of the equivalence of Morse property +with the asymptotic properties discussed earlier, this implies structural stability for asymp- +totically embedded groups. Another corollary of the local-to-global result is the algorithmic +recognizability of Morse actions. +We conclude the section by illustrating our technique by constructing Morse-Schottky ac- +tions of free groups on higher rank symmetric spaces. +4.1 +Stability of Morse actions +We consider isometric actions Γ ñ X of finitely generated groups. +Definition 4.1 (Morse action). We call an action Γ ñ X Θ-Morse if one (any) orbit map +Γ Ñ Γx Ă X is a Θ-Morse embedding with respect to a(ny) word metric on Γ. We call an action +Γ ñ X τmod-Morse if it is Θ-Morse for some τmod-Weyl convex compact subset Θ Ă ostpτmodq. +Remark 4.2 (Morse actions are τmod-regular and undistorted). (i) It follows immedi- +ately from the definition of Morse quasigeodesics that Θ-Morse actions are τmod-regular for the +simplex type τmod determined by Θ. +(ii) Morse subgroups of G are undistorted in the sense that the orbit maps are quasi-isometric +embeddings. In [KL1] we prove that Morse subgroups of G satisfy a stronger property: They +are coarse Lipschitz retracts of G. This retraction property is stronger than nondistortion: +Every finitely generated subgroup which is a coarse retract of G is undistorted in G, but there +are examples of undistorted subgroups which are not coarse retracts. For instance, the group +19 + +Φ :“ F2 ˆ F2 admits an undistorted embedding in the isometry group of X “ H2 ˆ H2. On the +other hand, pick an epimorphism φ : F2 Ñ Z and define the subgroup Γ ă Φ as the kernel of +the homomorphism +pγ1, γ2q ÞÑ φpγ1q ´ φpγ2q. +Then Γ is a finitely generated undistorted subgroup of Φ (see e.g. [OS, Theorem 2]), but is not +finitely presented (see e.g. [BR]). Hence, Γ ă G “ IsompH2q ˆ IsompH2q is undistorted but is +not a coarse Lipschitz retract. +We denote by HomτmodpΓ, Gq Ă HompΓ, Gq the subset of τmod-Morse actions Γ ñ X. +By analogy with local Morse quasigeodesics, we define local Morse group actions ρ : Γ ñ X +of a hyperbolic group (with a fixed finite generating set): +Definition 4.3. An action ρ is called S-locally pΘ, D, L, Aq-locally Morse, or pΘ, D, L, Aq- +locally Morse on the scale S, with respect to a base-point x P X, if the orbit map Γ Ñ Γ¨x Ă X +induces an S-local pΘ, D, L, Aq-local Morse embedding of the Cayley graph of Γ. +According to our local-to-global result for Morse embeddings, see Theorem 3.37, an action +of a word hyperbolic group is Morse if and only if it is local Morse on a sufficiently large scale. +Since this is a finite condition, it follows that the Morse property is stable under perturbation +of the action: +Theorem 4.4 (Morse is open for word hyperbolic groups). For any word hyperbolic +group Γ the subset HomτmodpΓ, Gq is open in HompΓ, Gq. More precisely, if ρ P HomτmodpΓ, Gq is +M-Morse with respect to a base-point x P X then there exists a neighborhood of ρ in HompΓ, Gq +consisting entirely of M1-Morse representations with respect to x, where M1 depends only on +M. +Proof. Let ρ : Γ ñ X be a Morse action. We fix a word metric on Γ and a base point x P X. +Then there exist data M “ pL, A, Θ, Dq such that the orbit map Γ Ñ Γx Ă X extends to a +pΘ, D, L, Aq-Morse map of the Cayley graph Y on Γ. +We relax the Morse parameters slightly, i.e. we consider pL, A, Θ, Dq-Morse quasigeodesics +as pL, A ` 1, Θ, D ` 1q-Morse quasigeodesics satisfying strict inequalities. For every scale S, +the orbit map Γ Ñ Γx Ă X, defines an pL, A ` 1, Θ, D ` 1, Sq-local Morse embedding Y Ñ X. +Due to Γ-equivariance, this is a finite condition in the sense that it is equivalent to a condition +involving only finitely many orbit points. Since we relaxed the Morse parameters, the same +condition is satisfied by all actions sufficiently close to ρ. +Theorem 3.37 provides a scale S such that all S-local pΘ, D`1, L, A`1q-Morse embeddings +Y Ñ X are M1-Morse for some Morse datum M1 depending only on pL, A ` 1, Θ, D ` 1, Sq. It +follows that actions sufficiently close to ρ are τmod-Morse. +Corollary 4.5. For every hyperbolic group Γ the space of faithful Morse representations +Hominj,τmodpΓ, Gq +is open in HomτmodpΓ, Gq. +20 + +Proof. Every hyperbolic group Γ has the unique maximal finite normal subgroup Φ Ÿ Γ (if Γ +is nonelementary then Φ is the kernel of the action of Γ on B8Γ). Since Morse actions are +properly discontinuous, the kernel of every Morse representation Γ Ñ G is contained in Φ. +Since HompΦ, Gq{G is finite, it follows that the set of faithful Morse representations is open in +HomτmodpΓ, Gq. +The result on the openness of the Morse condition for actions of word hyperbolic groups, +cf. Theorem 4.4, can be strengthened in the sense that the asymptotics of Morse actions vary +continuously: +Theorem 4.6 (Morse actions are structurally stable). The boundary map at infinity of +a Morse action depends continuously on the action. +Proof. According to Theorem 4.4 nearby actions are uniformly Morse. The assertion there- +fore follows from the fact that the ends of Morse quasirays vary uniformly continuously, cf. +Lemma 3.13. +Remark 4.7. (i) Note that since the boundary maps at infinity are embeddings, the Γ-actions +on the τmod-limit sets are topologically conjugate to each other and, for nearby actions, by a +homeomorphism close to the identity. +(ii) In rank one, our argument yields a different proof for Sullivan’s Structural Stability +Theorem [Su] for convex cocompact group actions on rank one symmetric spaces. Other proofs +can be found in [La, GW] (for Anosov subgroups in higher rank), [Co, Iz, Bo] for rank one +symmetric spaces. +Our next goal is to extend the topological conjugation from the limit set to the domains +of proper discontinuity. Recall that in [KLP4] we constructed domains of proper discontinuity +and cocompactness for τmod-Morse group actions on flag-manifolds Flagpνmodq “ G{Pνmod. Such +domains depend on a certain auxiliary datum, a balanced thickening Th Ă W, which is a Wτmod- +left invariant subset satisfying certain conditions; see [KLP4, sect. 3.4]. Let νmod Ă σmod be an +ι-invariant face such that Th is invariant under the action of Wνmod via the right multiplication +(this is automatic if νmod “ σmod since Wσmod “ teu). +The thickening Th Ă W defines a +thickening ThpΛτmodpΓqq Ă Flagpνmodq. One of the main results of [KLP4] (Theorem 1.7) is +that each τmod-Morse subgroup Γ ă G acts properly discontinuously and cocompactly on +ΩThpΓq :“ Flagpνmodq ´ ThpΛτmodpΓqq. +Theorem 4.8 (Stability of Morse quotient spaces). Suppose that ρn : Γ Ñ ρnpΓq “ Γn ă +G is a sequence of faithful τmod-Morse representations converging to a τmod-Morse embedding +ρ : Γ ãÑ G. Then: +1. The sequence of thickenings ThpΛτmodpΓnqq Hausdorff-converges to ThpΛτmodpΓqq. +2. If γn P Γ is a divergent sequence, then, after extraction, the sequence pρnpγnqq flag- +converges to a point in ΛτmodpΓq. +21 + +3. There is a sequence of equivariant diffeomorphisms hn : ΩThpΓq Ñ ΩThpΓnq converging +to the identity map uniformly on compacts. +4. In particular, the quotient-orbifolds ΩThpΓnq{Γn are diffeomorphic to ΩThpΓq{Γ for all +sufficiently large n. +Proof. 1. First of all, suppose that a sequence τn P Flagpτmodq converges to τ P Flagpτmodq. +Then, since Flagpνmodq “ G{Pνmod, there is a sequence gn P G, gn Ñ e, such that gnpτq “ τn. +Since +gnpThpτqq “ Thpgnτq “ Thpτnq, +it follows that we have Hausdorff-convergence of subsets Thpτnq Ñ Thpτq. +Moreover, this +convergence of subsets is uniform: There exists n0 “ npδq such that if dpτn, τq ă δ for all +n ě n0 then dpThpτnq, Thpτqq ă ǫ “ ǫpδq for all n ě n0. Here ǫ Ñ 0 as δ Ñ 0. Since the +sequence of limit sets ΛτmodpΓnq Hausdorff-converges to ΛτmodpΓq, it follows that the sequence +of thickenings ThpΛτmodpΓnqq Hausdorff-converges to ThpΛτmodpΓqq. This proves (1). +2. Consider a sequence of geodesic rays eξn in the Cayley graph Y of Γ such that γn lies +in an R-neighborhood of eξn for all n. Then, in view of the uniform M1-Morse property for +the representations ρn, each point ρnpγnqpxq belongs to the D1-neighborhood of the Weyl cone +V px, stpτnqq, where τn “ αnpξnq, αn : B8Γ Ñ ΛτmodpΓnq is the asymptotic embedding. Thus, by +the definition of flag-convergence, the sequences pρnpγnqq and pτnq have the same flag-limit in +Flagpτmodq. By Part 1, the sequence pτnq subconverges to a point in ΛτmodpΓq. Hence, the same +holds for pρnpγnqq. +3. The proof of this part is mostly standard, see [Iz] in the case when X is a hyperbolic space. +The quotient orbifold O “ ΩThpΓq{Γ has a natural pF, Gq-structure where F “ Flagpνmodq. +The orbifold O has finitely many components, let Z be one of them and let ˆZ Ă ΩThpΓq be a +component projecting to Z. It suffices to construct maps hn on each component ˆZ and then +extend these maps to maps hn of ΩThpΓq by ρn-equivariance. +The covering map ˆZ Ñ Z induces an epimorphism φ : π1pZq Ñ ΓZ, where ΓZ is the Γ- +stabilizer of ˆZ. Let dev : ˜Z Ñ ˆZ Ă ΩThpΓq be the developing map, where ˜Z Ñ Z is the +universal covering. By Ehresmann-Thurston holonomy theorem (see [Lo], [CEG], [Go], [K1, +sect. 7.1]), for all sufficiently large n, the homomorphism φn :“ ρn ˝ φ is the holonomy of +an pF, Gq-structure on Z. +Moreover, the developing maps devn : ˜Z Ñ F converge to dev +uniformly on compacts in the C8-topology. Since π1p ˆZq is contained in the kernel of φ, it +is also in the kernel of φn. +Hence, the maps devn descend to maps y +devn : ˆZ Ñ F. +The +sequence y +devn still converges to the identity embedding ˆZ ãÑ F uniformly on compacts. Pick +a compact fundamental set C Ă ˆZ for the ΓZ-action, i.e. a compact subset whose Γ-orbit +equals ˆZ. In view of Part 1 of the theorem, y +devnpCq Ă ΩThpΓnq for all sufficiently large n. +Therefore, we can assume that y +devnp ˆZq is contained in a component ˆZn of ΩThpΓnq. By the +compactness of the quotient-orbifolds, y +devn projects to a finite-to-one (smooth) orbi-covering +map cn : Z Ñ Zn :“ ˆZn{ρnpΓZq. Hence, y +devn : ˆZ Ñ ˆZn is a covering map as well. If ˆZn +were simply-connected, it would follow that y +devn is a diffeomorphism as required (and this is +22 + +how Izeki concludes his proof in [Iz]). We will prove that y +devn is a diffeomorphism by a direct +argument. +Suppose that each y +devn is not injective. Then, by the equivariance of these maps, after +extraction, there exist convergent sequences zn Ñ z, z1 +n Ñ z1 in ˆZ and a sequence γn P Γ such +that +ρnpγnqy +devnpznq “ y +devnpz1 +nq, +γnpznq ‰ z1 +n. +If the sequence pγnq were contained in a finite subset of Γ we would obtain a contradiction with +the uniform convergence on compacts y +devn Ñ id on ˆZ. Hence, after extraction, we may assume +that pγnq is a divergent sequence. We, therefore, obtain a dynamical relation between the points +z, z1 via the sequence pρnpγnqq. According to Part 2, the sequence pρnpγnqq flag-accumulates to +ΛτmodpΓq. The dynamical relation then contradicts fatness of the balanced thickening Th, see +[KLP4, sect. 5.2] and the proof of Theorem 6.8 in [KLP4]. +We conclude that the maps +y +devn : ˆZ Ñ ˆZn +are diffeomorphisms for all sufficiently large n. Since ρn : Γ Ñ Γn are isomorphisms, equivari- +ance of the developing maps implies that the maps hn : ΩThpΓq Ñ ΩThpΓnq are diffeomor- +phisms for sufficiently large n. +4. This part is an immediate corollary of Part 3. +Remark 4.9. (i) In the case when X is a hyperbolic space, the equivariant diffeomorphism hn : +ΩpΓq Ñ ΩpΓnq combined with the equivariant homeomorphism of the limit sets ΛpΓq Ñ ΛpΓnq +yield an equivariant homeomorphism B8X Ñ B8X, see [Tu, Iz]. Such an extension does not +exist in higher rank since, in general, there is no equivariant homeomorphism of thickened limit +sets ThpΛτmodpΓqq Ñ ThpΛτmodpΓnqq. This can be already seen for group actions on products of +hyperbolic planes. +(ii) An analogue of Theorem 4.8 holds when we replace the group actions on flag-manifolds +with actions on Finsler compactifications of the symmetric space and replace flag-manifold +thickenings ThpΛτmodq with Finsler thickenings ThF :upΛτmodq Ă BF :uX. Proving this requires +extending Ehresmann–Thurston holonomy theorem to the category of smooth manifolds with +corners and we will not pursue it here. +4.2 +Schottky actions +In this section we apply our local-to-global result for straight sequences (Theorem 3.18) to con- +struct Morse actions of free groups, generalizing and sharpening1 Tits’s ping-pong construction. +We consider two oriented τmod-regular geodesic lines a, b in X. Let τ˘a, τ˘b P Flagpτmodq +denote the simplices which they are τ-asymptotic to, and let θ˘a, θ˘b P σmod denote the types +of their forward/backward ideal endpoints in B8X. (Note that θ´a “ ιpθaq and θ´b “ ιpθbq.) +Let Θ be a compact convex subset of ostpτmodq Ă σmod, which is invariant under ι. +1In the sense that we obtain free subgroups which are not only embedded, but also asymptotically embedded +in G. +23 + +Definition 4.10 (Generic pair of geodesics). We call the pair of geodesics pa, bq generic if +the four simplices τ˘a, τ˘b are pairwise opposite. +Let α, β P G be axial isometries with axes a and b respectively and translating in the positive +direction along these geodesics. Then τ˘a and τ˘b are the attractive/repulsive fixed points of +α and β on Flagpτmodq. +For every pair of numbers m, n P N we consider the representation of the free group in two +generators +ρm,n : F2 “ xA, By Ñ G +sending the generator A to αm and B to βn. We regard it as an isometric action ρm,n : F2 ñ X. +Definition 4.11 (Schottky subgroup). A τmod-Schottky subgroup of G is a free τmod-asymp- +totically embedded subgroup of G. +If G has rank one, this definition amounts to the requirement that Γ is convex cocompact +and free. Equivalently, this is a discrete finitely generated subgroup of G which contains no +nontrivial elliptic and parabolic elements and has totally disconnected limit set (see see [K1]). +We note that this definition essentially agrees with the standard definition of Schottky groups +in rank 1 Lie groups, provided one allows fundamental domains at infinity for such groups to +be bounded by pairwise disjoint compact submanifolds which need not be topological spheres, +see [K1] for the detailed discussion. +Theorem 4.12 (Morse Schottky actions). If the pair of geodesics pa, bq is generic and if +θ˘a, θ˘b P intpΘq, then the action ρm,n is Θ-Morse for sufficiently large m, n. Thus, such ρm,n +is injective and its image is a τmod-Schottky subgroup of G. +Remark 4.13. In particular, these actions are faithful and undistorted, compare Remark 4.2. +Proof. Let S “ tA˘1, B˘1u be the standard generating set. We consider the sequences pγkq +in F2 with the property that γ´1 +k γk`1 P S and γk`1 ‰ γk´1 for all k. They correspond to the +geodesic segments in the Cayley tree of F2 associated to S which connect vertices. +Let x P X be a base point. In view of Lemma 3.8 we must show that the corresponding +sequences pγkxq in the orbit F2 ¨x are uniformly Θ-Morse. (Meaning e.g. that the maps R Ñ X +sending the intervals rk, k ` 1q to the points γkx are uniform Θ-Morse quasigeodesics.) As +in the proof of Theorem 3.34 we will obtain this by applying our local to global result for +straight spaced sequences (Theorem 3.18) to the associated midpoint sequences. Note that the +sequences pγkxq themselves cannot expected to be straight. +Taking into account the Γ-action, the uniform straightness of all midpoint sequences depends +on the geometry of a finite configuration in the orbit. It is a consequence of the following fact. +Consider the midpoints y˘m of the segments xα˘mpxq and z˘n of the segments xβ˘npxq. +Lemma 4.14. For sufficiently large m, n the quadruple ty˘m, z˘nu is arbitrarily separated and +Θ-regular. Moreover, for any of the four points, the segments connecting it to the other three +points have arbitrarily small ζ-angles with the segment connecting it to x. +24 + +Proof. The four points are arbitrarily separated from each other and from x because the axes +a and b diverge from each other due to our genericity assumption. +By symmetry, it suffices to verify the rest of the assertion for the point ym, i.e. we show that +the segments ymy´m and ymzn are Θ-regular for large m, n and that limmÑ8 =ζ +ympx, y´mq “ 0 +and limn,mÑ8 =ζ +ympx, znq “ 0. +The orbit points α˘mx and the midpoints y˘m are contained in a tubular neighborhood of the +axis a. Therefore, the segments ymx and ymy´m are Θ-regular for large m and =ympx, y´mq Ñ 0. +This implies that also =ζ +ympx, y´mq Ñ 0. +To verify the assertion for pym, znq we use that, due to genericity, the simplices τa and τb +are opposite and we consider the parallel set P “ Ppτa, τbq. Since the geodesics a and b are +forward asymptotic to P, it follows that the points x, ym, zn have uniformly bounded distance +from P. We denote their projections to P by ¯x, ¯ym, ¯zn. +Let Θ2 Ă intpΘq be an auxiliary Weyl convex subset such that θ˘a, θ˘b P intpΘ2q. We have +that ¯ym P V p¯x, stΘ2pτaqq for large m because the points ym lie in a tubular neighborhood of +the ray with initial point ¯x and asymptotic to a. Similarly, ¯zn P V p¯x, stΘ2pτbqq for large n. It +follows that ¯x P V p¯ym, stΘ2pτbqq and, using the convexity of Θ-cones (Proposition 2.1), that +¯zn P V p¯ym, stΘ2pτbqq. +The cone V pym, stΘ2pτbqq is uniformly Hausdorff close to the cone V p¯ym, stΘ2pτbqq because +the Hausdorff distance of the cones is bounded by the distance dpym, ¯ymq of their tips. Hence +there exist points x1, z1 +n P V pym, stΘ2pτbqq uniformly close to x, zn. Since dpym, x1q, dpym, z1 +nq Ñ +8 as m, n Ñ 8, it follows that the segments ymx and ymzn are Θ-regular for large m, n. +Furthermore, since =ζ +ympx1, z1 +nq “ 0 and =ympx, x1q Ñ 0 as well as =ympzn, z1 +nq Ñ 0, it follows +that =ζ +ympx, znq Ñ 0. +Proof of Theorem concluded. The lemma implies that for any given l, ǫ the midpoint triples +of the four point sequences pγkxq are pΘ, ǫq-straight and l-spaced if m, n are sufficiently large, +compare the quadruple condition (Definition 3.31). This means that the midpoint sequences of +all sequences pγkxq are pΘ, ǫq-straight and l-spaced for large m, n. Theorem 3.18 then implies +that the sequences pγkxq are uniformly Θ-Morse. +Remark 4.15. 1. Generalizing the above argument to free groups with finitely many gener- +ators, one can construct Morse Schottky subgroups for which the set θpΛq Ă σmod of types of +limit points is arbitrarily Hausdorff close to a given ι-invariant Weyl convex subset Θ. This +provides an alternative approach to the second main theorem in [Be] using coarse geometric +arguments. +2. In [DKL] Theorem 4.12 was generalized (by arguments similar to the its proof) to free +products of Morse subgroups of G. +4.3 +Algorithmic recognition of Morse actions +In this section, we describe an algorithm which has an isometric action ρ : Γ ñ X and a point +x P X as its input and terminates if and only if the action ρ is Morse (otherwise, the algorithm +25 + +runs forever). +We begin by describing briefly the Riley’s algorithm (see [Ri]) accomplishing a similar task, +namely, detecting geometrically finite actions on X “ H3. Suppose that we are given a finite +(symmetric) set of generators g1 “ 1, . . . , gm of a subgroup Γ Ă POp3, 1q and a base-point +x P X “ H3. The idea of the algorithm is to construct a finite sided Dirichlet fundamental +domain D for Γ (with the center at x): Every geometrically finite subgroup of POp3, 1q admits +such a domain. (The latter is false for geometrically finite subgroups of POpn, 1q, n ě 4, but is, +nevertheless, true for convex cocompact subgroups.) Given a finite sided convex fundamental +domain, one concludes that Γ is geometrically finite. Here is how the algorithm works: For each +k define the subset Sk Ă Γ represented by words of length ď k in the letters g1, . . . , gm. For +each g P Sk consider the half-space Bispx, gpxqq Ă X bounded by the bisector of the segment +xgpxq and containing the point x. Then compute the intersection +Dk “ +č +gPSk +Bispx, gpxqq. +Check if Dk satisfies the conditions of the Poincar´e’s Fundamental Domain theorem. If it does, +then D “ Dk is a finite sided fundamental domain of Γ. If not, increase k by 1 and repeat the +process. Clearly, this process terminates if and only if Γ is geometrically finite. +One can enhance the algorithm in order to detect if a geometrically finite group is convex +cocompact. Namely, after a Dirichlet domain D is constructed, one checks for the following: +1. If the ideal boundary of a Dirichlet domain D has isolated ideal points (they would +correspond to rank two cusps which are not allowed in convex cocompact groups). +2. If the ideal boundary of D contains tangent circular arcs with points of tangency fixed +by parabolic elements (coming from the “ideal vertex cycles”). Such points correspond to rank +1 cusps, which again are not allowed in convex cocompact groups. +Checking 1 and 2 is a finite process; after its completion, one concludes that Γ is convex +cocompact. +We refer the reader to [Gi1, Gi2, GiM, K2] and [KL2, sect. 1.8] for more details concerning +discreteness algorithms for groups acting on hyperbolic planes and hyperbolic 3-spaces. +We now consider group actions on general symmetric spaces. Let Γ be a hyperbolic group +with a fixed finite (symmetric) generating set; we equip the group Γ with the word metric +determined by this generating set. +For each n, let Ln denote the set of maps q : r0, 3ns X Z Ñ Γ which are restrictions of +geodesics ˜q : Z Ñ Γ, such that qp0q “ 1 P Γ. In view of the geodesic automatic structure on Γ +(see e.g. [Ep, Theorem 3.4.5]), the set Ln can be described via a finite state automaton. +Suppose that ρ : Γ ñ X is an isometric action on a symmetric space X; we fix a base-point +x P X and the corresponding orbit map f : Γ Ñ Γx Ă X. We also fix an ι-invariant face τmod +of the model spherical simplex σmod of X. The algorithm that we are about to describe will +detect that the action ρ is τmod-Morse. +26 + +Remark 4.16. If the face τmod is not fixed in advance, we would run algorithms for each face +τmod in parallel. +For the algorithm we will be using a special (countable) increasing family of Weyl convex +compact subsets Θ “ Θi Ă ostpτmodq Ă σmod which exhausts ostpτmodq; in particular, every +compact ι-invariant convex subset of ostpτmodq Ă σmod is contained in some Θi: +Θi :“ tv P σ : +min +αPΦτmod +αpvq ě 1 +i u, +(4.17) +where Φτmod is the subset of the set of simple roots Φ (with respect to σmod) which vanish on +the face τmod. Clearly, the sets Θi satisfy the required properties. Furthermore, we consider +only those L and D which are natural numbers. +Next, consider the sequence +pLi, Θi, Diq “ pi, Θi, Diq, i P N. +In order to detect τmod-Morse actions we will use the local characterization of Morse quasi- +geodesics given by Theorem 3.18 and Proposition 3.32. Due to the discrete nature of quasi- +geodesics that we will be considering, it suffices to assume that the additive quasi-isometry +constant A is zero. +Consider the functions +lpΘ, Θ1, δq, ǫpΘ, Θ1, δq +as in Theorem 3.18. Using these functions, for the sets Θ “ Θi, Θ1 “ Θi`1 and the constant +δ “ 1 we define the numbers +li “ lpΘ, Θ1, δq, ǫi “ ǫpΘ, Θ1, δq. +Next, for the numbers L “ Li, D “ Di and the sets Θ “ Θi, Θ1 “ Θi`1, consider the +numbers +si “ spLi, 0, Θi, Θi`1, Di, ǫi`1, li`1q +as in Proposition 3.32. According to this proposition, every pLi, 0, Θi, Diq-Morse quasigeodesic +satisfies the pΘi`1, ǫi`1, li`1, sq-quadruple condition for all s ě si. We note that, a priori, the +sequence si need not be increasing. +We set S1 “ s1 and define a monotonic sequence Si +recursively by +Si`1 “ maxpSi, si`1q. +Then every pΘi, Di, Li, 0q-Morse quasigeodesic also satisfies the pΘi`1, ǫi`1, li`1, Si`1q-quadruple +condition. +We are now ready to describe the algorithm. For each i P N we compute the numbers +li, ǫi and, then, Si, as above. We then consider finite discrete paths in Γ, q P LSi, and the +corresponding discrete paths in X, pptq “ qptqx, t P r0, 3Sis X Z. The number of paths q (and, +hence, p) for each i is finite, bounded by the growth function of the group Γ. +27 + +For each discrete path p we check the pΘi, ǫi, li, Siq-quadruple condition. If for some i “ i˚, +all paths p satisfy this condition, the algorithm terminates: It follows from Theorem 3.18 that +the map f sends all normalized discrete biinfinite geodesics in Γ to Morse quasigeodesics in +X. Hence, the action Γ ñ X is Morse in this case. Conversely, suppose that the action of Γ +is pΘ, D, L, 0q-Morse. Then f sends all isomeric embeddings ˜q : Z Ñ Γ to pΘ, D, L, 0q-Morse +quasigeodesics ˜p in X. In view of the properties of the sequence +pLi, Θi, Diq, +it follows that for some i, +pL, Θ, Dq ď pLi, Θi, Diq, +i.e., L ď Li, Θ Ă Θi, D ď Di; hence, all the biinfinite discrete paths ˜p are pΘi, Di, Li, 0q- +Morse quasigeodesic. By the definition of the numbers li, ǫi, Si, it then follows that all the +discrete paths p “ f ˝ q, q P LSi satisfy the pΘi`1, ǫi`1, li`1, Si`1q-quadruple condition. Thus, +the algorithm will terminate at the step i ` 1 in this case. +Therefore, the algorithm terminates if and only if the action is Morse (for some parameters). +If the action is not Morse, the algorithm will run forever. +Remark 4.18. Applied to a rank one symmetric space X and a hyperbolic group Γ without +a nontrivial normal finite subgroup, the above algorithm verifies if the given representation +ρ : Γ Ñ IsompXq is faithful with convex-cocompact image. We could not find this result in the +existing literature; cf. however [GK]. +5 +Appendix: Further properties of Morse quasigeodesics +This is the only part of the paper not contained in [KLP1]. Here we collect various properties +of Morse quasigeodesics that we found to be useful elsewhere in our work. +5.1 +Finsler geometry of symmetric spaces +In [KL1], see also [KLP5], we considered a certain class of G-invariant “polyhedral” Finsler +metrics on X. Their geometric and asymptotic properties turned out to be well adapted to +the study of geometric and dynamical properties of regular subgroups. They provide a Finsler +geodesic combing of X which is, in many ways, more suitable for analyzing the asymptotic +geometry of X than the geodesic combing given by the standard Riemannian metric on X. +These Finsler metrics also play a basic role in the present paper. We briefly recall their definition +and some basic properties, and refer to [KL1, §5.1] for more details. +Let ¯θ P intpτmodq be a type spanning the face type τmod. The ¯θ-Finsler distance d +¯θ on X is +the G-invariant pseudo-metric defined by +d +¯θpx, yq :“ max +θpξq“¯θ +` +bξpxq ´ bξpyq +˘ +28 + +for x, y P X, where the maximum is taken over all ideal points ξ P B8X with type θpξq “ ¯θ. +It is positive, i.e. a (non-symmetric) metric, if and only if the radius of σmod with respect to ¯θ +is ă π +2. This is in turn equivalent to ¯θ not being contained in a factor of a nontrivial spherical +join decomposition of σmod, and is always satisfied e.g. if X is irreducible. +If d¯θ is positive, it is equivalent to the Riemannian metric. In general, if it is only a pseudo- +metric, it is still equivalent to the Riemannian metric d on uniformly regular pairs of points. +More precisely, if the pair of points x, y is Θ-regular, then +L´1dpx, yq ď d +¯θpx, yq ď Ldpx, yq +with a constant L “ LpΘq ě 1. +Regarding symmetry of the Finsler distance, one has the identity +dι¯θpy, xq “ d +¯θpx, yq +and hence d¯θ is symmetric if and only if ι¯θ “ ¯θ. We refer to d¯θ as a Finsler metric of type τmod. +The d¯θ-balls in X are convex but not strictly convex. (Their intersections with flats through +their centers are polyhedra.) Accordingly, d +¯θ-geodesics connecting two given points x, y are not +unique. To simplify notation, xy will stand for some d +¯θ-geodesic connecting x and y. The union +of all d¯θ-geodesic xy equals the τmod-diamond ♦τmodpx, yq, that is, a point lies on a d¯θ-geodesic +xy if and only if it is contained in ♦τmodpx, yq, see [KLP5]. Finsler geometry thus provides an +alternative description of diamonds. Note that with this description, the diamond ♦τmodpx, yq +is also defined when the segment xy is not τmod-regular. Such a degenerate τmod-diamond is +contained in a smaller totally-geodesic subspace, namely in the intersection of all τmod-parallel +sets containing the points x, y. The description of geodesics and diamonds also implies that the +unparameterized d +¯θ-geodesics depend only on the face type τmod, and not on ¯θ. We will refer to +d¯θ-geodesics as τmod-Finsler geodesics. Note that Riemannian geodesics are Finsler geodesics. +We will call a Θ-regular τmod-Finsler geodesic a Θ-Finsler geodesic. If xy is a Θ-regular (Rie- +mannian) segment, then the union of Θ-Finsler geodesics xy equals the Θ-diamond ♦Θpx, yq. +Every τmod-Finsler ray in X is contained in a τmod-Weyl cone, and we will use the notation +xτ for a τmod-Finsler ray contained V px, stpτqq. Similarly, every τmod-Finsler line is contained +in a τmod-parallel set, and we denote by τ´τ` an oriented τmod-Finsler line forward/backward +asymptotic to two antipodal simplices τ˘ P Flagpτmodq and contained in Ppτ´, τ`q. +Examples of Θ-regular Finsler geodesics can be obtained as follows. Let pxiq be a (finite +or infinite) sequence contained in a parallel set Ppτ´, τ`q such that each Riemannian segment +xixi`1 is τ`-longitudinal and Θ1-regular. Then the concatenation of these geodesic segments is +Conversely, every Θ-regular Finsler geodesic c : I Ñ X can be approximated by a piecewise- +Riemannian Finsler geodesic c1: Pick a number s ą 0 and consider a maximal s-separated +subset J Ă I. Then take c1 to be the concatenation of Riemannian geodesic segments cpiqcpjq +for consecutive pairs i, j P J. In view of this approximation procedure, the String of Diamonds +Theorem (Theorem 3.30) holds if instead of Riemannian geodesic segments xixi`1 we allow +Θ-regular Finsler segments. +29 + +5.2 +Stability of diamonds +Diamonds can be regarded as Finsler-geometric replacements of geodesic segments in nonposi- +tively curved symmetric spaces of higher rank. +Riemannian geodesic segments in Hadamard manifolds (and, more generally, CATp0q metric +spaces) depend uniformly continuously on their tips: By convexity of the distance function we +have, +dHauspxy, x1y1q ď maxpdpx, x1q, dpy, y1qq. +In [KLP2, Prop. 3.70] we proved that diamonds ♦τmod depend continuously on their tips. +Below we establish uniform control on how much sufficiently large Θ-diamonds vary with +their tips. +Lemma 5.1. For d1 ą d ą 0 there exists C “ CpΘ, Θ1, d, d1q such that the following holds: +If a segment x´x` Ă X is Θ-regular with length ě C and y˘ P Bpx˘, dq, then the segment +y´y` is Θ1-regular and ♦Θpx´, x`q Ă Nd1p♦Θ1py´, y`qq. +Proof. The Θ1-regularity of y´y` for sufficiently large C follows from the ∆-triangle inequality. +Suppose that there exists no constant C for which also the second assertion holds. Then +there are sequences of points x˘ +n with dpx´ +n , x` +n q Ñ `8, y˘ +n with dpx˘ +n , y˘ +n q ď d, xn P ♦Θpx´ +n , x` +n q +and yn P ♦Θ1py´ +n , y` +n q with dpxn, ♦Θ1py´ +n , y` +n qq “ dpxn, ynq “ d1. We may assume convergence +xn Ñ x8 and yn Ñ y8 in X. +After extraction, at least one of the sequences px˘ +n q diverges. There are two cases to consider. +Suppose first that both sequences px˘ +n q diverge. Then they are uniformly τmod-regular and, +after extraction, we have τmod-flag convergence x˘ +n , y˘ +n Ñ τ˘ P Flagpτmodq. The limit simplices +τ˘ are antipodal (because xn Ñ x8). We observe that +dpxn, B♦Θ1px´ +n , x` +n qq, dpyn, B♦Θ1py´ +n , y` +n qq Ñ `8. +It follows that the sequences of diamonds ♦Θ1px´ +n , x` +n q and ♦Θ1py´ +n , y` +n q both Hausdorff converge +to the τmod-parallel set P “ Ppτ´, τ`q. It holds that x8 P P because xn P ♦Θpx´ +n , x` +n q. On the +other hand, dpx8, Pq “ d1 because dpxn, ♦Θ1py´ +n , y` +n qq “ d1, a contradiction. +Second, suppose that only one of the sequences px˘ +n q diverges, say, after extraction, x´ +n Ñ x´ +8 +and y´ +n Ñ y´ +8 in X to limit points with dpx´ +8, y´ +8q ď d, and x` +n Ñ τ` P Flagpτmodq. Now the +distance of xn from the boundary of the Θ1-Weyl cone with tip x` +n and containing xn goes +to infinity and it follows that ♦Θ1px´ +n , x` +n q Ñ V px´ +8, stΘ1pτ`qq and, similarly, ♦Θ1py´ +n , y` +n q Ñ +V py´ +8, stΘ1pτ`qq. The asymptotic limit Weyl cones have Hausdorff distance dpx´ +8, y´ +8q. On the +other hand, x8 P V px´ +8, stΘ1pτ`qq and dpx8, V py´ +8, stΘ1pτ`qqq “ d1, again a contradiction. +This shows that also (ii) holds for sufficiently large C. +We reformulate this result in terms of Finsler geodesics: +Lemma 5.2. There exists C “ CpΘ, Θ1, d, d1q such that the following holds: If x´x` is a Θ- +Finsler geodesic in X with dpx´, x`q ě C and y˘ are points with dpy˘, x˘q ď d, then every +30 + +point x on x´x` lies within distance d1 of a point y on a Θ1-Finsler geodesic y´y`. +Note that we do not claim here that one can take the same Finsler geodesic y´y` for all +points x on x´x`. +We now apply this stabilty result to Morse quasigeodesics. One, somewhat annoying, feature +of the definition of Θ-Morse quasigeodesics p : I Ñ X is that pprt1, t2sq is not required to be +uniformly close to a Θ-diamond spanned by ppt1q, ppt2q. (One reason is because the segment +ppt1qppt2q need not be Θ-regular.) Nevertheless, Lemma 5.1 implies: +Lemma 5.3. For every Morse datum M “ pΘ, B, L, Aq and Θ1 ą Θ, there exists C “ CpM, Θ1q +and D1 such that whenever dpx1, x2q ě C, the segment x1x2 “ ppt1qppt2q is Θ1-regular and +pprt1, t2sq lies in the D1-neighborhood of the Θ1-diamond ♦Θ1px1, x2q. +5.3 +Finsler approximation of Morse quasigeodesics +The next theorem establishes that every (sufficiently long) Morse quasigeodesic is uniformly +close to a Finsler geodesic with the same end-points. In this theorem, for convenience of the +notation, we will be allowing Morse quasigeodesics p to be defined on closed intervals I in the +extended real line; this is just a shorthand for a map I1 “ I XR Ñ X such that, as I1 Q t Ñ ˘8, +pptq Ñ pp˘8q P Flagpτmodq. When we say that such maps p, c are within distance D1 from each +other, this simply means that their restrictions to I1 are within distance ď D1. +Theorem 5.4 (Finsler approximation theorem). For every Morse datum M “ pΘ, D, L, Aq, +Θ1 ą Θ, and a positive number S, there exist C “ CpM, Θ1, Sq, D1 “ D1pM, Θ1, Sq satisfying +the following. +Let p : I “ rt´, t`s Ñ X Y Flagpτmodq be a M-Morse quasigeodesic between the points +x˘ “ ppt˘q P X Y Flagpτmodq such that dpx´, x`q ě C. Then there exists a Θ1-Finsler geodesic +x´x` equipped with a monotonic parameterization c : I Ñ x´x` such that: +(a) The maps p, c : I Ñ X are within distance ď D1 from each other. +(b) x´x` is an S-spaced piecewise-Riemannian geodesic, i.e. the Riemannian length of each +Riemannian segments of x´x` is ě S. +Proof. We will prove this in the case when both x˘ are in X since the proofs when one or both +points x˘ are in Flagpτmodq are similar: One replaces diamonds with Weyl cones or parallel +sets. +By the definition of an M-Morse quasigeodesic, for all subintervals rs´, s`s Ă rt´, t`s, there +exists a Θ-diamond +♦Θpy1 +´, y1 +`q +whose D-neighborhood contains pprs´, s`sq, and for y˘ “ pps˘q, we have +dpy˘, y1 +˘q ď D. +31 + +Therefore, applying the first part of Lemma 5.1, we conclude that the Riemannian segment +y´y` is Θ1-regular provided that dpy´, y`q ě C1 “ C1pM, Θ1q. In view of the quasigeodesic +property of p, the last inequality follows from the separation condition +s` ´ s´ ě s “ spM, Θ1q. +This, of course, also applies to rs´, s`s “ rt´, t`s and, hence, using the second part of Lemma +5.1, we obtain +ppIq Ă ND +` +♦Θpx1 +´, x1 +`q +˘ +Ă ND`D1 p♦Θ1px´, x`qq , +where D1 “ D1pM, Θ1q. We let +¯y˘ P ♦1 :“ ♦Θ1px´, x`q “ V px´, stΘ1pτ`qq X V px`, stΘ1pτ´qq +denote the nearest-point projections of y˘ “ pps˘q. +As long as s` ´ s´ ě s1pM, Θ1q, the +Riemannian segments ¯y´¯y` are also Θ1-regular and have length ě S. Furthermore, as in the +proof of Proposition 3.32, we can choose s1 such that each segment ¯y´¯y` is τ`-longitudinal. +We assume, from now on, that t` ´ t´ ě s2pM, Θ1q, which is achieved by assuming that +L´1pdpx´, x`q ´ Aq ě s1pM, Θ1q. +Take a maximal s1-separated subset J Ă I containing t˘. For each j P J define the point +zj :“ ppjq P ♦1. +Then for all consecutive i, j P J, s1 ď |j ´ i| ď 2s1 we have +L´1s1 ´ pA ` 2D ` 2D1q ď dpzi, zjq ď 2Ls1 ` pA ` 2D ` 2D1q. +(5.5) +We then let c denote the concatenation of Riemannian segments zizj for consecutive i, j P J, +where we use the affine parameterization of ri, js Ñ zizj. Thus, c is a Θ1-Finsler geodesic. We +now take the smallest s2 ě s1pM, Θ1q satisfying +S ď L´1s2 ´ pA ` 2D ` 2D1q, +the inequalities (5.5) imply that c satisfies both requirements of the approximation theorem +with +D1 “ 2Ls2 ` pA ` 2D ` 2D1q ` pD ` D1q ` p2Ls2 ` Aq. +Remark 5.6. In the case when the domain of p is unbounded, one can prove a bit sharper +result, namely, one can take Θ1 “ Θ. Compare [KL3, sect. 6]. +5.4 +Altering Morse quasigeodesics +Below we consider certain modifications of M-Morse quasigeodesics p in X represented as +concatenations p “ p´ ‹ p0 ‹ p`, where x˘ are the end-points of p0, and y˘, x˘ are the end- +points of p˘. (As in the previous section, we will be allowing y˘ to be in X Y Flagpτmodq.) +32 + +These modifications will have the form p1 “ p1 +´ ‹ p1 +0 ‹ p1 +`, where p1 +˘ and p1 +0 are all Morse. We +will see that, under certain assumptions, the entire p1 is again Morse (for suitable Morse datum +M1). +We begin by analyzing extensions of p to biinfinite paths. +Lemma 5.7 (Extension lemma). Suppose that +p˘ Ă V˘ “ V px˘, stpτ˘qq. +Whenever y˘ is in X, we let c˘ be Θ-regular Finsler rays contained in V˘ and connecting y˘ +to τ˘. Then, for every Θ1 ą Θ, there exists a Morse datum M1 containing Θ1 such that the +concatenation +ˆp “ c´ ‹ p ‹ c` +is M1-Morse, provided that dpx˘, y˘q ě C “ CpM, Θ1q. +Proof. We fix an auxiliary subset Θ1 satisfying Θ ă Θ1 ă Θ1. We let S “ SpΘ1, Θ1, 1q, ǫ “ +ǫpΘ1, Θ1, 1q be constants as in the string of diamonds theorem (Theorem 3.30). +According to Theorem 5.4, there exists a Θ1-regular Finsler geodesic +¯c “ y´¯x´ ‹ ¯x´¯x` ‹ ¯x`y` +within distance D1 “ D1pM, Θ1, Sq from the path p, such that ¯c is the concatenation of segments +of length ě S and dpx˘, ¯x˘q ď D1. We let z˘y˘ denote the subsegments of ¯x˘y˘ containing +y˘. +Since dpx˘, ¯x˘q ď D1, for each ǫ ą 0 and a sufficiently large C1 “ C1pD1, Θ1q, the inequality +dpx˘, y˘q ě C1 implies +=ζ +y˘px˘, ¯x˘q ď ǫ. +Therefore, +=ζ +y˘pz˘, τ˘q ě π ´ ǫ +and, hence, the piecewise-geodesic path +ˆc “ c´ ‹ ¯c ‹ c` +is pΘ1, ǫq-straight and S-spaced. Hence, by Theorem 3.30, the concatenation ˆc is M1-Morse, +where M1 “ pΘ1, 1, L, Aq. Since the path ˆp is within distance D1 from ˆc, it is M1-Morse, where +M1 “ M1 ` D1. +The next lemma was proven in [DKL, Thm. 4.11] in the case when p, p1 are finite paths. +The proof in the case of (bi)infinite paths is the same and we omit it. +Lemma 5.8 (Replacement lemma). Suppose that p1 “ p1 +´ ‹p1 +0 ‹p1 +` is a concatenation of M- +Morse quasigeodesics in X, such that the end-points of p˘, p1 +˘ and p0, p1 +0 are the same. Then for +every Θ1 ą Θ there exists a Morse datum M1 containing Θ1 such that the path p1 is M1-Morse. +33 + +In the following lemmata we will modify the path p by altering p˘ and keeping p0 unchanged +or moving it by a small amount (“wiggling the head and the tail of p”). +Lemma 5.9 (Wiggle lemma, I). Suppose that the paths p˘, p1 +˘ are both infinite. We let p1 +˘ +be M-Morse quasigeodesics with finite terminal points x˘ and set p1 :“ p1 +´ ‹ p0 ‹ p1 +`. Given +Θ1 ą Θ there exists ǫ “ ǫpM, Θ1q ą 0 and a Morse datum M1 containing Θ1 such that if +µ :“ maxp=ζ +x˘pp1 +˘p˘8q, p˘p˘8qqq ă ǫ, +then p1 is M1-Morse. +Proof. We fix an auxiliary compact Weyl-convex subset Θ1 Ă ostpτmodq such that Θ ă Θ1 ă Θ1. +Set τ˘ “ p˘p˘8q, τ 1 +˘ “ p1 +˘p˘8q. +According to Lemma 5.8, there exists a Morse datum M1 containing Θ1 such that for any +Θ1-regular Finsler geodesic rays c˘ :“ x˘τ˘, the concatenation c´ ‹ p0 ‹ c` is M1-Morse. +Let M2 ą M1 ` 1 be a Morse datum containing Θ1 and let S ą 0 be such that if a path +q in X is S-locally M1 ` 1-Morse then q is M2-Morse (see Theorem 3.34). Let ǫ be such that +for x P X, τ, τ 1 P Flagpτmodq, if =ζ +xpτ, τ 1q ă ǫ then each Θ1-regular Finsler segment of length +ď S in V px, stpτ 1qq is within unit distance from a Θ1-regular Finsler segment of length ď S in +V px, stpτqq. We assume now that µ ă ǫ. +Since p1 +˘ are M-Morse rays, they are within distance D1 “ D1pM, Θ1q from Θ1-regular +Finsler rays c1 +˘ “ x˘τ 1 +˘ connecting x˘ and τ 1 +˘. Define a new path c1 :“ c1 +´ ‹ p0 ‹ c1 +`. +By our choice of ǫ, the Θ1-regular Finsler subsegment s1 +˘ “ x˘y1 +˘ of c1 +˘ of length S is +within unit distance from a Θ1-regular Finsler subsegment s˘ “ x˘y˘ of c˘ of length S, where +c˘ “ x˘τ˘ is a Θ1-Finsler geodesic connecting x˘ to τ˘. +The concatenation +s´ ‹ p0 ‹ s` +is M1-Morse, and, since c1 +˘ are Θ1-Finsler geodesic, the path c1 is S-locally M1 ` 1-Morse. By +our choice of S, the path c1 is M2-Morse. Since c1 is within distance D1 from p1, the path p1 is +M2 ` D1-Morse. Lastly, we set M1 :“ M2 ` D1. +We generalize this lemma by allowing finite Morse quasigeodesics. We continue with the +setting of Lemma 5.9; we now allow paths p˘ and p1 +˘ to be finite, connecting y˘, x˘ and y1 +˘, x˘ +respectively. +(Some of y˘, y1 +˘ might be in Flagpτmodq.) +However, we will assume that the +distances dpx˘, y˘q, dpx1 +˘, y˘q are sufficiently large, ě C. +Lemma 5.10 (Wiggle lemma, II). Given Θ1 ą Θ there exist C ě 0, ǫ ą 0 and a Morse +datum M1 containing Θ1 such that if +µ :“ maxp=ζ +x˘py1 +˘, y˘qq ă ǫ, +and +ν :“ minpdpx˘, y˘q, dpx˘, y1 +˘qq ě C +then p1 is M1-Morse. +34 + +Proof. Pick an auxiliary compact Weyl-convex subset Θ2, Θ ă Θ2 ă Θ1. +We define biinfinite geodesic extensions ˆp, ˆp1 as in Lemma 5.7, by extending (if necessary) +the paths p˘, p1 +˘ via Θ-Finsler geodesics y˘τ˘ and y1 +˘τ 1 +˘. According to Lemma 5.7, there exists +C ą 0and a Morse datum M2 (containing Θ2), both depending on M and Θ2, such that the path +ˆp is M2-Morse. The same lemma applied to the paths ˆp1 +˘ implies that they are also M2-Morse. +By the construction, +µ :“ =ζ +x˘py1 +˘, y˘q “ =ζ +x˘pτ 1 +˘, τ˘q. +Now, claim follows from Lemma 5.9. +Lastly, we prove a general Wiggle Lemma where we allow to perturb the entire path p. We +consider concatenations +p “ p´ ‹ p0 ‹ p`, +p1 “ p1 +´ ‹ p1 +0 ‹ p1 +` +of M-Morse quasigeodesics, where we assume that p0, p1 +0 are within distance D0 from each +other. The paths p˘ connect y˘, x˘ and p1 +˘ connect y1 +˘, x1 +˘. +Lemma 5.11 (Wiggle lemma, III). Given Θ1 ą Θ there exist C ě 0, ǫ ą 0 and a Morse +datum M1 containing Θ1 such that if +µ :“ maxp=ζ +x˘py1 +˘, y˘qq ă ǫ, +and +ν :“ minpdpx˘, y˘q, dpx1 +˘, y1 +˘qq ě C +then p1 is M1-Morse. +Proof. As before, we fix an auxiliary compact Weyl-convex subset Θ3, Θ ă Θ3 ă Θ1. Then p1 +˘ +are within distance D3 “ D3pM, Θ3q from Θ3-regular Finsler geodesics c˘ :“ y1 +˘x˘. We apply +Lemma 5.10 to the pair of paths +p, p2 :“ c´ ‹ p0 ‹ c`. +It follows that p2 is M3-Morse for some Morse datum M3 containing Θ1 provided that µ ď +ǫ “ ǫpM, Θ3, Θ1q and ν ě C “ CpM, Θ3, Θ1q. Since the paths p2 and p1 are wihin distance +D1 :“ maxpD0, D3q from each other, the path p1 is M1 :“ M3 ` D1-Morse. +References +[BR] +G. Baumslag, J. 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Tukia, On isomorphisms of geometrically finite Moebius groups, Mathematical Pub- +lications of IHES, Vol. 61 (1985), pp. 171–214. +Addresses: +M.K.: Department of Mathematics, +University of California, Davis +CA 95616, USA +email: kapovich@math.ucdavis.edu +B.L.: Mathematisches Institut +Universit¨at M¨unchen +manicures Theresienstr. 39 +D-80333, M¨unchen, Germany +email: b.l@lmu.de +J.P.: Departament de Matem`atiques, +Universitat Aut`onoma de Barcelona, +08193 Bellaterra, Spain +email: porti@mat.uab.cat +38 + diff --git a/iNE3T4oBgHgl3EQfggp5/content/tmp_files/load_file.txt b/iNE3T4oBgHgl3EQfggp5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..942427f2e72b010d3ad5ee48ee0951fbb5d769cc --- /dev/null +++ b/iNE3T4oBgHgl3EQfggp5/content/tmp_files/load_file.txt @@ -0,0 +1,1937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf,len=1936 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='04562v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='DG] 11 Jan 2023 Morse actions of discrete groups on symmetric spaces: Local-to-global principle Michael Kapovich, Bernhard Leeb, Joan Porti January 12, 2023 Abstract Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse “Schottky subgroups” of higher rank semisimple Lie groups via arguments not based on Tits’ ping-pong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Preliminaries 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 Basic notions of geometry of symmetric spaces .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 31 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 Altering Morse quasigeodesics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 32 1 Introduction This is a sequel to our paper [KLP5] and mostly consists of the material of section 7 of our ear- lier paper [KLP1] (the only additional material appears in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 and the appendix to the paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We recall that quasigeodesics in Gromov hyperbolic spaces can be recog- nized locally by looking at sufficiently large finite pieces, see [CDP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In our earlier papers [KLP4, KLP5, KLP2, KL1, KL2], for higher rank symmetric spaces X (of noncompact type) we introduced an analogue of hyperbolic quasigeodesics, which we call Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Morse quasigeodesics are defined relatively to a certain face τmod of the model spherical face σmod of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In addition to the quasiisometry constants L, A, τmod-Morse quasigeodesics come equipped with two other parameters, a positive number D and a Weyl-convex subset Θ of the open star of τmod in the modal spherical chamber σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In [KLP1, KLP5, KLP2] we also defined τmod-Morse maps Y Ñ X from Gromov-hyperbolic spaces to symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' These maps are defined by the property that they send geodesics to uniformly τmod-Morse quasigeodesics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' τmod-Morse quasigeodesics with a fixed set of parameters, pΘ, D, L, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The main result of this paper is a local characterization of Morse quasigeodesics in X: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 (Local-to-global principle for Morse quasigeodesics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For L, A, Θ, Θ1, D there exist S, L1, A1, D1 such that every S-local pΘ, D, L, Aq-local Morse quasigeodesic in X is a pΘ1, D1, L1, A1q–Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Here S-locality of a certain property of a map means that this property is satisfied for restrictions of this map to subintervals of length S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We refer to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Based on this principle, we prove in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7 a local-to-global principle for Morse maps from hyperbolic metric spaces to symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We prove several consequences of these local-to-global principles: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The structural stability of Morse subgroups of G, generalizing Sullivan’s Structural Sta- bility Theorem in rank one [Su] (see also [KKL] for a detailed proof);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' see Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2 While structural stability for Anosov subgroups was known earlier (Labourie and Guichard– Wienhard), our method is more general and applies to a wider class of discrete subgroups, see [KL4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 (Openness of the space of Morse actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For a word hyperbolic group Γ, the subset of τmod-Morse actions is open in HompΓ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 (Structural stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Γ be word hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then for τmod-Morse actions ρ : Γ ñ X, the boundary embedding αρ : B8Γ Ñ Flagpτmodq depends continuously on the action ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, actions sufficiently close to a faithful Morse action are again discrete and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We supplement this structural stability theorem with a stability theorem on domains of proper discontinuity, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The locality of the Morse property implies that Morse subgroups are algorithmically recognizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 (Semidecidability of Morse property of group actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Γ be word hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then there exists an algorithm whose inputs are homomorphisms ρ : Γ Ñ G (defined on generators of Γ) and which terminates if and only if ρ defines a τmod-Morse action Γ ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the action is not Morse, the algorithm runs forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that in view of [K2], there are no algorithms (in the sense of BSS computability) which would recognize if a representation Γ Ñ IsompH3q is not geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We illustrate our techniques by constructing Morse-Schottky actions of free groups on higher rank symmetric spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Unlike all previously known constructions, our proof does not rely on ping-pong arguments, but is purely geometric and proceeds by constructing equivariant quasi-isometric embeddings of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The key step is the observation that a certain local straightness property for sufficiently spaced sequences of points in the symmetric space implies the global Morse property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This observation is also at the heart of the proof of the local-to-global principle for Morse actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since [KLP1] was originally posted in 2014, several improvements on the material of section 7 of [KLP1] and, hence, of the present paper were made: (a) Different forms of Combination Theorems for Anosov subgroups were proven in [DKL, DK1, DK2] written in collaboration with Subhadip Dey by the 1st and the 2nd author and, subsequently, by the 1st author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The first one was a generalization of the technique in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 the present paper, but the other two generalizations are based on a form of the ping-pong argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (b) Explicit estimates in the local-to-global principle for Morse quasigeodesics and, hence, Morse embeddings, were obtained by Max Riestenberg in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Riestenberg’s estimates are based on replacing certain limiting arguments used in the present paper with differential-geometric and Lie-theoretic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3 Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The notions of Morse quasigeodesics and actions are discussed in detail in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In that section, among other things, we establish local-to-global principles for Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In section 4 we apply local-to-global principles to discrete subgroups of Lie groups: We show that Morse actions are structurally stable and algorithmically recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We also construct Morse-Schottky actions of free groups on symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In section 5 (the appendix to the paper) we prove further properties of Morse quasigeodesics that we found to be useful in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The first author was supported by NSF grants DMS-12-05312 and DMS-16-04241, by KIAS (the Korea Institute for Advanced Study) through the KIAS scholar program, and by a Simons Foundation Fellowship, grant number 391602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The last author was supported by grants Mineco MTM2012-34834 and AGAUR SGR2009-1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The three authors are also grateful to the GEAR grant which partially supported the IHP trimester in Winter of 2012 (DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures and representation varieties” (the GEAR Network), and to the Max Planck Institute for Mathematics in Bonn, where some of this work was done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 Basic notions of geometry of symmetric spaces Throughout the paper we will be using definitions, notations and results of our earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We refer the reader to our earlier papers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' [KLP4, KLP5, KLP2, KL1, KL2] for the vari- ous notions related to symmetric spaces, such as polyhedral Finsler metrics on symmetric spaces ([KL1]), the opposition involution ι of σmod, model faces τmod of σmod and the associated τmod-flag manifolds Flagpτmodq (sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 of [KLP5]), type map θ : B8X Ñ σmod, open Schu- bert cells Cpτq Ă Flagpτmodq (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 of [KLP5]), ∆-valued distances d∆ on X (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6 of [KLP5]), Θ-regular geodesic segments (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 of [KLP5]), parallel sets, stars, open stars and Θ-stars, stpτq, ostpτq, and stΘpτq, Weyl sectors V px, τq (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 of [KLP5]), Weyl cones V px, stpτqq and Θ-cones V px, stΘpτqq, diamonds ♦τmodpx, yq and Θ-diamonds ♦Θpx, yq (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5 of [KLP5]), τmod-regular sequences and groups (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 of [KLP5]), τmod-convergence subgroups, flag-convergence, the Finsler interpretation of flag-convergence (see [KL1, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2] and [KLP5]), τmod-limit sets ΛτmodpΓq Ă Flagpτmodq (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5 of [KLP5]), visual limit set (page 4 of [KLP5]), uniformly τmod-regular sequences and subgroups (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6 of [KLP5]), Morse subgroups (section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 of [KLP5]) and, more generally, Morse quasigeodesics and Morse maps (Definitions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='31, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='33 of [KLP2]), antipodal limit sets (Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 of [KLP5]) and antipodal maps to flag-manifolds (Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11 of [KLP2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In the paper we will be frequently using convexity of Θ-cones in X: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10 in [KLP5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every Weyl-convex subset Θ Ă stpτmodq, 4 for every x P X and τ P Flagpτmodq, the cone V px, stΘpτqq Ă X is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 Standing notation and conventions We will use the notation X for a symmetric space of noncompact type, G for a semisimple Lie group acting isometrically and transitively on X, and K is a maximal compact sub- group of G, so that X is diffeomorphic to G{K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will assume that G is commensurable with the isometry group IsompXq in the sense that we allow finite kernel and cokernel for the natural map G Ñ IsompXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, the image of G in IsompXq contains the identity component IsompXqo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We let τmod Ď σmod be a fixed ι-invariant face type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will use the notation xn f ÝÑ τ P Flagpτmodq for the flag-convergence of a τmod-regular sequence xn P X to a simplex τ P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will be using the notation Θ, Θ1 for an ι-invariant, compact, Weyl-convex (see Defini- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7 in [KLP5]) subset of the open star ostpτmodq Ă σmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will always assume that Θ ă Θ1, meaning that Θ Ă intpΘ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Constants L, A, D, ǫ, δ, l, a, s, S are meant to be always strictly positive and L ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 ζ-angles We fix as auxiliary datum a ι-invariant type ζ “ ζmod P intpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (We will omit the subscript in ζmod in order to avoid cumbersome notation for ζ-angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') For a simplex τ Ă B8X of type τmod, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' τ P Flagpτmodq, we define ζpτq P τ as the ideal point of type ζmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Given two such simplices τ˘ P Flagpτmodq and a point x P X, define the ζ-angles =ζ xpτ´, τ`q “ =ζ xpτ´, ξ`q :“ =xpξ´, ξ`q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2) where ξ˘ “ ζpτ˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Similarly, define the ζ-Tits angle =ζ Titspτ´, τ`q “ =ζ Titspτ´, ξ`q :“ =xpξ´, ξ`q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3) where x belongs to a flat F Ă X such that τ´, τ` Ă BTitsF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then simplices τ˘ (of the same type) are antipodal iff =ζ Titspτ´, τ`q “ π for some, equivalently, every, choice of ζ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We observe that the ideal points ζ˘ are opposite, =Titspζ´, ζ`q “ π, if and only if they can be seen under angle » π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=', close to π) from some point in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More precisely, there exists ǫpζmodq such that: 5 If =xpζ´, ζ`q ą π ´ ǫpζmodq for some point x then ζ˘ are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This follows from the angle comparison =xpζ´, ζ`q ď =Titspζ´, ζ`q and the fact that the Tits distance between ideal points of the fixed type ζmod takes only finitely many values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For a τmod-regular unit tangent vector v P TX we denote by τpvq Ă B8X the unique simplex of type τmod such that ray ρv with the initial direction v represents an ideal point in ostpτpvqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We put ζpvq “ ζpτpvqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that ζpvq depends continuously on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For a τmod-regular segment xy in X we let τpxyq “ τpvq, where v is the unit vector tangent to xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then, for a τmod-regular segments xy, xz and τ P Flagpτmodq, we define the ζ-angles =ζ xpy, τq “ =ζ xpτpxyq, τq, =ζ xpy, zq “ =ζ xpτpxyq, τpxzqq Thus, the ζ-angle depends not on y, z but rather on the simplices τpxyq, τpxzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' These ζ- angles will play the role of angles the between diamonds ♦τmodpx, yq and ♦τmodpx, zq, meeting at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that if X has rank 1, then the ζ-angles are just the ordinary Riemannian angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 Distances to parallel sets versus angles In this section we collect some geometric facts regarding parallel sets in symmetric spaces, primarily dealing with estimation of distances from points in X to parallel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The constants and functions in this section are not explicit and their existence is proven by compactness arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For explicit computations here and in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18, we refer the reader to the PhD thesis of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We first prove a lemma (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6) which strengthens Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='46 of [KLP5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that τ˘ are antipodal simplices in BTitsX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then every geodesic ray γ asymptotic to a point ξ P ostpτ`q, is strongly asymptotic to a geodesic ray in Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If ξ belongs to the interior of the simplex τ`, then the assertion follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='46 of [KLP5]: Weyl sectors V px1, τq and V px2, τq are strongly asymptotic if and only if x1 and x2 lie in the same horocycle at τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We now consider the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose, that ξ belongs to an open simplex intpτ 1q, such that τ is a face of τ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then there exists an apartment a Ă BTitsX containing both ξ (and, hence, τ 1 as well as τ) and the simplex τ´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let F Ă X be the maximal flat with B8F “ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then F contains a geodesic asymptotic to points in τ´ and τ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, F is contained in Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' On the other hand, by the same Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='46 of [KLP5], applied to the simplex τ 1, we conclude that γ is strongly asymptotic to a geodesic ray in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The following lemma provides a quantitative strengthening of the conclusion of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6: 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Θ be a compact subset of ostpτ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then those rays xξ with θpξq P Θ are uni- formly strongly asymptotic to Ppτ´, τ`q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' dp¨, Ppτ´, τ`qq decays to zero along them uniformly in terms of dpx, Ppτ´, τ`qq and Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that the assertion of lemma is false, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=', there exists ǫ ą 0, a sequence Ti P R` diverging to infinity, and a sequence of rays ρi “ xiξi with ξi P Θ and dpxi, Ppτ´, τ`qq ď d, so that dpy, Ppτ´, τ`qq ě ǫ, @y P ρpr0, Tisq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8) Using the action of the stabilizer of Ppτ´, τ`q, we can assume that the points xi belong to a certain compact subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, the sequence of rays xiξi subconverges to a ray xξ with dpx, Ppτ´, τ`qq ď d and ξ P Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8) then implies that the entire limit ray xξ is contained outside of the open ǫ-neighborhood of the parallel set Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' However, in view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6, the ray xξ is strongly asymptotic to a geodesic in Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We next relate distances from points x P X to parallel sets and the ζ-angles at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that the simplices τ˘, equivalently, the ideal points ζ˘ “ ζpτ˘q (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3), are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then =ζ xpτ´, τ`q “ =xpζ´, ζ`q “ π if and only if x lies in the parallel set Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, =ζ xpτ´, τ`q » π if and only if x is close to Ppτ´, τ`q, and both quantities control each other near the parallel set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More precisely: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (i) If dpx, Ppτ´, τ`qq ď d, then =ζ xpτ´, τ`q ě π ´ ǫpdq with ǫpdq Ñ 0 as d Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (ii) For sufficiently small ǫ, ǫ ď ǫ1pζmodq, we have: The inequality =ζ xpτ´, τ`q ě π´ǫ implies that dpx, Ppτ´, τ`qq ď dpǫq for some function dpǫq which converges to 0 as ǫ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The intersection of parabolic subgroups Pτ´ X Pτ` preserves the parallel set Ppτ´, τ`q and acts transitively on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Compactness and the continuity of =¨pζ´, ζ`q therefore imply that π ´ =¨pζ´, ζ`q attains on the boundary of the tubular r-neighborhood of Ppτ´, τ`q a strictly positive maximum and minimum, which we denote by φ1prq and φ2prq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, φiprq Ñ 0 as r Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We have the estimate: π ´ φ1pdpx, Ppτ´, τ`qqq ď =xpζ´, ζ`q ď π ´ φ2pdpx, Ppτ´, τ`qqq The functions φiprq are (weakly) monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This follows from the fact that, along rays asymptotic to ζ´ or ζ`, the angle =¨pζ´, ζ`q is monotonically increasing and the distance dp¨, Ppτ´, τ`qq is monotonically decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The estimate implies the assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The control of dp¨, Ppτ´, τ`qq and =¨pζ´, ζ`q “spreads” along the Weyl cone V px, stpτ`qq, since the latter is asymptotic to the parallel set Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Moreover, the control improves, if one enters the cone far into a τmod-regular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More precisely: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let y P V px, stΘpτ`qq be a point with dpx, yq ě l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (i) If dpx, Ppτ´, τ`qq ď d, then dpy, Ppτ´, τ`qq ď D1pd, Θ, lq ď d 7 with D1pd, Θ, lq Ñ 0 as l Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (ii) For sufficiently small ǫ, ǫ ď ǫ1pζmodq, we have: If =xpζ´, ζ`q ě π ´ ǫ, then =ypζ´, ζ`q ě π ´ ǫ1pǫ, Θ, lq ě π ´ ǫpdpǫqq with ǫ1pǫ, Θ, lq Ñ 0 as l Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The distance from Ppτ´, τ`q takes its maximum at the tip x of the cone V px, stpτ`qq, because it is monotonically decreasing along the rays xξ for ξ P stpτ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This yields the right- hand bounds d and, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9 twice, ǫpdpǫqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Those rays xξ with uniformly τmod-regular type θpξq P Θ are uniformly strongly asymptotic to Ppτ´, τ`q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' dp¨, Ppτ´, τ`qq decays to zero along them uniformly in terms of d and Θ, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This yields the decay D1pd, Θ, lq Ñ 0 as l Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The decay of ǫ1 follows by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3 Morse maps In this section we investigate the Morse property of sequences and maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The main aim of this section is to establish a local criterion for being Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' To do so we introduce a local notion of straightness for sequences of points in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Morse sequences are in general not straight, but they become straight after suitable modification, namely by sufficiently coarsifying them and then passing to the sequence of successive midpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Conversely, the key result is that sufficiently spaced straight sequences are Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We conclude that there is a local-to-global characterization of the Morse property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 Morse quasigeodesics Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 (Morse quasigeodesic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A pΘ, D, L, Aq-Morse quasigeodesic in X is an pL, Aq-quasigeodesic p : I Ñ X (defined on an interval I Ă R) such that for all t1, t2 P I the subpath p|rt1,t2s is D-close to a Θ-diamond ♦Θpx1, x2q with dpxi, pptiqq ď D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will refer to a quadruple pΘ, D, L, Aq as a Morse datum and abbreviate M “ pΘ, D, L, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Set M `D1 “ pΘ, D`D1, L, A`2D1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We say that M contains Θ if M has the form pΘ, D, L, Aq for some D ě 0, L ě 1, A ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The following lemma is immediate from the definiton of a M-Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 (Perturbation lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If p, p1 are paths in X such that p is M-Morse and dpp, p1q ď D1 then p1 is M ` D1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A Morse quasigeodesic p is called a Morse ray if its domain is a half-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If I “ R then a Morse quasigeodesic is called a Morse quasiline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 8 Morse quasirays do in general not converge at infinity (in the visual compactification of X), but they τmod-converge at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This is a consequence of: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 (Conicality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Every Morse quasiray p : r0, 8q Ñ X is uniformly Hausdorff close to a subset of a cone V ppp0q, stΘpτqq for a unique simplex τ of type τmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The subpaths p|r0,t0s are uniformly Hausdorff close to Θ-diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' These subconverge to a cone V px, stΘpτqq x uniformly close to pp0q and τ a simplex of type τmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This establishes the existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since ppnq f ÝÑ τ, the uniqueness of τ follows from the uniqueness of τmod-limits, see [KLP5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 (End of Morse quasiray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call the unique simplex given by the previous lemma the end of the Morse quasiray p : r0, 8q Ñ X and denote it by pp`8q P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hausdorff close Morse quasirays have the same end by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 we will prove uniform continuity of ends of Morse quasirays with respect to the topology of coarse convergence of quasirays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 Morse maps We now turn to Morse maps with more general domains (than just intervals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Y be a Gromov-hyperbolic geodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A map f : Y Ñ X is called M-Morse if it sends geodesics in Y to M-Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, every Morse map is a quasiisometric embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' While this definition makes sense for general metric spaces, in [KLP2] we proved that the domain of a Morse map is necessarily hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More generally, one can define Morse maps on quasigeodesic metric spaces: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6 (Quasigeodesic metric space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A metric space Z is called pl, aq-quasigeodesic if all pairs of points in Y can be connected by pl, aq-quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A space is called quasi- geodesic if it is pl, aq-quasigeodesic for some pair of parameters l, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Every quasigeodesic space is quasiisometric to a geodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Namely, if Z is pλ, αq- quasigeodesic space then it is quasiisometric to its pλ ` αq-Rips complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The quasigeodesic spaces considered in this paper are discrete groups equipped with word metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7 (Morse embedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let pΘ, D, L, Aq be a Morse datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' An pΘ, D, L, A, l, aq-Morse embedding (or a map) from an pl, aq-quasigeodesic space Z into X is a map f : Z Ñ X which sends pl, aq-quasigeodesics in Z to pΘ, D, L, Aq-Morse quasigeodesics in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 9 Of course, every pl, aq-quasigeodesic metric space is also pl1, a1q-quasigeodesic space for any l1 ě l, a1 ě a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The next lemma shows that this choice of quasigeodesic constants is essentially irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let f : Z Ñ X be a map from a Gromov-hyperbolic pl, aq-quasigeodesic space Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If f is M “ pΘ, D, L, A, l, aq-Morse then for any pl1, a1q, it sends pl1, a1q-quasigeodesics in Z to M1 “ pΘ, D1, L1, A1q-Morse quasigeodesics in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Here the datum M1 depends only on M, l1, a1 and the hyperbolicity constant δ of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This is a consequence of the definition of Morse quasigeodesics, and the Morse Lemma applied to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Notice that the parameter Θ in the Morse datum M1 is the same as in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, we arrive to Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A map f : Z Ñ X of a quasigeodesic hyperbolic space Z is called Θ-Morse if it sends uniform quasigeodesics in Z to Θ-Morse uniform quasigeodesics in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This notion depends only on the quasi-isometry class of Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' the precomposition of a Θ-Morse embedding with a quasi-isometry is again Θ-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For this to be true we have to require control on the images of quasigeodesics of arbitrarily bad (but uniform) quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Γ be a hyperbolic group with fixed a finite generating set S, and let Y be the Cayley graph of Γ with respect to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For x P X, an isometric action Γ ñ X determines the orbit map ox : Γ Ñ Γx Ă X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Every such map extends to the Cayley graph Y of Γ, sending edges to geodesics in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' An isometric action Γ ñ X or a representation ρ : Γ Ñ G, is called M-Morse (with respect to a base-point x P X) if the (extended) orbit map ox : Y Ñ X is M-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Similarly, a subgroup Γ ă G is Morse if the inclusion homomorphism Γ ãÑ G is Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The Morse property of an action and the parameter Θ, of course, does not depend on the choice of a generating set of Γ and a base-point x, but the triple pD, L, Aq does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, it makes sense to talk about a Θ-Morse and τmod-Morse actions of hyperbolic groups, where Θ Ă ostpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In [KLP5, KLP2, KL1] we gave many alternative definitions of Morse actions, including the equivalence of this definition to the notion of Anosov subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 Continuity at infinity Let X, Y be proper metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We fix a base point y P Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A sequence of maps fn : Y Ñ X is said to coarsely converge to a map f : Y Ñ X if there exists C ă 8 such that for every R there exists N “ NpC, Rq for which dpfn|B, f|Bq ď C, where B “ Bpy, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 10 Note the difference of this definition with the notion of uniform convergence on compacts: Since we are working in the coarse setting, requiring the distance between maps to be less than ǫ close to zero is pointless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of the Arzela–Ascoli theorem, the space of pL, Aq-coarse Lipschitz maps Y Ñ X sending y to a fixed bounded subset of X, is coarsely sequentially compact: Every sequence contains a coarsely converging subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In the next lemma we assume that Y is a geodesic δ-hyperbolic space and X is a symmetric space of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The lemma itself is an immediate consequence of the perturbation lemma, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that pn : R` Ñ X is a sequence of M-Morse rays which coarsely converges to a map p : R` Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then p is M1-Morse, where M1 “ M ` C and the constant C is the one appearing in the definition of coarse convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, a coarse limit of a sequence of (uniformly) Morse quasigeodesics is again Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For the next lemma, we equip the flag manifold F “ Flagpτmodq with some background metric dF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that pn : R` Ñ X is a sequence of M-Morse rays coarsely converging to a M-Morse ray p : R` Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then the sequence τn :“ pnp8q of ends of the quasirays pn converges to τ “ pp8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Moreover, the latter convergence is uniform in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every ǫ ą 0 there exists n0 depending only on M and C and NpR, Cq (appearing in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11) such that for all n ě n0, dFpτn, τq ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that the claim is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then in view of coarse compactness of the space of M-Morse maps sending y to a fixed compact subset of X, there exists a sequence ppnq as in the lemma, coarsely converging to p, such that the sequence pnp8q “ τn converges to τ 1 ‰ pp8q “ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the coarse convergence pn Ñ p, there exists C ă 8 and a sequence tn Ñ 8 such that dppnptnq, pptnqq ď C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the definition of Morse quasigeodesics, there exists a sequence of cones V pxn, stpτnqq (with xn in a bounded subset B Ă X) such that the image of pn is contained in the D-neighborhood of V pxn, stpτnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, the sequence ppnptnqq flag- converges to τ 1, while ppptnqq flag-converges to τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to [KLP5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='23], altering a sequence by a uniformly bounded amount, does not change the flag-limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, the sequence ppptnqq also flag-converges to τ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, τ “ τ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 A Morse Lemma for straight sequences In order to motivate the results of this section we recall the following sufficient condition for a piecewise-geodesic path in a Hadamard manifold Y of curvature ď ´1 to be quasigeodesic (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' [KaLi]): 11 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that c is a piecewise-geodesic path in Y whose angles at the vertices are ě α ą 0 and whose edges are longer than L, where α and L satisfy coshpL{2q sinpα{2q ě ν ą 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='15) Then c is an pLpνq, Apνqq-quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By considering c with vertices on a horocycle in the hyperbolic plane, one see that the inequality in this proposition is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If L is sufficiently large and α is sufficiently close to π then c is (uniformly) quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In higher rank, we do not have an analogue of the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='15), instead, we will be generalizing the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' However, angles in the corollary will be replaced with ζ-angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will show (in a String of Diamonds Theorem, theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30) that if a piecewise-geodesic path c in X has sufficiently long edges and ζ-angles between consecutive segments sufficiently close to π, then c is M-Morse for a suitable Morse datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In the following, we consider finite or infinite sequences pxnq of points in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='17 (Straight and spaced sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call a sequence pxnq pΘ, ǫq-straight if the segments xnxn`1 are Θ-regular and =ζ xnpxn´1, xn`1q ě π ´ ǫ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call it l-spaced if the segments xnxn`1 have length ě l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that every straight sequence can be extended to a biinfinite straight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Straightness is a local condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The goal of this section is to prove the following local- to-global result asserting that sufficiently straight and spaced sequences satisfy a higher rank version of the Morse Lemma (for quasigeodesics in hyperbolic space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 (Morse Lemma for straight spaced sequences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For Θ, Θ1, δ there exist l, ǫ such that: Every pΘ, ǫq-straight l-spaced sequence pxnq is δ-close to a parallel set Ppτ´, τ`q with sim- plices τ˘ of type τmod, and it moves from τ´ to τ` in the sense that its nearest point projection ¯xn to Ppτ´, τ`q satisfies ¯xn˘m P V p¯xn, stΘ1pτ˘qq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='19) for all n and m ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='20 (Global spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As a corollary of this theorem, we will show that straight spaced sequences are quasigeodesic: dpxn, xn`mq ě clm ´ 2δ with a constant c “ cpΘ1q ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' See Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, by interpolating the sequence pxnq via geodesic segments we obtain a Morse quasigeodesic in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 is a higher-rank generalization of two familiar facts from geometry of Gromov-hyperbolic geodesic metric spaces: The fact that local quasigeodesics (with suitable parameters) are global quasigeodesics and the Morse lemma stating that quasigeodesics stay uniformly close to geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In the higher rank, quasigeodesics, of course, need not be close to geodesics, but, instead (under the straightness assumption), are close to diamonds/Weyl cones/parallel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' One can obviously strengthen the Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='16 by stating that for each ǫ ă π there exists L0pǫq such that if α ě π ´ ǫ and L ě L0pǫq then c is a uniform quasigeodesic in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A similar strengthening is false for symmetric spaces of rank ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For instance, when W – S3 and ǫ “ 2π{3, then no matter what Θ, Θ1 and l are, the conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 fails already for sequences contained in a single flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In order to prove the theorem, we start by considering half-infinite sequences and prove that they keep moving away from an ideal simplex of type τmod if they do so initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='21 (Moving away from an ideal simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Given a face τ Ă BTitsX of type τmod and distinct points x, y P X, define the angle =ζ xpτ, yq :“ =xpz, yq where z is a point (distinct from x) on the geodesic ray xξ, where ξ P τ is the point of type ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We say that a sequence pxnq moves ǫ-away from a simplex τ of type τmod if =ζ xnpτ, xn`1q ě π ´ ǫ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='22 (Moving away from ideal simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For small ǫ and large l, ǫ ď ǫ0 and l ě lpǫ, Θq, the following holds: If the sequence pxnqně0 is pΘ, ǫq-straight l-spaced and if =ζ x0pτ, x1q ě π ´ 2ǫ, then pxnq moves ǫ-away from τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10(ii), the unit speed geodesic segment c : r0, t1s Ñ X from pp0q to pp1q moves ǫpdp2ǫqq-away from τ at all times, and ǫ1p2ǫ, Θ, lq-away at times ě l, which includes the final time t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For lpǫ, Θq sufficiently large, we have ǫ1p2ǫ, Θ, lq ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then c moves ǫ-away from τ at time t1, which means that =ζ x1pτ, x0q ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Straightness at x1 and the triangle inequality yield that again =ζ x1pτ, x2q ě π ´ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' One proceeds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that there do exist simplices τ satisfying the hypothesis of the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For instance, one can extend the initial segment x0x1 backwards to infinity and choose τ “ τpx1x0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Now we look at biinfinite sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 13 We assume in the following that pxnqnPZ is pΘ, ǫq-straight l-spaced for small ǫ and large l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As a first step, we study the asymptotics of such sequences and use the argument for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='22 to find a pair of opposite ideal simplices τ˘ such that pxnq moves from τ´ towards τ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='23 (Moving towards ideal simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For small ǫ and large l, ǫ ď ǫ0 and l ě lpǫ, Θq, the following holds: There exists a pair of opposite simplices τ˘ of type τmod such that the inequality =ζ xnpτ¯, xn˘1q ě π ´ 2ǫ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='24) holds for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every n define a compact set C¯ n Ă Flagpτmodq C˘ n “ tτ˘ : =ζ xnpτ˘, xn¯1q ě π ´ 2ǫu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='22, straightness at xn`1 implies that C´ n Ă C´ n`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence the family tC´ n unPZ form a nested sequence of nonempty compact subsets and therefore have nonempty intersection containing a simplex τ´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Analogously, there exists a simplex τ` which belongs to C` n for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It remains to show that the simplices τ´, τ` are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Using straightness and the triangle inequality, we see that =ζ xnpτ´, τ`q ě π ´ 5ǫ for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, if 5ǫ ă ǫpζq, then the simplices τ´, τ` are antipodal in view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The pair of opposite simplices pτ´, τ`q which we found determines a parallel set in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The second step is to show that pxnq is uniformly close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='25 (Close to parallel set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For small ǫ and large l, ǫ ď ǫpδq and l ě lpΘ, δq, the sequence pxnq is δ-close to Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The statement follows from the combination of the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4) (in the second part of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='23) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The third and final step of the proof is to show that the nearest point projection p¯xnq of pxnq to Ppτ´, τ`q moves from τ´ towards τ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='26 (Projection moves towards ideal simplices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For small ǫ and large l, ǫ ď ǫ0 and l ě lpǫ, Θ, Θ1q, the segments ¯xn¯xn`1 are Θ1-regular and =ζ ¯xnpτ´, ¯xn`1q “ π for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the previous lemma, pxnq is δ0-close to Ppτ´, τ`q if ǫ0 is sufficiently small and l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since xnxn`1 is Θ-regular, the triangle inequality for ∆-lengths yields that the segment ¯xn¯xn`1 is Θ1-regular, again if l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 14 Let ξ` denote the ideal endpoint of the ray extending this segment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' ¯xn`1 P ¯xnξ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then xn`1 is 2δ0-close to the ray xnξ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We obtain that =ζ Titspτ´, ξ`q ě =ζ xnpτ´, ξ`q » =ζ xnpτ´, xn`1q » π where the last step follows from inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The discreteness of Tits distances between ideal points of fixed type ζ implies that in fact =ζ Titspτ´, ξ`q “ π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' the ideal points ζpτ´q and ζpξ`q are antipodal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' But the only simplex opposite to τ´ in B8Ppτ´, τ`q is τ`, so τpξ`q “ τ` and =ζ ¯xnpτ´, ¯xn`1q “ =ζ ¯xnpτ´, ξ`q “ π, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It suffices to consider biinfinite sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The conclusion of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='26 is equivalent to ¯xn`1 P V p¯xn, stΘ1pτ`qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Combining Lem- mas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='25 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='26, we thus obtain the theorem for m “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The convexity of Θ1-cones, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1, implies that V p¯xn`1, stΘ1pτ`qq Ă V p¯xn, stΘ1pτ`qq, and the assertion follows for all m ě 1 by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The conclusion of the theorem implies flag-convergence x˘n Ñ τ˘ as n Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' However, the sequences pxnqnP˘N do in general not converge at infinity, but accumulate at compact subsets of stΘ1pτ˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5 Lipschitz retractions to straight paths Consider a (possibly infinite) closed interval J in R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we will assume that J has integer or infinite bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that p : J X Z Ñ P “ Ppτ´, τ`q Ă X is an l-separated, λ-Lipschitz, pΘ, 0q- straight coarse sequence pointing away from τ´ and towards τ`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We extend p to a piecewise- geodesic map p : J Ñ P by sending intervals rn, n ` 1s to geodesic segments ppnqppn ` 1q via affine maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We retain the name p for the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There exists L “ Lpl, λ, Θq and an L-Lipschitz retraction of X to p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=', an L-Lipschitz map r : X Ñ J so that r ˝ p “ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, p : J X Z Ñ X is a p¯L, ¯Aq- quasigeodesic, where ¯L, ¯A depend only on l, λ, Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It suffices to prove existence of a retraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since P is convex in X, it suffices to construct a map P Ñ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Pick a generic point ξ “ ξ` P τ` and let bξ : P Ñ R denote the Busemann function normalized so that bξpppzqq “ 0 for some z P J X Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then the Θ-regularity 15 assumption on p implies that the slope of the piecewise-linear function bξ ˝ p : J Ñ R is strictly positive, bounded away from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The assumption that p is l-separated λ-Lipschitz implies that l ď |p1ptq| ď λ for each t (where the derivative exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The straightness assumption on p implies that the function h :“ bξ ˝ p : J Ñ R is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By combining these observations, we conclude that h is an L-biLipschitz homeomorphism for some L “ Lpl, λ, Θq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lastly, we define r : P Ñ J, r “ h´1 ˝ bξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since bξ is 1-Lipschitz, the map r is L-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the construction, r ˝ p “ Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that p : JXZ Ñ X is a l-spaced, λ-Lipschitz, pΘ, ǫq-straight sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Pick some Θ1 such that Θ Ă intpΘ1q and let δ “ δpl, Θ, Θ1, ǫq be the constant as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then for L “ Lpl ´ 2δ, λ ` 2δ, Θ1q we have: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There exists an pL, 2δq-coarse Lipschitz retraction X Ñ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The map p is a pΘ1, D1, L1, A1q-quasigeodesic with D1, L1, A1 depending only on l, λ, Θ, Θ1, ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The statement immediately follows the above lemma combined with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Reformulating in terms of piecewise-geodesic paths, we obtain Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30 (String of diamonds theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For any pair of Weyl convex subsets Θ ă Θ1 and a number D ě 0 there exist positive numbers ǫ, S, L, A depending on the datum pΘ, Θ1, Dq such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that c is an arc-length parameterized piecewise-geodesic path (finite or infinite) in X obtained by concatenating geodesic segments xixi`1 satisfying for all i: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Each segment xixi`1 is Θ-regular and has length ě S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' =ζ xipxi´1, xi`1q ě π ´ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then the path c is pΘ1, D, L, Aq-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6 Local Morse quasigeodesics According to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30, sufficiently straight and spaced straight piecewise-geodesic paths are Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In this section we will now prove that, conversely, the Morse property implies straightness in a suitable sense, namely that for sufficiently spaced quadruples the associated midpoint triples are arbitrarily straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (For the quadruples themselves this is in general not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='31 (Quadruple condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For points x, y P X we let midpx, yq denote the midpoint of the geodesic segment xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A map p : I Ñ X satisfies the pΘ, ǫ, l, sq-quadruple condition if for all t1, t2, t3, t4 P I with t2 ´ t1, t3 ´ t2, t4 ´ t3 ě s the triple of midpoints pmidpt1, t2q, midpt2, t3q, midpt3, t4qq 16 is pΘ, ǫq-straight and l-spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32 (Morse implies quadruple condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For L, A, Θ, Θ1, D, ǫ, l exists a scale s “ spL, A, Θ, Θ1, D, ǫ, lq such that every pΘ, D, L, Aq-Morse quasigeodesic satisfies the pΘ1, ǫ, l, s1q-quadruple condition for every s1 ě s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let p : I Ñ X be an pL, A, Θ, Dq-Morse quasigeodesic, and let t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' , t4 P I such that t2 ´ t1, t3 ´ t2, t4 ´ t3 ě s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We abbreviate pi :“ pptiq and mi “ midppi, pi`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Regarding straightness, it suffices to show that the segment m2m1 is Θ1-regular and that =ζ m2pp2, m1q ď ǫ 2 provided that s is sufficiently large in terms of the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the Morse property, there exists a diamond ♦Θpx1, x3q such that dpx1, p1q, dpx3, p3q ď D and p2 P NDp♦Θpx1, x3qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The diamond spans a unique parallel set Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (Necessarily, x3 P V px1, stΘpτ`qq and x1 P V px3, stΘpτ´qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') We denote by ¯pi and ¯mi the projections of pi and mi to the parallel set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We first observe that m2 (and m3) is arbitrarily close to the parallel set if s is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If this were not true, a limiting argument would produce a geodesic line at strictly positive finite Hausdorff distance P p0, Ds from Ppτ´, τ`q and asymptotic to ideal points in stΘpτ˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' However, all lines asymptotic to ideal points in stΘpτ˘q are contained in Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Next, we look at the directions of the segments ¯m2 ¯m1 and ¯m2¯p2 and show that they have the same τ-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since ¯p2 is 2D-close to V p¯p1, stΘpτ`qq, we have that the point ¯p1 is 2D-close to V p¯p2, stΘpτ´qq, and hence also ¯m1 is 2D-close to V p¯p2, stΘpτ´qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There- fore, ¯p1, ¯m1 P V p¯p2, stΘ1pτ´qq if s is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Similarly, ¯m2 P V p¯p2, stΘ1pτ`qq and hence ¯p2 P V p ¯m2, stΘ1pτ´qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The convexity of Θ1-cones, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1, implies that also ¯m1 P V p ¯m2, stΘ1pτ´qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, =ζ ¯m2p¯p2, ¯m1q “ 0 if s is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since m2 is arbitrarily close to the parallel set if s is sufficiently large, it follows by another limiting argument that =ζ m2pp2, m1q ď ǫ 2 if s is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Regarding the spacing, we use that ¯m1 P V p¯p2, stΘ1pτ´qq and ¯m2 P V p¯p2, stΘ1pτ`qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It follows that dp ¯m1, ¯m2q ě c ¨ pdp ¯m1, ¯p2q ` dp¯p2, ¯m2qq with a constant c “ cpΘ1q ą 0, and hence that dpm1, m2q ě l if s is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32 tell that the Morse property for quasigeodesics is equiv- alent to straightness (of associated spaced sequences of points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since straightness is a local condition, this leads to a local to global result for Morse quasigeodesics, namely that the Morse property holds globally if it holds locally up to a sufficiently large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='33 (Local Morse quasigeodesic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' An S-local pΘ, D, L, Aq-Morse quasigeode- sic in X is a map p : I Ñ X such that for all t0 the subpath p|rt0,t0`Ss is a pΘ, D, L, Aq-Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that local Morse quasigeodesics are uniformly coarse Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 17 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34 (Local-to-global principle for Morse quasigeodesics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For L, A, Θ, Θ1, D exist S, L1, A1, D1 such that every S-local pΘ, D, L, Aq-local Morse quasigeodesic in X is an pΘ1, D1, L1, A1q-Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We choose an auxiliary Weyl convex subset Θ2 such that Θ ă Θ2 ă Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let p : I Ñ X be an S-local pΘ, D, L, Aq-local Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We consider its coarsi- fication on a (large) scale s and the associated midpoint sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we put ps n “ ppnsq and ms n “ midpps n, ps n`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Whereas the coarsification itself does in general not become arbitrarily straight as the scale s increases, this is true for its midpoint sequence due to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We want it to be sufficiently straight and spaced so that we can apply to it the Morse Lemma from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore we first fix an auxiliary constant δ, and further auxiliary con- stants l, ǫ as determined by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 in terms of Θ1, Θ2 and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32 applied to the pΘ, D, L, Aq-Morse quasigeodesics p|rt0,t0`Ss yields that pms nq is pΘ2, ǫq-straight and l-spaced if S ě 3s and the scale s is large enough depending on L, A, Θ, Θ2, D, ǫ, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Now we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 to pms nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It yields a nearby sequence p ¯ms nq, dp ¯ms n, ms nq ď δ, with the following property: For all n1 ă n2 ă n3 the segments ¯ms n1 ¯ms n3 are uniformly regular and the points ms n2 are δ-close to the diamonds ♦Θ1p ¯ms n1, ¯ms n3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since the subpaths p|rns,pn`1qss filling in pps nq are pL, Aq-quasigeodesics (because S ě s), and it follows that for all t1, t2 P I the subpaths p|rt1,t2s are D1-close to Θ1-diamonds with D1 depending on L, A, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 also implies a global spacing for the sequence pms nq, compare Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='20, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' dpms n, ms n1q ě c ¨ |n ´ n1| with a positive constant c depending on Θ1, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence p is a global pL1, A1q-quasigeodesic with L1, A1 depending on L, A, s, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Combining this information, we obtain that p is an pΘ1, D1, L1, A1q-Morse quasigeodesic for certain constants L1, A1 and D1 depending on L, A, Θ, Θ1 and D, provided that the scale S is sufficiently large in terms of the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7 Local-to-global principle for Morse maps We now deduce from our local-to-global result for Morse quasigeodesics, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34, a local- to-global result for Morse embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We restrict to the setting of maps of Gromov-hyperbolic pl, aq-quasigeodesic metric spaces Z to symmetric spaces X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='35 (Local Morse embedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call a map f : Z Ñ X an S-local pΘ, D, L, Aq-Morse map if for any pl, aq-quasigeodesic q : I Ñ Z defined on an interval I of length ď S the image path f ˝ q is a pΘ, D, L, Aq-Morse quasigeodesic in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='36 (Local-to-global principle for Morse embeddings of Gromov hyper- bolic spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For l, a, L, A, Θ, Θ1, D exists a scale S and a datum pD1, L1, A1q such that every S-local pΘ, D, L, Aq-Morse embedding from an pl, aq-quasigeodesic Gromov hyperbolic space into X is a pΘ1, D1, L1, A1q-Morse embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 18 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let f : Z Ñ X denote the local Morse embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It sends every pl, aq-quasigeodesic q : I Ñ Z to an S-local pΘ, D, L, Aq-Morse quasigeodesic p “ f ˝ q in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34, p is pL1, A1, Θ1, D1q-Morse if S ě Spl, a, L, A, Θ, Θ1, Dq, where L1, A1, D1 depend on the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Below is a reformulation of this theorem in the case of geodesic Gromov-hyperbolic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Z be a δ-hyperbolic geodesic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' An R-ball Bpz, Rq in Z need not be convex, but it is δ-quasiconvex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, the restriction of the metric from Z to Bpz, Rq results in a p1, δq-quasigeodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='37 (Local-to-global principle for Morse embeddings of geodesic spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For L, A, Θ, Θ1, D, δ exists a scale R and a datum pD1, L1, A1q such that if Z is a δ-hyperbolic geodesic metric space and the restriction of f to any R-ball is pΘ, D, L, A, 1, δq-Morse, then f : Z Ñ X is pΘ1, D1, L1, A1q-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4 Group-theoretic applications As a consequence of the local-to-global criterion for Morse maps, in this section we establish that the Morse property for isometric group actions is an open condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, for two nearby Morse actions, the actions on their τmod-limit sets are also close, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' conjugate by an equivariant homeomorphism close to identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of the equivalence of Morse property with the asymptotic properties discussed earlier, this implies structural stability for asymp- totically embedded groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Another corollary of the local-to-global result is the algorithmic recognizability of Morse actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We conclude the section by illustrating our technique by constructing Morse-Schottky ac- tions of free groups on higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 Stability of Morse actions We consider isometric actions Γ ñ X of finitely generated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 (Morse action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call an action Γ ñ X Θ-Morse if one (any) orbit map Γ Ñ Γx Ă X is a Θ-Morse embedding with respect to a(ny) word metric on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call an action Γ ñ X τmod-Morse if it is Θ-Morse for some τmod-Weyl convex compact subset Θ Ă ostpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 (Morse actions are τmod-regular and undistorted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (i) It follows immedi- ately from the definition of Morse quasigeodesics that Θ-Morse actions are τmod-regular for the simplex type τmod determined by Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (ii) Morse subgroups of G are undistorted in the sense that the orbit maps are quasi-isometric embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In [KL1] we prove that Morse subgroups of G satisfy a stronger property: They are coarse Lipschitz retracts of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This retraction property is stronger than nondistortion: Every finitely generated subgroup which is a coarse retract of G is undistorted in G, but there are examples of undistorted subgroups which are not coarse retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For instance, the group 19 Φ :“ F2 ˆ F2 admits an undistorted embedding in the isometry group of X “ H2 ˆ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' On the other hand, pick an epimorphism φ : F2 Ñ Z and define the subgroup Γ ă Φ as the kernel of the homomorphism pγ1, γ2q ÞÑ φpγ1q ´ φpγ2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then Γ is a finitely generated undistorted subgroup of Φ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' [OS, Theorem 2]), but is not finitely presented (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' [BR]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, Γ ă G “ IsompH2q ˆ IsompH2q is undistorted but is not a coarse Lipschitz retract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We denote by HomτmodpΓ, Gq Ă HompΓ, Gq the subset of τmod-Morse actions Γ ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By analogy with local Morse quasigeodesics, we define local Morse group actions ρ : Γ ñ X of a hyperbolic group (with a fixed finite generating set): Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' An action ρ is called S-locally pΘ, D, L, Aq-locally Morse, or pΘ, D, L, Aq- locally Morse on the scale S, with respect to a base-point x P X, if the orbit map Γ Ñ Γ¨x Ă X induces an S-local pΘ, D, L, Aq-local Morse embedding of the Cayley graph of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to our local-to-global result for Morse embeddings, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='37, an action of a word hyperbolic group is Morse if and only if it is local Morse on a sufficiently large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since this is a finite condition, it follows that the Morse property is stable under perturbation of the action: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 (Morse is open for word hyperbolic groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For any word hyperbolic group Γ the subset HomτmodpΓ, Gq is open in HompΓ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More precisely, if ρ P HomτmodpΓ, Gq is M-Morse with respect to a base-point x P X then there exists a neighborhood of ρ in HompΓ, Gq consisting entirely of M1-Morse representations with respect to x, where M1 depends only on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let ρ : Γ ñ X be a Morse action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We fix a word metric on Γ and a base point x P X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then there exist data M “ pL, A, Θ, Dq such that the orbit map Γ Ñ Γx Ă X extends to a pΘ, D, L, Aq-Morse map of the Cayley graph Y on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We relax the Morse parameters slightly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we consider pL, A, Θ, Dq-Morse quasigeodesics as pL, A ` 1, Θ, D ` 1q-Morse quasigeodesics satisfying strict inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every scale S, the orbit map Γ Ñ Γx Ă X, defines an pL, A ` 1, Θ, D ` 1, Sq-local Morse embedding Y Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Due to Γ-equivariance, this is a finite condition in the sense that it is equivalent to a condition involving only finitely many orbit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since we relaxed the Morse parameters, the same condition is satisfied by all actions sufficiently close to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='37 provides a scale S such that all S-local pΘ, D`1, L, A`1q-Morse embeddings Y Ñ X are M1-Morse for some Morse datum M1 depending only on pL, A ` 1, Θ, D ` 1, Sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It follows that actions sufficiently close to ρ are τmod-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every hyperbolic group Γ the space of faithful Morse representations Hominj,τmodpΓ, Gq is open in HomτmodpΓ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 20 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Every hyperbolic group Γ has the unique maximal finite normal subgroup Φ Ÿ Γ (if Γ is nonelementary then Φ is the kernel of the action of Γ on B8Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since Morse actions are properly discontinuous, the kernel of every Morse representation Γ Ñ G is contained in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since HompΦ, Gq{G is finite, it follows that the set of faithful Morse representations is open in HomτmodpΓ, Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The result on the openness of the Morse condition for actions of word hyperbolic groups, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4, can be strengthened in the sense that the asymptotics of Morse actions vary continuously: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6 (Morse actions are structurally stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The boundary map at infinity of a Morse action depends continuously on the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 nearby actions are uniformly Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The assertion there- fore follows from the fact that the ends of Morse quasirays vary uniformly continuously, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (i) Note that since the boundary maps at infinity are embeddings, the Γ-actions on the τmod-limit sets are topologically conjugate to each other and, for nearby actions, by a homeomorphism close to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (ii) In rank one, our argument yields a different proof for Sullivan’s Structural Stability Theorem [Su] for convex cocompact group actions on rank one symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Other proofs can be found in [La, GW] (for Anosov subgroups in higher rank), [Co, Iz, Bo] for rank one symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Our next goal is to extend the topological conjugation from the limit set to the domains of proper discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Recall that in [KLP4] we constructed domains of proper discontinuity and cocompactness for τmod-Morse group actions on flag-manifolds Flagpνmodq “ G{Pνmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Such domains depend on a certain auxiliary datum, a balanced thickening Th Ă W, which is a Wτmod- left invariant subset satisfying certain conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' see [KLP4, sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let νmod Ă σmod be an ι-invariant face such that Th is invariant under the action of Wνmod via the right multiplication (this is automatic if νmod “ σmod since Wσmod “ teu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The thickening Th Ă W defines a thickening ThpΛτmodpΓqq Ă Flagpνmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' One of the main results of [KLP4] (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7) is that each τmod-Morse subgroup Γ ă G acts properly discontinuously and cocompactly on ΩThpΓq :“ Flagpνmodq ´ ThpΛτmodpΓqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 (Stability of Morse quotient spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that ρn : Γ Ñ ρnpΓq “ Γn ă G is a sequence of faithful τmod-Morse representations converging to a τmod-Morse embedding ρ : Γ ãÑ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The sequence of thickenings ThpΛτmodpΓnqq Hausdorff-converges to ThpΛτmodpΓqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If γn P Γ is a divergent sequence, then, after extraction, the sequence pρnpγnqq flag- converges to a point in ΛτmodpΓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There is a sequence of equivariant diffeomorphisms hn : ΩThpΓq Ñ ΩThpΓnq converging to the identity map uniformly on compacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, the quotient-orbifolds ΩThpΓnq{Γn are diffeomorphic to ΩThpΓq{Γ for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' First of all, suppose that a sequence τn P Flagpτmodq converges to τ P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then, since Flagpνmodq “ G{Pνmod, there is a sequence gn P G, gn Ñ e, such that gnpτq “ τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since gnpThpτqq “ Thpgnτq “ Thpτnq, it follows that we have Hausdorff-convergence of subsets Thpτnq Ñ Thpτq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Moreover, this convergence of subsets is uniform: There exists n0 “ npδq such that if dpτn, τq ă δ for all n ě n0 then dpThpτnq, Thpτqq ă ǫ “ ǫpδq for all n ě n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Here ǫ Ñ 0 as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since the sequence of limit sets ΛτmodpΓnq Hausdorff-converges to ΛτmodpΓq, it follows that the sequence of thickenings ThpΛτmodpΓnqq Hausdorff-converges to ThpΛτmodpΓqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Consider a sequence of geodesic rays eξn in the Cayley graph Y of Γ such that γn lies in an R-neighborhood of eξn for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then, in view of the uniform M1-Morse property for the representations ρn, each point ρnpγnqpxq belongs to the D1-neighborhood of the Weyl cone V px, stpτnqq, where τn “ αnpξnq, αn : B8Γ Ñ ΛτmodpΓnq is the asymptotic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, by the definition of flag-convergence, the sequences pρnpγnqq and pτnq have the same flag-limit in Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By Part 1, the sequence pτnq subconverges to a point in ΛτmodpΓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, the same holds for pρnpγnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The proof of this part is mostly standard, see [Iz] in the case when X is a hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The quotient orbifold O “ ΩThpΓq{Γ has a natural pF, Gq-structure where F “ Flagpνmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The orbifold O has finitely many components, let Z be one of them and let ˆZ Ă ΩThpΓq be a component projecting to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It suffices to construct maps hn on each component ˆZ and then extend these maps to maps hn of ΩThpΓq by ρn-equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The covering map ˆZ Ñ Z induces an epimorphism φ : π1pZq Ñ ΓZ, where ΓZ is the Γ- stabilizer of ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let dev : ˜Z Ñ ˆZ Ă ΩThpΓq be the developing map, where ˜Z Ñ Z is the universal covering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By Ehresmann-Thurston holonomy theorem (see [Lo], [CEG], [Go], [K1, sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1]), for all sufficiently large n, the homomorphism φn :“ ρn ˝ φ is the holonomy of an pF, Gq-structure on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Moreover, the developing maps devn : ˜Z Ñ F converge to dev uniformly on compacts in the C8-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since π1p ˆZq is contained in the kernel of φ, it is also in the kernel of φn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, the maps devn descend to maps y devn : ˆZ Ñ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The sequence y devn still converges to the identity embedding ˆZ ãÑ F uniformly on compacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Pick a compact fundamental set C Ă ˆZ for the ΓZ-action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' a compact subset whose Γ-orbit equals ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of Part 1 of the theorem, y devnpCq Ă ΩThpΓnq for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, we can assume that y devnp ˆZq is contained in a component ˆZn of ΩThpΓnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the compactness of the quotient-orbifolds, y devn projects to a finite-to-one (smooth) orbi-covering map cn : Z Ñ Zn :“ ˆZn{ρnpΓZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, y devn : ˆZ Ñ ˆZn is a covering map as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If ˆZn were simply-connected, it would follow that y devn is a diffeomorphism as required (and this is 22 how Izeki concludes his proof in [Iz]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will prove that y devn is a diffeomorphism by a direct argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that each y devn is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then, by the equivariance of these maps, after extraction, there exist convergent sequences zn Ñ z, z1 n Ñ z1 in ˆZ and a sequence γn P Γ such that ρnpγnqy devnpznq “ y devnpz1 nq, γnpznq ‰ z1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the sequence pγnq were contained in a finite subset of Γ we would obtain a contradiction with the uniform convergence on compacts y devn Ñ id on ˆZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, after extraction, we may assume that pγnq is a divergent sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We, therefore, obtain a dynamical relation between the points z, z1 via the sequence pρnpγnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to Part 2, the sequence pρnpγnqq flag-accumulates to ΛτmodpΓq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The dynamical relation then contradicts fatness of the balanced thickening Th, see [KLP4, sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2] and the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 in [KLP4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We conclude that the maps y devn : ˆZ Ñ ˆZn are diffeomorphisms for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since ρn : Γ Ñ Γn are isomorphisms, equivari- ance of the developing maps implies that the maps hn : ΩThpΓq Ñ ΩThpΓnq are diffeomor- phisms for sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This part is an immediate corollary of Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (i) In the case when X is a hyperbolic space, the equivariant diffeomorphism hn : ΩpΓq Ñ ΩpΓnq combined with the equivariant homeomorphism of the limit sets ΛpΓq Ñ ΛpΓnq yield an equivariant homeomorphism B8X Ñ B8X, see [Tu, Iz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Such an extension does not exist in higher rank since, in general, there is no equivariant homeomorphism of thickened limit sets ThpΛτmodpΓqq Ñ ThpΛτmodpΓnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This can be already seen for group actions on products of hyperbolic planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (ii) An analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 holds when we replace the group actions on flag-manifolds with actions on Finsler compactifications of the symmetric space and replace flag-manifold thickenings ThpΛτmodq with Finsler thickenings ThF :upΛτmodq Ă BF :uX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proving this requires extending Ehresmann–Thurston holonomy theorem to the category of smooth manifolds with corners and we will not pursue it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 Schottky actions In this section we apply our local-to-global result for straight sequences (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18) to con- struct Morse actions of free groups, generalizing and sharpening1 Tits’s ping-pong construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We consider two oriented τmod-regular geodesic lines a, b in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let τ˘a, τ˘b P Flagpτmodq denote the simplices which they are τ-asymptotic to, and let θ˘a, θ˘b P σmod denote the types of their forward/backward ideal endpoints in B8X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (Note that θ´a “ ιpθaq and θ´b “ ιpθbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') Let Θ be a compact convex subset of ostpτmodq Ă σmod, which is invariant under ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1In the sense that we obtain free subgroups which are not only embedded, but also asymptotically embedded in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 23 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10 (Generic pair of geodesics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We call the pair of geodesics pa, bq generic if the four simplices τ˘a, τ˘b are pairwise opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let α, β P G be axial isometries with axes a and b respectively and translating in the positive direction along these geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then τ˘a and τ˘b are the attractive/repulsive fixed points of α and β on Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every pair of numbers m, n P N we consider the representation of the free group in two generators ρm,n : F2 “ xA, By Ñ G sending the generator A to αm and B to βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We regard it as an isometric action ρm,n : F2 ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11 (Schottky subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' A τmod-Schottky subgroup of G is a free τmod-asymp- totically embedded subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If G has rank one, this definition amounts to the requirement that Γ is convex cocompact and free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Equivalently, this is a discrete finitely generated subgroup of G which contains no nontrivial elliptic and parabolic elements and has totally disconnected limit set (see see [K1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We note that this definition essentially agrees with the standard definition of Schottky groups in rank 1 Lie groups, provided one allows fundamental domains at infinity for such groups to be bounded by pairwise disjoint compact submanifolds which need not be topological spheres, see [K1] for the detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='12 (Morse Schottky actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the pair of geodesics pa, bq is generic and if θ˘a, θ˘b P intpΘq, then the action ρm,n is Θ-Morse for sufficiently large m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, such ρm,n is injective and its image is a τmod-Schottky subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In particular, these actions are faithful and undistorted, compare Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let S “ tA˘1, B˘1u be the standard generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We consider the sequences pγkq in F2 with the property that γ´1 k γk`1 P S and γk`1 ‰ γk´1 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' They correspond to the geodesic segments in the Cayley tree of F2 associated to S which connect vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let x P X be a base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 we must show that the corresponding sequences pγkxq in the orbit F2 ¨x are uniformly Θ-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (Meaning e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' that the maps R Ñ X sending the intervals rk, k ` 1q to the points γkx are uniform Θ-Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34 we will obtain this by applying our local to global result for straight spaced sequences (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18) to the associated midpoint sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that the sequences pγkxq themselves cannot expected to be straight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Taking into account the Γ-action, the uniform straightness of all midpoint sequences depends on the geometry of a finite configuration in the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It is a consequence of the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Consider the midpoints y˘m of the segments xα˘mpxq and z˘n of the segments xβ˘npxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For sufficiently large m, n the quadruple ty˘m, z˘nu is arbitrarily separated and Θ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Moreover, for any of the four points, the segments connecting it to the other three points have arbitrarily small ζ-angles with the segment connecting it to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The four points are arbitrarily separated from each other and from x because the axes a and b diverge from each other due to our genericity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By symmetry, it suffices to verify the rest of the assertion for the point ym, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we show that the segments ymy´m and ymzn are Θ-regular for large m, n and that limmÑ8 =ζ ympx, y´mq “ 0 and limn,mÑ8 =ζ ympx, znq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The orbit points α˘mx and the midpoints y˘m are contained in a tubular neighborhood of the axis a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, the segments ymx and ymy´m are Θ-regular for large m and =ympx, y´mq Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This implies that also =ζ ympx, y´mq Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' To verify the assertion for pym, znq we use that, due to genericity, the simplices τa and τb are opposite and we consider the parallel set P “ Ppτa, τbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since the geodesics a and b are forward asymptotic to P, it follows that the points x, ym, zn have uniformly bounded distance from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We denote their projections to P by ¯x, ¯ym, ¯zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Θ2 Ă intpΘq be an auxiliary Weyl convex subset such that θ˘a, θ˘b P intpΘ2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We have that ¯ym P V p¯x, stΘ2pτaqq for large m because the points ym lie in a tubular neighborhood of the ray with initial point ¯x and asymptotic to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Similarly, ¯zn P V p¯x, stΘ2pτbqq for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It follows that ¯x P V p¯ym, stΘ2pτbqq and, using the convexity of Θ-cones (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1), that ¯zn P V p¯ym, stΘ2pτbqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The cone V pym, stΘ2pτbqq is uniformly Hausdorff close to the cone V p¯ym, stΘ2pτbqq because the Hausdorff distance of the cones is bounded by the distance dpym, ¯ymq of their tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence there exist points x1, z1 n P V pym, stΘ2pτbqq uniformly close to x, zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since dpym, x1q, dpym, z1 nq Ñ 8 as m, n Ñ 8, it follows that the segments ymx and ymzn are Θ-regular for large m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, since =ζ ympx1, z1 nq “ 0 and =ympx, x1q Ñ 0 as well as =ympzn, z1 nq Ñ 0, it follows that =ζ ympx, znq Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof of Theorem concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The lemma implies that for any given l, ǫ the midpoint triples of the four point sequences pγkxq are pΘ, ǫq-straight and l-spaced if m, n are sufficiently large, compare the quadruple condition (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This means that the midpoint sequences of all sequences pγkxq are pΘ, ǫq-straight and l-spaced for large m, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 then implies that the sequences pγkxq are uniformly Θ-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Generalizing the above argument to free groups with finitely many gener- ators, one can construct Morse Schottky subgroups for which the set θpΛq Ă σmod of types of limit points is arbitrarily Hausdorff close to a given ι-invariant Weyl convex subset Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This provides an alternative approach to the second main theorem in [Be] using coarse geometric arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In [DKL] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='12 was generalized (by arguments similar to the its proof) to free products of Morse subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 Algorithmic recognition of Morse actions In this section, we describe an algorithm which has an isometric action ρ : Γ ñ X and a point x P X as its input and terminates if and only if the action ρ is Morse (otherwise, the algorithm 25 runs forever).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We begin by describing briefly the Riley’s algorithm (see [Ri]) accomplishing a similar task, namely, detecting geometrically finite actions on X “ H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that we are given a finite (symmetric) set of generators g1 “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' , gm of a subgroup Γ Ă POp3, 1q and a base-point x P X “ H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The idea of the algorithm is to construct a finite sided Dirichlet fundamental domain D for Γ (with the center at x): Every geometrically finite subgroup of POp3, 1q admits such a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (The latter is false for geometrically finite subgroups of POpn, 1q, n ě 4, but is, nevertheless, true for convex cocompact subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') Given a finite sided convex fundamental domain, one concludes that Γ is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Here is how the algorithm works: For each k define the subset Sk Ă Γ represented by words of length ď k in the letters g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' , gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For each g P Sk consider the half-space Bispx, gpxqq Ă X bounded by the bisector of the segment xgpxq and containing the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then compute the intersection Dk “ č gPSk Bispx, gpxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Check if Dk satisfies the conditions of the Poincar´e’s Fundamental Domain theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If it does, then D “ Dk is a finite sided fundamental domain of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If not, increase k by 1 and repeat the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Clearly, this process terminates if and only if Γ is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' One can enhance the algorithm in order to detect if a geometrically finite group is convex cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Namely, after a Dirichlet domain D is constructed, one checks for the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the ideal boundary of a Dirichlet domain D has isolated ideal points (they would correspond to rank two cusps which are not allowed in convex cocompact groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the ideal boundary of D contains tangent circular arcs with points of tangency fixed by parabolic elements (coming from the “ideal vertex cycles”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Such points correspond to rank 1 cusps, which again are not allowed in convex cocompact groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Checking 1 and 2 is a finite process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' after its completion, one concludes that Γ is convex cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We refer the reader to [Gi1, Gi2, GiM, K2] and [KL2, sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8] for more details concerning discreteness algorithms for groups acting on hyperbolic planes and hyperbolic 3-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We now consider group actions on general symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let Γ be a hyperbolic group with a fixed finite (symmetric) generating set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we equip the group Γ with the word metric determined by this generating set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For each n, let Ln denote the set of maps q : r0, 3ns X Z Ñ Γ which are restrictions of geodesics ˜q : Z Ñ Γ, such that qp0q “ 1 P Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of the geodesic automatic structure on Γ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' [Ep, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5]), the set Ln can be described via a finite state automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that ρ : Γ ñ X is an isometric action on a symmetric space X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we fix a base-point x P X and the corresponding orbit map f : Γ Ñ Γx Ă X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We also fix an ι-invariant face τmod of the model spherical simplex σmod of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The algorithm that we are about to describe will detect that the action ρ is τmod-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 26 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the face τmod is not fixed in advance, we would run algorithms for each face τmod in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For the algorithm we will be using a special (countable) increasing family of Weyl convex compact subsets Θ “ Θi Ă ostpτmodq Ă σmod which exhausts ostpτmodq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' in particular, every compact ι-invariant convex subset of ostpτmodq Ă σmod is contained in some Θi: Θi :“ tv P σ : min αPΦτmod αpvq ě 1 i u, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='17) where Φτmod is the subset of the set of simple roots Φ (with respect to σmod) which vanish on the face τmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Clearly, the sets Θi satisfy the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, we consider only those L and D which are natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Next, consider the sequence pLi, Θi, Diq “ pi, Θi, Diq, i P N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In order to detect τmod-Morse actions we will use the local characterization of Morse quasi- geodesics given by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Due to the discrete nature of quasi- geodesics that we will be considering, it suffices to assume that the additive quasi-isometry constant A is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Consider the functions lpΘ, Θ1, δq, ǫpΘ, Θ1, δq as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Using these functions, for the sets Θ “ Θi, Θ1 “ Θi`1 and the constant δ “ 1 we define the numbers li “ lpΘ, Θ1, δq, ǫi “ ǫpΘ, Θ1, δq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Next, for the numbers L “ Li, D “ Di and the sets Θ “ Θi, Θ1 “ Θi`1, consider the numbers si “ spLi, 0, Θi, Θi`1, Di, ǫi`1, li`1q as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to this proposition, every pLi, 0, Θi, Diq-Morse quasigeodesic satisfies the pΘi`1, ǫi`1, li`1, sq-quadruple condition for all s ě si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We note that, a priori, the sequence si need not be increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We set S1 “ s1 and define a monotonic sequence Si recursively by Si`1 “ maxpSi, si`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then every pΘi, Di, Li, 0q-Morse quasigeodesic also satisfies the pΘi`1, ǫi`1, li`1, Si`1q-quadruple condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We are now ready to describe the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For each i P N we compute the numbers li, ǫi and, then, Si, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We then consider finite discrete paths in Γ, q P LSi, and the corresponding discrete paths in X, pptq “ qptqx, t P r0, 3Sis X Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The number of paths q (and, hence, p) for each i is finite, bounded by the growth function of the group Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 27 For each discrete path p we check the pΘi, ǫi, li, Siq-quadruple condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If for some i “ i˚, all paths p satisfy this condition, the algorithm terminates: It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18 that the map f sends all normalized discrete biinfinite geodesics in Γ to Morse quasigeodesics in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, the action Γ ñ X is Morse in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Conversely, suppose that the action of Γ is pΘ, D, L, 0q-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then f sends all isomeric embeddings ˜q : Z Ñ Γ to pΘ, D, L, 0q-Morse quasigeodesics ˜p in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of the properties of the sequence pLi, Θi, Diq, it follows that for some i, pL, Θ, Dq ď pLi, Θi, Diq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=', L ď Li, Θ Ă Θi, D ď Di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' hence, all the biinfinite discrete paths ˜p are pΘi, Di, Li, 0q- Morse quasigeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the definition of the numbers li, ǫi, Si, it then follows that all the discrete paths p “ f ˝ q, q P LSi satisfy the pΘi`1, ǫi`1, li`1, Si`1q-quadruple condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, the algorithm will terminate at the step i ` 1 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, the algorithm terminates if and only if the action is Morse (for some parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If the action is not Morse, the algorithm will run forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Applied to a rank one symmetric space X and a hyperbolic group Γ without a nontrivial normal finite subgroup, the above algorithm verifies if the given representation ρ : Γ Ñ IsompXq is faithful with convex-cocompact image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We could not find this result in the existing literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' however [GK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5 Appendix: Further properties of Morse quasigeodesics This is the only part of the paper not contained in [KLP1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Here we collect various properties of Morse quasigeodesics that we found to be useful elsewhere in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 Finsler geometry of symmetric spaces In [KL1], see also [KLP5], we considered a certain class of G-invariant “polyhedral” Finsler metrics on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Their geometric and asymptotic properties turned out to be well adapted to the study of geometric and dynamical properties of regular subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' They provide a Finsler geodesic combing of X which is, in many ways, more suitable for analyzing the asymptotic geometry of X than the geodesic combing given by the standard Riemannian metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' These Finsler metrics also play a basic role in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We briefly recall their definition and some basic properties, and refer to [KL1, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let ¯θ P intpτmodq be a type spanning the face type τmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The ¯θ-Finsler distance d ¯θ on X is the G-invariant pseudo-metric defined by d ¯θpx, yq :“ max θpξq“¯θ ` bξpxq ´ bξpyq ˘ 28 for x, y P X, where the maximum is taken over all ideal points ξ P B8X with type θpξq “ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It is positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' a (non-symmetric) metric, if and only if the radius of σmod with respect to ¯θ is ă π 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This is in turn equivalent to ¯θ not being contained in a factor of a nontrivial spherical join decomposition of σmod, and is always satisfied e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' if X is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If d¯θ is positive, it is equivalent to the Riemannian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In general, if it is only a pseudo- metric, it is still equivalent to the Riemannian metric d on uniformly regular pairs of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' More precisely, if the pair of points x, y is Θ-regular, then L´1dpx, yq ď d ¯θpx, yq ď Ldpx, yq with a constant L “ LpΘq ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Regarding symmetry of the Finsler distance, one has the identity dι¯θpy, xq “ d ¯θpx, yq and hence d¯θ is symmetric if and only if ι¯θ “ ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We refer to d¯θ as a Finsler metric of type τmod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The d¯θ-balls in X are convex but not strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (Their intersections with flats through their centers are polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') Accordingly, d ¯θ-geodesics connecting two given points x, y are not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' To simplify notation, xy will stand for some d ¯θ-geodesic connecting x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The union of all d¯θ-geodesic xy equals the τmod-diamond ♦τmodpx, yq, that is, a point lies on a d¯θ-geodesic xy if and only if it is contained in ♦τmodpx, yq, see [KLP5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Finsler geometry thus provides an alternative description of diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that with this description, the diamond ♦τmodpx, yq is also defined when the segment xy is not τmod-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Such a degenerate τmod-diamond is contained in a smaller totally-geodesic subspace, namely in the intersection of all τmod-parallel sets containing the points x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The description of geodesics and diamonds also implies that the unparameterized d ¯θ-geodesics depend only on the face type τmod, and not on ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will refer to d¯θ-geodesics as τmod-Finsler geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that Riemannian geodesics are Finsler geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will call a Θ-regular τmod-Finsler geodesic a Θ-Finsler geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' If xy is a Θ-regular (Rie- mannian) segment, then the union of Θ-Finsler geodesics xy equals the Θ-diamond ♦Θpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Every τmod-Finsler ray in X is contained in a τmod-Weyl cone, and we will use the notation xτ for a τmod-Finsler ray contained V px, stpτqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Similarly, every τmod-Finsler line is contained in a τmod-parallel set, and we denote by τ´τ` an oriented τmod-Finsler line forward/backward asymptotic to two antipodal simplices τ˘ P Flagpτmodq and contained in Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Examples of Θ-regular Finsler geodesics can be obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let pxiq be a (finite or infinite) sequence contained in a parallel set Ppτ´, τ`q such that each Riemannian segment xixi`1 is τ`-longitudinal and Θ1-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then the concatenation of these geodesic segments is Conversely, every Θ-regular Finsler geodesic c : I Ñ X can be approximated by a piecewise- Riemannian Finsler geodesic c1: Pick a number s ą 0 and consider a maximal s-separated subset J Ă I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then take c1 to be the concatenation of Riemannian geodesic segments cpiqcpjq for consecutive pairs i, j P J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of this approximation procedure, the String of Diamonds Theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30) holds if instead of Riemannian geodesic segments xixi`1 we allow Θ-regular Finsler segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 29 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2 Stability of diamonds Diamonds can be regarded as Finsler-geometric replacements of geodesic segments in nonposi- tively curved symmetric spaces of higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Riemannian geodesic segments in Hadamard manifolds (and, more generally, CATp0q metric spaces) depend uniformly continuously on their tips: By convexity of the distance function we have, dHauspxy, x1y1q ď maxpdpx, x1q, dpy, y1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In [KLP2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='70] we proved that diamonds ♦τmod depend continuously on their tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Below we establish uniform control on how much sufficiently large Θ-diamonds vary with their tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For d1 ą d ą 0 there exists C “ CpΘ, Θ1, d, d1q such that the following holds: If a segment x´x` Ă X is Θ-regular with length ě C and y˘ P Bpx˘, dq, then the segment y´y` is Θ1-regular and ♦Θpx´, x`q Ă Nd1p♦Θ1py´, y`qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The Θ1-regularity of y´y` for sufficiently large C follows from the ∆-triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that there exists no constant C for which also the second assertion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then there are sequences of points x˘ n with dpx´ n , x` n q Ñ `8, y˘ n with dpx˘ n , y˘ n q ď d, xn P ♦Θpx´ n , x` n q and yn P ♦Θ1py´ n , y` n q with dpxn, ♦Θ1py´ n , y` n qq “ dpxn, ynq “ d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We may assume convergence xn Ñ x8 and yn Ñ y8 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' After extraction, at least one of the sequences px˘ n q diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There are two cases to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose first that both sequences px˘ n q diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then they are uniformly τmod-regular and, after extraction, we have τmod-flag convergence x˘ n , y˘ n Ñ τ˘ P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The limit simplices τ˘ are antipodal (because xn Ñ x8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We observe that dpxn, B♦Θ1px´ n , x` n qq, dpyn, B♦Θ1py´ n , y` n qq Ñ `8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It follows that the sequences of diamonds ♦Θ1px´ n , x` n q and ♦Θ1py´ n , y` n q both Hausdorff converge to the τmod-parallel set P “ Ppτ´, τ`q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It holds that x8 P P because xn P ♦Θpx´ n , x` n q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' On the other hand, dpx8, Pq “ d1 because dpxn, ♦Θ1py´ n , y` n qq “ d1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Second, suppose that only one of the sequences px˘ n q diverges, say, after extraction, x´ n Ñ x´ 8 and y´ n Ñ y´ 8 in X to limit points with dpx´ 8, y´ 8q ď d, and x` n Ñ τ` P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Now the distance of xn from the boundary of the Θ1-Weyl cone with tip x` n and containing xn goes to infinity and it follows that ♦Θ1px´ n , x` n q Ñ V px´ 8, stΘ1pτ`qq and, similarly, ♦Θ1py´ n , y` n q Ñ V py´ 8, stΘ1pτ`qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The asymptotic limit Weyl cones have Hausdorff distance dpx´ 8, y´ 8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' On the other hand, x8 P V px´ 8, stΘ1pτ`qq and dpx8, V py´ 8, stΘ1pτ`qqq “ d1, again a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This shows that also (ii) holds for sufficiently large C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We reformulate this result in terms of Finsler geodesics: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' There exists C “ CpΘ, Θ1, d, d1q such that the following holds: If x´x` is a Θ- Finsler geodesic in X with dpx´, x`q ě C and y˘ are points with dpy˘, x˘q ď d, then every 30 point x on x´x` lies within distance d1 of a point y on a Θ1-Finsler geodesic y´y`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Note that we do not claim here that one can take the same Finsler geodesic y´y` for all points x on x´x`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We now apply this stabilty result to Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' One, somewhat annoying, feature of the definition of Θ-Morse quasigeodesics p : I Ñ X is that pprt1, t2sq is not required to be uniformly close to a Θ-diamond spanned by ppt1q, ppt2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (One reason is because the segment ppt1qppt2q need not be Θ-regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') Nevertheless, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1 implies: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every Morse datum M “ pΘ, B, L, Aq and Θ1 ą Θ, there exists C “ CpM, Θ1q and D1 such that whenever dpx1, x2q ě C, the segment x1x2 “ ppt1qppt2q is Θ1-regular and pprt1, t2sq lies in the D1-neighborhood of the Θ1-diamond ♦Θ1px1, x2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='3 Finsler approximation of Morse quasigeodesics The next theorem establishes that every (sufficiently long) Morse quasigeodesic is uniformly close to a Finsler geodesic with the same end-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In this theorem, for convenience of the notation, we will be allowing Morse quasigeodesics p to be defined on closed intervals I in the extended real line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' this is just a shorthand for a map I1 “ I XR Ñ X such that, as I1 Q t Ñ ˘8, pptq Ñ pp˘8q P Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' When we say that such maps p, c are within distance D1 from each other, this simply means that their restrictions to I1 are within distance ď D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 (Finsler approximation theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For every Morse datum M “ pΘ, D, L, Aq, Θ1 ą Θ, and a positive number S, there exist C “ CpM, Θ1, Sq, D1 “ D1pM, Θ1, Sq satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let p : I “ rt´, t`s Ñ X Y Flagpτmodq be a M-Morse quasigeodesic between the points x˘ “ ppt˘q P X Y Flagpτmodq such that dpx´, x`q ě C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then there exists a Θ1-Finsler geodesic x´x` equipped with a monotonic parameterization c : I Ñ x´x` such that: (a) The maps p, c : I Ñ X are within distance ď D1 from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (b) x´x` is an S-spaced piecewise-Riemannian geodesic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' the Riemannian length of each Riemannian segments of x´x` is ě S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will prove this in the case when both x˘ are in X since the proofs when one or both points x˘ are in Flagpτmodq are similar: One replaces diamonds with Weyl cones or parallel sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the definition of an M-Morse quasigeodesic, for all subintervals rs´, s`s Ă rt´, t`s, there exists a Θ-diamond ♦Θpy1 ´, y1 `q whose D-neighborhood contains pprs´, s`sq, and for y˘ “ pps˘q, we have dpy˘, y1 ˘q ď D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 31 Therefore, applying the first part of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1, we conclude that the Riemannian segment y´y` is Θ1-regular provided that dpy´, y`q ě C1 “ C1pM, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In view of the quasigeodesic property of p, the last inequality follows from the separation condition s` ´ s´ ě s “ spM, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' This, of course, also applies to rs´, s`s “ rt´, t`s and, hence, using the second part of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='1, we obtain ppIq Ă ND ` ♦Θpx1 ´, x1 `q ˘ Ă ND`D1 p♦Θ1px´, x`qq , where D1 “ D1pM, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We let ¯y˘ P ♦1 :“ ♦Θ1px´, x`q “ V px´, stΘ1pτ`qq X V px`, stΘ1pτ´qq denote the nearest-point projections of y˘ “ pps˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As long as s` ´ s´ ě s1pM, Θ1q, the Riemannian segments ¯y´¯y` are also Θ1-regular and have length ě S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Furthermore, as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='32, we can choose s1 such that each segment ¯y´¯y` is τ`-longitudinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We assume, from now on, that t` ´ t´ ě s2pM, Θ1q, which is achieved by assuming that L´1pdpx´, x`q ´ Aq ě s1pM, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Take a maximal s1-separated subset J Ă I containing t˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' For each j P J define the point zj :“ ppjq P ♦1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then for all consecutive i, j P J, s1 ď |j ´ i| ď 2s1 we have L´1s1 ´ pA ` 2D ` 2D1q ď dpzi, zjq ď 2Ls1 ` pA ` 2D ` 2D1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5) We then let c denote the concatenation of Riemannian segments zizj for consecutive i, j P J, where we use the affine parameterization of ri, js Ñ zizj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Thus, c is a Θ1-Finsler geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We now take the smallest s2 ě s1pM, Θ1q satisfying S ď L´1s2 ´ pA ` 2D ` 2D1q, the inequalities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='5) imply that c satisfies both requirements of the approximation theorem with D1 “ 2Ls2 ` pA ` 2D ` 2D1q ` pD ` D1q ` p2Ls2 ` Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' In the case when the domain of p is unbounded, one can prove a bit sharper result, namely, one can take Θ1 “ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Compare [KL3, sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4 Altering Morse quasigeodesics Below we consider certain modifications of M-Morse quasigeodesics p in X represented as concatenations p “ p´ ‹ p0 ‹ p`, where x˘ are the end-points of p0, and y˘, x˘ are the end- points of p˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (As in the previous section, we will be allowing y˘ to be in X Y Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') 32 These modifications will have the form p1 “ p1 ´ ‹ p1 0 ‹ p1 `, where p1 ˘ and p1 0 are all Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We will see that, under certain assumptions, the entire p1 is again Morse (for suitable Morse datum M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We begin by analyzing extensions of p to biinfinite paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7 (Extension lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that p˘ Ă V˘ “ V px˘, stpτ˘qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Whenever y˘ is in X, we let c˘ be Θ-regular Finsler rays contained in V˘ and connecting y˘ to τ˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then, for every Θ1 ą Θ, there exists a Morse datum M1 containing Θ1 such that the concatenation ˆp “ c´ ‹ p ‹ c` is M1-Morse, provided that dpx˘, y˘q ě C “ CpM, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We fix an auxiliary subset Θ1 satisfying Θ ă Θ1 ă Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We let S “ SpΘ1, Θ1, 1q, ǫ “ ǫpΘ1, Θ1, 1q be constants as in the string of diamonds theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='4, there exists a Θ1-regular Finsler geodesic ¯c “ y´¯x´ ‹ ¯x´¯x` ‹ ¯x`y` within distance D1 “ D1pM, Θ1, Sq from the path p, such that ¯c is the concatenation of segments of length ě S and dpx˘, ¯x˘q ď D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We let z˘y˘ denote the subsegments of ¯x˘y˘ containing y˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since dpx˘, ¯x˘q ď D1, for each ǫ ą 0 and a sufficiently large C1 “ C1pD1, Θ1q, the inequality dpx˘, y˘q ě C1 implies =ζ y˘px˘, ¯x˘q ď ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Therefore, =ζ y˘pz˘, τ˘q ě π ´ ǫ and, hence, the piecewise-geodesic path ˆc “ c´ ‹ ¯c ‹ c` is pΘ1, ǫq-straight and S-spaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='30, the concatenation ˆc is M1-Morse, where M1 “ pΘ1, 1, L, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since the path ˆp is within distance D1 from ˆc, it is M1-Morse, where M1 “ M1 ` D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The next lemma was proven in [DKL, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11] in the case when p, p1 are finite paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The proof in the case of (bi)infinite paths is the same and we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8 (Replacement lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that p1 “ p1 ´ ‹p1 0 ‹p1 ` is a concatenation of M- Morse quasigeodesics in X, such that the end-points of p˘, p1 ˘ and p0, p1 0 are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then for every Θ1 ą Θ there exists a Morse datum M1 containing Θ1 such that the path p1 is M1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 33 In the following lemmata we will modify the path p by altering p˘ and keeping p0 unchanged or moving it by a small amount (“wiggling the head and the tail of p”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9 (Wiggle lemma, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Suppose that the paths p˘, p1 ˘ are both infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We let p1 ˘ be M-Morse quasigeodesics with finite terminal points x˘ and set p1 :“ p1 ´ ‹ p0 ‹ p1 `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Given Θ1 ą Θ there exists ǫ “ ǫpM, Θ1q ą 0 and a Morse datum M1 containing Θ1 such that if µ :“ maxp=ζ x˘pp1 ˘p˘8q, p˘p˘8qqq ă ǫ, then p1 is M1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We fix an auxiliary compact Weyl-convex subset Θ1 Ă ostpτmodq such that Θ ă Θ1 ă Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Set τ˘ “ p˘p˘8q, τ 1 ˘ “ p1 ˘p˘8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='8, there exists a Morse datum M1 containing Θ1 such that for any Θ1-regular Finsler geodesic rays c˘ :“ x˘τ˘, the concatenation c´ ‹ p0 ‹ c` is M1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let M2 ą M1 ` 1 be a Morse datum containing Θ1 and let S ą 0 be such that if a path q in X is S-locally M1 ` 1-Morse then q is M2-Morse (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Let ǫ be such that for x P X, τ, τ 1 P Flagpτmodq, if =ζ xpτ, τ 1q ă ǫ then each Θ1-regular Finsler segment of length ď S in V px, stpτ 1qq is within unit distance from a Θ1-regular Finsler segment of length ď S in V px, stpτqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We assume now that µ ă ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since p1 ˘ are M-Morse rays, they are within distance D1 “ D1pM, Θ1q from Θ1-regular Finsler rays c1 ˘ “ x˘τ 1 ˘ connecting x˘ and τ 1 ˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Define a new path c1 :“ c1 ´ ‹ p0 ‹ c1 `.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By our choice of ǫ, the Θ1-regular Finsler subsegment s1 ˘ “ x˘y1 ˘ of c1 ˘ of length S is within unit distance from a Θ1-regular Finsler subsegment s˘ “ x˘y˘ of c˘ of length S, where c˘ “ x˘τ˘ is a Θ1-Finsler geodesic connecting x˘ to τ˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The concatenation s´ ‹ p0 ‹ s` is M1-Morse, and, since c1 ˘ are Θ1-Finsler geodesic, the path c1 is S-locally M1 ` 1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By our choice of S, the path c1 is M2-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since c1 is within distance D1 from p1, the path p1 is M2 ` D1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lastly, we set M1 :“ M2 ` D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We generalize this lemma by allowing finite Morse quasigeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We continue with the setting of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' we now allow paths p˘ and p1 ˘ to be finite, connecting y˘, x˘ and y1 ˘, x˘ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' (Some of y˘, y1 ˘ might be in Flagpτmodq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=') However, we will assume that the distances dpx˘, y˘q, dpx1 ˘, y˘q are sufficiently large, ě C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10 (Wiggle lemma, II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Given Θ1 ą Θ there exist C ě 0, ǫ ą 0 and a Morse datum M1 containing Θ1 such that if µ :“ maxp=ζ x˘py1 ˘, y˘qq ă ǫ, and ν :“ minpdpx˘, y˘q, dpx˘, y1 ˘qq ě C then p1 is M1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' 34 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Pick an auxiliary compact Weyl-convex subset Θ2, Θ ă Θ2 ă Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We define biinfinite geodesic extensions ˆp, ˆp1 as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7, by extending (if necessary) the paths p˘, p1 ˘ via Θ-Finsler geodesics y˘τ˘ and y1 ˘τ 1 ˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='7, there exists C ą 0and a Morse datum M2 (containing Θ2), both depending on M and Θ2, such that the path ˆp is M2-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The same lemma applied to the paths ˆp1 ˘ implies that they are also M2-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' By the construction, µ :“ =ζ x˘py1 ˘, y˘q “ =ζ x˘pτ 1 ˘, τ˘q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Now, claim follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lastly, we prove a general Wiggle Lemma where we allow to perturb the entire path p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We consider concatenations p “ p´ ‹ p0 ‹ p`, p1 “ p1 ´ ‹ p1 0 ‹ p1 ` of M-Morse quasigeodesics, where we assume that p0, p1 0 are within distance D0 from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' The paths p˘ connect y˘, x˘ and p1 ˘ connect y1 ˘, x1 ˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='11 (Wiggle lemma, III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Given Θ1 ą Θ there exist C ě 0, ǫ ą 0 and a Morse datum M1 containing Θ1 such that if µ :“ maxp=ζ x˘py1 ˘, y˘qq ă ǫ, and ν :“ minpdpx˘, y˘q, dpx1 ˘, y1 ˘qq ě C then p1 is M1-Morse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' As before, we fix an auxiliary compact Weyl-convex subset Θ3, Θ ă Θ3 ă Θ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Then p1 ˘ are within distance D3 “ D3pM, Θ3q from Θ3-regular Finsler geodesics c˘ :“ y1 ˘x˘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' We apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='10 to the pair of paths p, p2 :“ c´ ‹ p0 ‹ c`.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' It follows that p2 is M3-Morse for some Morse datum M3 containing Θ1 provided that µ ď ǫ “ ǫpM, Θ3, Θ1q and ν ě C “ CpM, Θ3, Θ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=' Since the paths p2 and p1 are wihin distance D1 :“ maxpD0, D3q 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M¨unchen, Germany email: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='l@lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='de J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content=': Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Spain email: porti@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='uab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} +page_content='cat 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfggp5/content/2301.04562v1.pdf'} diff --git a/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/2301.01147v1.pdf.txt b/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/2301.01147v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f090ecd9772b73ee0a13d8065f3b23867266d242 --- /dev/null +++ b/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/2301.01147v1.pdf.txt @@ -0,0 +1,2851 @@ +4Seasons: Benchmarking Visual SLAM and Long-Term +Localization for Autonomous Driving in Challenging +Conditions +Patrick Wenzel1*, Nan Yang2†, Rui Wang3†, Niclas Zeller4† and Daniel Cremers1 +1Department of Computer Science, Technical University of Munich, Germany. +2Reality Labs at Meta, Redmond, United States. +3Microsoft Mixed Reality & AI Lab, Zurich, Switzerland. +4Karlsruhe University of Applied Sciences, Karlsruhe, Germany. +*Corresponding author(s). E-mail(s): patrick.wenzel@tum.de; +†Work done at Technical University of Munich. +Abstract +In this paper, we present a novel visual SLAM and long-term localization benchmark for autonomous +driving in challenging conditions based on the large-scale 4Seasons dataset. The proposed bench- +mark provides drastic appearance variations caused by seasonal changes and diverse weather and +illumination conditions. While significant progress has been made in advancing visual SLAM on +small-scale datasets with similar conditions, there is still a lack of unified benchmarks represen- +tative of real-world scenarios for autonomous driving. We introduce a new unified benchmark +for jointly evaluating visual odometry, global place recognition, and map-based visual localiza- +tion performance which is crucial to successfully enable autonomous driving in any condition. +The data has been collected for more than one year, resulting in more than 300 km of record- +ings in nine different environments ranging from a multi-level parking garage to urban (including +tunnels) to countryside and highway. We provide globally consistent reference poses with up to +centimeter-level accuracy obtained from the fusion of direct stereo-inertial odometry with RTK +GNSS. We evaluate the performance of several state-of-the-art visual odometry and visual localiza- +tion baseline approaches on the benchmark and analyze their properties. The experimental results +provide new insights into current approaches and show promising potential for future research. +Our benchmark and evaluation protocols will be available at https://www.4seasons-dataset.com/. +Keywords: Autonomous Driving, Benchmark, Long-Term Visual Localization, SLAM, Visual Odometry, +Camera Pose Estimation +1 Introduction +During the last decade, research on visual odome- +try (VO) and simultaneous localization and map- +ping (SLAM) has made tremendous strides [13, 16, +40, 41], particularly in the context of autonomous +driving [14, 39, 69, 73]. One reason for this +progress has been the publication of large-scale +datasets tailored for benchmarking these meth- +ods [8, 10, 20]. Nonetheless, existing algorithms +have significant limitations. Most approaches are +1 +arXiv:2301.01147v1 [cs.CV] 31 Dec 2022 + +2 +Wenzel et al. +Fig. +1: +4Seasons +benchmark +dataset +overview. Top: overlaid maps recorded at differ- +ent times and environmental conditions. The 3D +points from the reference map (black) align well +with the 3D points from the query map (blue), +indicating that the reference poses are accurate. +Bottom: sample images demonstrating the diver- +sity of our benchmark. The first row shows a +collection from the same scene across different +weather and lighting conditions: snowy, cloudy, +sunny, and night. The second row depicts the +variety of scenarios within the benchmark: inner +city, suburban, countryside, and a parking garage. +tailored to work well on small-scale datasets which +exhibit limited challenging conditions. +Therefore, the next logical step towards pro- +gressing research in the direction of visual SLAM +is to make it robust under dynamically chang- +ing and challenging conditions. This includes VO, +e.g. at night or rain, as well as long-term place +recognition and localization against a pre-built +map. In this regard, the advent of deep learning +has exhibited itself to be a promising poten- +tial in complementing the performance of visual +SLAM [12, 28, 30, 58]. Therefore, it has become +all the more important to have datasets that are +commensurate with handling the challenges of any +real-world environment while also being capable +of discerning the performance of state-of-the-art +approaches. +To accommodate this demand, we present a +cross-season and multi-weather benchmark, par- +ticularly focusing on visual SLAM and long-term +localization for autonomous driving. This bench- +mark is based on the versatile large-scale 4Seasons +dataset [72]. To the best of our knowledge, we pro- +vide the first large-scale cross-season benchmark +dataset comprising stereo images, corresponding +high frame-rate inertial measurement unit (IMU), +and accurate RTK GNSS measurements to evalu- +ate sequential localization methods. By traversing +the same route under different conditions and over +a long-term time horizon, we capture variety in +illumination and weather, as well as in the appear- +ance of the scenes. For each scenario, we provide +multiple traversals exhibiting different environ- +mental conditions, as described in Table 5. The +recordings show vastly different variations in the +scene geometry including dynamic objects, road- +works, construction sites, and seasonal changes. +To acquire accurate reference poses of large-scale +scenes, we use a custom stereo-inertial sensor +together with a RTK GNSS system to obtain up +to centimeter-accurate poses. Figure 1 visualizes +two overlaid 3D reconstructions of the same scene +recorded at different times. Moreover, the figure +depicts sample images of the dataset used to eval- +uate six degrees of freedom (6DoF) localization +against a prior map using query images taken +from a variety of challenging conditions. We pro- +vide reference poses for a subset of the recordings, +and withhold the remaining for an online evalua- +tion benchmark suite. We design a benchmark to +measure the impact of long-term environmental +changes on the performance of visual SLAM and +localization for autonomous driving. +The main contributions of this paper are the +extensive benchmark suite for evaluating the long- +term visual localization problem for autonomous +driving, the evaluation of state-of-the-art baseline +SLAM and visual localization algorithms, and the +interpretation of the results. +This work extends our paper published at +GCPR 2020 [72] through the following additional +contributions: +• We propose a large-scale cross-season and multi- +weather benchmark suite for long-term visual +SLAM in automotive applications. It allows the +joint evaluation of visual odometry, global place +recognition, and map-based visual localization +approaches. +• We release plenty of additional sequences cover- +ing nine different types of environments, ranging +from a multi-level parking garage to urban +(including tunnels) to countryside and highway. + +Wenzel et al. +3 +• We provide an extensive evaluation of state- +of-the-art baseline approaches for visual SLAM +and visual localization on the presented bench- +mark. +To foster research, our benchmark and evalu- +ation protocols will be available at https://www. +4seasons-dataset.com/. +2 Related Work +There exists a variety of benchmarks and datasets +focusing on VO and SLAM for autonomous driv- +ing. Here, we divide these datasets into the ones +which focus only on VO as well as those covering +different weather conditions and therefore aiming +towards long-term SLAM. +2.1 Visual Odometry Datasets & +Benchmarks +The most popular benchmark for autonomous +driving probably is KITTI [20]. This multi-sensor +dataset covers a wide range of tasks including +not only VO, but also 3D object detection and +tracking, scene flow estimation as well as seman- +tic scene understanding. The dataset contains +diverse scenarios ranging from urban to country- +side to highway. Nevertheless, all scenarios are +only recorded once and under similar weather +conditions. Ground truth is obtained based on a +high-end inertial navigation system (INS). +Another dataset containing LiDAR, IMU, and +image data at a large scale is the M´alaga Urban +dataset [6]. However, in contrast to KITTI, no +accurate 6DoF ground truth is provided, and +therefore it does not allow for an appropriate +quantitative evaluation. Moreover, only a few +places are visited multiple times. +Other popular datasets for the evaluation +of VO and visual-inertial odometry (VIO) algo- +rithms that are not related to autonomous driv- +ing include [59] (handheld RGB-D), [7] (UAV +stereo-inertial), [15] (handheld mono), and [55] +(handheld stereo-inertial). +2.2 Long-Term SLAM Datasets & +Benchmarks +More related to our work are datasets containing +multiple traversals of the same environment over +a long period. Concerning SLAM for autonomous +driving, the Oxford RobotCar Dataset [38] repre- +sents a kind of pioneer work. This dataset consists +of large-scale sequences recorded multiple times +in the same environment for one year. Hence, +it covers large variations in the appearance and +structure of the scene. However, the diversity of +the scenarios is only limited to an urban envi- +ronment. Also, the ground truth provided for the +dataset is not accurate up to centimeter-level [38, +57]. Other existing datasets are lacking sequen- +tial structure [33], only provide a certain adverse +condition [42], or focus on AR scenarios [50]. +The work by [52] proposes three comple- +mentary benchmark datasets based on exist- +ing datasets, namely RobotCar Seasons (based +on [38]), Aachen Day-Night (based on [51]), and +CMU Seasons (based on [5]) that have been used +for benchmarking visual localization approaches. +The ground truth of the RobotCar Seasons [52] +dataset is obtained based on structure from +motion (SfM) and LiDAR point cloud align- +ment. However, due to inaccurate GNSS mea- +surements [38], a globally consistent ground truth +up to centimeter-level accuracy can not be guar- +anteed. Furthermore, this dataset only provides +one reference traversal in the overcast condi- +tion. In contrast, we provide globally consistent +reference models for all training traversals cov- +ering a wide variety of conditions. Hence, every +traversal can be used as a reference model that +allows further research on, e.g. analyzing suitable +reference-query pairs for long-term localization +and mapping. +Global place recognition datasets such as +Pittsburgh [63], Tokyo 24/7 [64], and Mapil- +lary Street-Level Sequences [71] provide only +coarse-scale location information. Other related +localization +datasets +include +12-Scenes +[67], +InLoc +[61], +Cambridge +Landmarks +[32], +and +CrowdDriven [27]. +2.3 Other Datasets +Examples of further multipurpose autonomous +driving datasets that also can be used for VO +are [8, 10, 26, 70]. +As stated in Section 1, our proposed bench- +mark dataset differentiates from previous related +work in terms of being both large-scale (similar +to [20]) and having high variations in appear- +ance and conditions (similar to [38]). Furthermore, + +4 +Wenzel et al. +accurate reference poses based on the fusion of +direct stereo VIO and RTK GNSS are provided. +To the best of our knowledge, we are the first to +introduce a public, modular benchmark for evalu- +ating visual SLAM, global place recognition, and +map-based visual localization approaches under +challenging conditions for autonomous driving. +3 System Overview +This section presents the sensor setup which is +used for data recording (Section 3.1). Further- +more, we describe the calibration of the entire +sensor suite (Section 3.2) as well as our approach +to obtain up to centimeter-accurate global 6DoF +reference poses (Section 3.3). +3.1 Sensor Setup +The hardware setup consists of a custom stereo- +inertial sensor for 6DoF pose estimation, as well +as a high-end RTK GNSS receiver for global posi- +tioning and global pose refinement. Figure 2 shows +our test vehicle equipped with the sensor system +used for data acquisition. +3.1.1 Stereo-Inertial Sensor +The core of the sensor system is our custom stereo- +inertial sensor. This sensor consists of a pair of +monochrome industrial-grade global shutter cam- +eras (Basler acA2040-35gm) and lenses with a +fixed focal length of f = 3.5 mm (Stemmer Imag- +ing CVO GMTHR23514MCN). The cameras are +mounted on a highly-rigid aluminum rail with a +stereo baseline of 30 cm. On the same rail, a Pre- +cision MEMS IMU (Analog Devices ADIS16465) +is mounted. The cameras and the IMU are trig- +gered over an external clock generated by an +FPGA. Here, the trigger accounts for exposure +compensations, meaning that the time between +the centers of the exposure interval for two con- +secutive images is always kept constant (1/[frame +rate]) independent of the exposure time itself. +Furthermore, based on the FPGA, the IMU is +properly synchronized with the cameras. In the +dataset, we record stereo sequences with a frame +rate of 30 fps. We perform pixel binning with a +factor of two and crop the image to a resolution of +800 × 400. This results in a field of view of approx- +imately 77° horizontally and 43° vertically. The +IMU is recorded at a frequency of 2000 Hz. During +recording, we guarantee an equal exposure time +for the left and the right image of each stereo pair +as well as a smooth exposure transition in highly +dynamic lighting conditions, as it is favorable to +visual SLAM. We provide those exposure times for +each frame. +3.1.2 GNSS Receiver +For global positioning and to compensate drift in +the VIO system, we utilize an RTK GNSS receiver +(mosaic-X5) from Septentrio in combination with +an Antcom Active G8 GNSS antenna. The GNSS +receiver provides a horizontal position accuracy +of up to 6 mm by utilizing RTK correction sig- +nals. While the high-end GNSS receiver is used +for accurate positioning, we use a second receiver +connected to the time-synchronization FPGA to +obtain GNSS timestamps for the sensors. +3.2 Calibration +3.2.1 Aperture and Focus Adjustment +The lenses used in the stereo system have both +adjustable aperture and focus. Therefore, before +performing the geometric calibration of all sensors, +we manually adjust both cameras for a matching +average brightness and a minimum focus blur [25], +across a structured planar target in 10 m distance. +3.2.2 Stereo Camera and IMU +For the intrinsic and extrinsic calibration of the +stereo cameras, as well as the extrinsic calibra- +tion and time-synchronization of the IMU, we +use Kalibr1 [45]. The stereo cameras are mod- +eled using the Kannala-Brandt model [31], a +generic camera model consisting of a total of eight +parameters. We validated the calibration accuracy +of each recording by performing a feature-based +epipolar-line consistency check. +3.2.3 GNSS Antenna +Since the GNSS antenna does not have any orien- +tation but has an isotropic reception pattern, only +the 3D translation vector between one of the cam- +eras and the antenna within the camera frame has +to be known. This vector was measured manually +for our sensor setup. +1https://github.com/ethz-asl/kalibr + +Wenzel et al. +5 +(a) Test vehicle. +(b) Sensor system. +Fig. 2: Recording setup. Test vehicle and sensor system used for dataset recording. The sensor system +consists of a custom stereo-inertial sensor with a stereo baseline of 30 cm and a high-end RTK GNSS +receiver from Septentrio. +3.3 Ground Truth Generation +Reference poses (i.e. ground truth) for VO and +SLAM should provide high accuracy in both local +relative 6DoF transformations and global posi- +tioning. To fulfill the first requirement, we extend +the state-of-the-art stereo direct sparse VO [69] +by integrating IMU measurements [68], achieving +a stereo-inertial SLAM system offering average +tracking drift around 0.6 % of the traveled dis- +tance. +To fulfill the second requirement, the poses +estimated by our stereo-inertial system are fused +with the RTK GNSS measurements using a global +pose graph. We first estimate a Sim(3) transfor- +mation to globally align the camera positions in +the VIO coordinate system to those in the GNSS +coordinate system using the Kabsch–Umeyama +algorithm [65]. A transformation in Sim(3) is esti- +mated instead of in SE(3) to account for the global +scale drift in the VIO system. Denoting the Lie- +algebra of SE(3) as se(3), each aligned camera +pose ξVIO +wi +∈ se(3) is added to the pose graph as +a se(3) node, where ξwi defines a transformation +from the i-th camera coordinate system to the +world coordinate system. The camera connections +from the VIO sliding window (one connection cor- +responds to two cameras co-observing a part of +the scene) are added as se(3) − se(3) edges, with +the relative poses ξVIO +ji +as the measurements. If +a camera pose has a valid corresponding GNSS +pose, that is, the GNSS pose is available and +the observed standard deviation of the position +is smaller than a predefined threshold, the GNSS +pose ti ∈ R3 is added to the pose graph as a +fixed R3 node and an se(3) − R3 edge is added. +The energy function defined for the pose graph +optimization is thus defined as: +E(ξwi, . . . , ξwn) = +� +ξVIO +ji +∈ε +(ξVIO +ji +◦ ξ−1 +wi ◦ ξwj)⊤Σ−1 +ji (ξVIO +ji +◦ ξ−1 +wi ◦ ξwj)+ +ω +� +ti∈ν +(ti − (ξwi ◦ ξcg)[t])⊤Σ−1 +i (ti − (ξwi ◦ ξcg)[t]), +(1) +where ε is the set of VIO camera connections, ν is +the set of valid RTK GNSS poses. Σji ∈ R6×6 and +Σi ∈ R3×3 are the covariance matrices from the +VIO and GNSS systems. ξcg denotes the extrin- +sic calibration between the camera and the GNSS +antenna. A scale term ω is added to balance +the two different domains. The ◦-operator defines +the concatenation of poses defined as se(3) and +therefore is defined as follows: +ξi ◦ ξj := log(exp(ξi) · exp(ξj)), +(2) +where log(·) defines the logarithm and exp(·) +the exponential map of the SE(3) Lie-algebra. +ξ[t] denotes the translation part in se(3). The +energy function is optimized using the Leven- +berg–Marquardt algorithm in [35]. + +GNsS antenna +stereo-inertial sensor6 +Wenzel et al. +One crucial aspect of the dataset is that +the reference poses that we provide are accu- +rate enough, even though some recorded sequences +contain challenging conditions in partially GNSS- +denied environments. Although the stereo-inertial +sensor system has an average drift around 0.6 %, +this cannot be guaranteed for all cases. Hence, +for the reference poses in our dataset, we report +whether a pose can be considered to be reliable +by measuring the distance to the corresponding +RTK GNSS measurement. For all poses, with- +out corresponding RTK GNSS measurement we +do not guarantee a certain accuracy. Nevertheless, +due to the highly accurate stereo-inertial odom- +etry system, these poses can be considered accu- +rate in most cases, even in environments without +GNSS, e.g. tunnels, or areas with tall buildings. +We provide details about the pose accuracy in +Section 4.2.1. +4 Benchmark Setup +To overcome the shortcomings of existing bench- +marks and datasets for autonomous driving, as +discussed in Section 2, we define the following +requirements for an appropriate benchmark. +• Accuracy: we provide up to centimeter-accurate +6DoF poses obtained by fusing VIO measure- +ments with RTK GNSS correction data. +• Large-scale: we provide large-scale sequences +(trajectories longer than 10 km) to allow for +extensive evaluations of SLAM and visual local- +ization under challenging conditions. +• Diversity: besides large-scale, we also provide +both short-term and long-term changes within +the recorded scenes. This is important to eval- +uate the generalization capabilities of recent +learning-based methods. +• Multitask: the benchmark can be used to eval- +uate visual odometry, global place recognition, +and map-based visual localization under chal- +lenging conditions. +Based on these properties, we propose a novel +large-scale dataset that is used as an extensive +benchmark suite for evaluating multitasking chal- +lenges related to autonomous driving under chang- +ing conditions. The sequences have been collected +in the metropolitan area of Munich, Germany. The +different scenes are described in the next section. +Fig. 3: Data collection map. This figure shows +the map of the covered area of our benchmark +dataset. We provide sequences at a large scale +and a huge variety of different environments. A +detailed visualization of each scenario’s trajectory +is shown in Figure 15. +4.1 Scenarios +This section describes the different sequences we +have collected for the dataset. The sequences +involve different scenarios – ranging from urban +driving to a parking garage and rural areas. +We provide complex trajectories, which include +partially overlapping routes, and multiple loops +within a sequence. For each scenario, we have col- +lected multiple traversals covering a large range +of variations in the structure and environmental +appearance due to weather, illumination, dynamic +objects, and seasonal effects. In total, our bench- +mark dataset consists of nine different scenarios. +Figure 3 shows the covered area, including +highlighted traces. Each scenario is visualized in +a separate color. We now describe each scene in +more detail. +1. Office Loop. A loop around an industrial area +of the city. +2. Highway. A drive along the A9 three-lane +highway in the northern part of Munich. + +St 2053 +FS 20 +ohho +5 +A9 +Landkreis Fr +Unterschleisheim +Naturschutzgebiet +Malertshofer +Holz +hing-ord +B13 +Garching-Word +und Fo +Berglhoiz +3heim +4 +Garching bei +B 471 +Hochbruck +Munchen +Hochbruck +Garching-Sud上 +Schweizerholz, +eigher +5t 2053 +Frottmaninger +Heide +Kreuz Mun nen-Nord +Munchen Weuherberg +Monchen-Frotmaning-Nord +Naturschutzgebiet +Sudliche +Frottmaninger +und Hartelholz +Heide +chen-Nord. +St 2053 +B13 +euzMu +Munchen-Frd +maning-Sug +Harthof: +99 +9 +Freim.n +AmHart +Muniche +Euro-Industriepar! +3 +Unterfohring +C +kfurter Ring +Ring +.Schwabing +esenfeld +Milbertshofen +Nord +M3 +Oberfohrir +Schwabing +johanneskirchen +6 +Herzogpar +Englschalk +laxvorstadt +Dennin +Daglfing +Bogenhausen +Zamdorf +lehel +Munchen-Steinhousen +Munchen-Zamdo +Munchen-Dog! +Munchen +Haidhausen +rstadWenzel et al. +7 +Table 1: Statistics of the 4Seasons benchmark. This table shows the different scenarios and record- +ings along with the weather condition, seasons, and time of the day from our benchmark. We provide a +variety of scenarios and short-term to long-term changes. The recordings in this table are used for the +benchmark evaluation. The ground truth (GNSS/IMU, point clouds, and reference poses) is withheld. +Benchmark type (VO = visual odometry, GPR = global place recognition, MBVL = map-based visual +localization) defines the benchmark a sequence is used for. All recordings with ground truth are shown +in Table 5. +Scenario +Recording +Weather +(cloudy, rainy, snowy, sunny) +Season +(winter, spring, summer, fall) +Daytime +(morning, afternoon, evening, night) Benchmark Type +Map Accuracy +Horizontal RMSE +(GNSS-Ref. Pose) +Map Accuracy +% of Accurate Poses +office loop 1 test +2020-03-03 12-12-32 cloudy +spring +afternoon +GPR +12.29 cm +59.91 % +office loop 2 test +2020-03-26 15-03-02 cloudy/sunny +spring +afternoon +VO +5.14 cm +90.22 % +office loop 3 test +2021-05-10 19-25-54 cloudy +spring +evening +VO + MBVL +5.78 cm +92.06 % +highway 1 test +2020-10-08 10-19-46 sunny +fall +morning +VO +8.04 cm +73.65 % +highway 2 test +2021-02-25 13-11-30 sunny +winter +afternoon +VO +4.80 cm +74.31 % +neighborhood 1 test +2020-03-26 14-54-05 cloudy +spring +afternoon +GPR +2.20 cm +87.38 % +neighborhood 2 test +2021-05-10 18-26-26 cloudy +spring +evening +VO + MBVL +1.51 cm +87.42 % +business campus 1 test 2021-01-07 13-03-56 cloudy/snowy +winter +afternoon +VO + MBVL +3.39 cm +97.36 % +countryside 1 test +2020-03-26 14-30-52 cloudy +spring +afternoon +GPR +2.53 cm +91.75 % +countryside 2 test +2021-01-07 14-03-57 cloudy/snowy +winter +afternoon +VO + MBVL +2.36 cm +92.21 % +city loop 1 test +2020-03-03 12-28-45 cloudy +spring +afternoon +GPR +5.36 cm +83.62 % +city loop 2 test +2021-02-25 11-27-40 sunny +winter +morning +VO + MBVL +3.36 cm +81.40 % +old town 1 test +2020-10-08 12-11-19 cloudy +fall +afternoon +GPR +7.19 cm +94.26 % +old town 2 test +2021-05-10 19-51-14 cloudy +spring +evening +VO +1.84 cm +96.04 % +old town 3 test +2021-05-10 21-18-00 cloudy +spring +night +VO + MBVL +4.94 cm +92.07 % +maximilianeum 1 test +2021-02-25 12-16-32 sunny +winter +afternoon +VO +1.90 cm +80.13 % +maximilianeum 2 test +2021-05-10 20-59-00 cloudy +spring +night +VO +12.46 cm +76.46 % +parking garage 1 test +2020-06-12 10-29-20 sunny +summer +morning +VO + MBVL +0.75 cm +35.06 % +parking garage 2 test +2021-05-10 19-18-36 cloudy +spring +evening +GPR +4.54 cm +40.75 % +3. Neighborhood. Traversal through a neigh- +borhood at the outskirts of the city, covering +detached houses with gardens and trees in the +street. +4. Business Campus. Several loops around a +campus in a business area. +5. Countryside. Rural area around agricultural +fields that exhibits very homogeneous and +repetitive structures. +6. City Loop. A large-scale loop at a ring road +within the city of Munich, including a tunnel. +7. Old Town. Loop around the urban city center +with tall buildings, much traffic, and dynamic +objects. +8. Maximilianeum. The Maximilianeum is a +famous palatial building in Munich which is +located at the eastern end of a royal avenue +with paving stones and a tram route. +9. Parking +Garage. +A +three-level +parking +garage to benchmark combined indoor and +outdoor environments. +The VIO traces for each scenario are shown +in Figure 15. We provide reference poses and 3D +models as sparse point clouds generated by our +ground truth generation pipeline (c.f . Figure 4) +along with the corresponding raw image frames +and raw IMU measurements. Figure 5 shows an +example of the optimized trajectory, which depicts +the accuracy of the provided reference poses. +Table 1 shows all the sequences with withheld +ground truth used for benchmarking. +The benchmark dataset presents a challenge to +current approaches to visual SLAM and long-term +localization because it contains data from different +seasons and weather conditions, as well as from +different times of day, as shown in Figure 14. +4.2 Reference Pose Validation +The top part of Figure 1 shows two overlaid +point clouds from different runs across the same +scene. Note that despite the weather and sea- +sonal differences, the point clouds align very well. +This shows that our reference poses are suffi- +ciently accurate for benchmarking long-term local- +ization. Furthermore, a qualitative assessment of +the point-to-point correspondences is shown in +Figure 6. The figure shows a subset of very accu- +rate pixel-wise correspondences across different +seasons (fall/winter) in the top and different illu- +mination conditions (sunny/night) in the bottom. + +8 +Wenzel et al. +Fig. 4: 3D models of different scenarios contained in the dataset. The figure shows an office loop +around an industrial area (left), multiple loops around a business campus with high buildings (middle), +and a stretch recorded in a multi-level parking garage (right). The green lines encode the GNSS trajec- +tories, and the red lines encode the VIO trajectories. Top: shows the trajectories before the fusion using +pose graph optimization. Bottom: shows the results after the pose graph optimization. Note that after +the pose graph optimization, the reference trajectories are well aligned. +These point-to-point correspondences are a result +of our up to centimeter-accurate global reference +poses. This makes them suitable as training pairs +for learning-based algorithms. Recently, there has +been an increasing demand for pixel-wise cross- +season correspondences, which are needed to learn +dense feature descriptors [12, 47, 57]. However, +there is still a lack of datasets to satisfy this +demand. The KITTI [20] dataset does not pro- +vide cross-season data. The Oxford RobotCar +Dataset [38] provides cross-season data, however, +since the ground truth is not accurate enough, +the paper does not recommend benchmarking +localization and mapping approaches. +Recently, RobotCar Seasons [52] was proposed +to overcome the inaccuracy of the provided ground +truth. However, similar to the authors of [57], we +found that it is still challenging to obtain accurate +cross-season pixel-wise matches due to pose incon- +sistencies. Furthermore, this dataset only provides +images captured from three synchronized cameras +mounted on a car, pointing to the rear-left, rear, +and rear-right, respectively. Moreover, another +limitation of the dataset is that it only provides +relatively small segments and no long trajecto- +ries. Furthermore, a significant portion of it suffers +from strong motion blur and low image quality. +4.2.1 Pose Accuracy +One potential limitation of our benchmark dataset +is that we can only guarantee a certain pose accu- +racy when GNSS is available. Naturally, GNSS is +unreliable in urban canyons or tunnels. Therefore, +for the benchmark evaluation, we only consider +poses as reference poses if GNSS is available and +the observed standard deviation of the position is +less than 5 cm. Please note that we only require +accurate reference poses for the evaluation of +visual localization. The evaluation of VO is based +on the accumulated drift over time, i.e. it is only +required that the start and end positions for each +segment of a sequence are accurate. Furthermore, +we provide quantitative measures of the quality of +the maps. We report the percentage of accurate +reference poses for each trajectory. Moreover, we + +11Wenzel et al. +9 +Fig. 5: Reference poses validation. This figure +shows two additional 3D models of the scenar- +ios collected. Note that these two sequences are +quite large (more than 10 km and 6 km, respec- +tively). Top: before the fusion using pose graph +optimization. Bottom: results after optimization. +The green lines encode the GNSS trajectories, the +red lines show the VIO trajectories (before fusion) +and the fused trajectories (after fusion). The left +part of the figure shows a zoomed-in view of a tun- +nel, where the GNSS signal becomes very noisy, as +highlighted in the red boxes. Besides, due to the +large size of the sequence, the accumulated track- +ing error leads to a significant deviation of the +VIO trajectory from the GNSS recordings. Our +pose graph optimization, by depending globally on +GNSS positions and locally on VIO relative poses, +successfully eliminates global VIO drifts and local +GNSS positioning flaws. +report the overall map accuracy in terms of hor- +izontal RMSE between the GNSS poses and the +refined poses after pose graph optimization. +The percentage of accurate poses for each test +sequence can be seen in Table 1 and Table 5 for the +training sequences. For a qualitative visual anal- +ysis, we show accurate pixel-wise correspondences +in Figure 6, indicating that the reference poses +are sufficiently accurate. We do not claim that our +poses are consistently centimeter-accurate, how- +ever, by analyzing the map accuracy we can assure +the quality of the poses used for benchmarking. +4.3 Data Source +We +release +(distorted +& +undistorted) +8-bit +grayscale images, IMU measurements, and sensor +calibration, including the calibration sequences, +for all sequences (training and testing). In addi- +tion, RTK GNSS measurements, in NMEA for- +mat, VO point clouds, and reference poses are +released only for training sequences. For the test- +ing sequences, such data is withheld for evalua- +tion. Moreover, we specify the distance between +Fig. +6: +Accurate +pixel-wise +correspon- +dences, making cross-season training pos- +sible. Qualitative assessment of the accuracy of +our data collection and geometric reconstruction +method for a sample of four different conditions +(from top left in clockwise order: cloudy, snowy, +night, sunny) across the same scene. Each same +colored point in the four images corresponds to the +same geometric point in the world. The cameras +corresponding to these images have different poses +in the global frame of reference. Please note that +the points are not matched, but rather a result +of our accurate reference poses and geometric +reconstruction. +the refined reference poses and the raw RTK +GNSS measurements. +5 Benchmark Tasks +In this section, we define the benchmark evalua- +tion metrics, tasks, and their evaluation protocols +for visual odometry, global place recognition, and +map-based visual localization. Visual localization +consists of retrieving the 6DoF pose of a query +within an existing 3D model and can be inter- +preted as a two-step approach. First, global image +retrieval is performed to obtain a rough esti- +mate of the query pose w.r.t. a map. Second, +local feature matching is used to refine the pose +estimate. +For the evaluation, in each task, we consider a +set of estimated 6DoF poses Test +i +∈ SE(3), as well +as a set of reference, poses Tref +i +∈ SE(3). While the +reference poses are always defined w.r.t. a global +world frame, the estimated poses are defined either +w.r.t. the same global world frame (for global place +recognition and map-based visual localization) or +to a selected local frame2 (visual odometry). +2Can be for instance the camera frame of the first recorded +left camera image. + +10 +Wenzel et al. +5.1 Visual Odometry in Challenging +Conditions +Visual odometry aims to accurately estimate +the +relative +6DoF +camera +pose +based +on +recorded images. To benchmark the task of VO +there already exists various datasets [15, 19, +59]. All of these existing datasets consist of +sequences recorded at rather homogeneous con- +ditions (indoors, or sunny/overcast outdoor con- +ditions). However, methods specially developed +for autonomous driving use cases must perform +robustly under almost any condition. We believe +that the proposed benchmark will contribute to +improving the performance of VO under diverse +weather and lighting conditions in an automotive +environment. Therefore, instead of replacing exist- +ing benchmarks and datasets, we aim to provide +an extension that is more focused on challenging +conditions in autonomous driving. As we pro- +vide frame-wise accurate poses for large portions +of the sequences, metrics well known from other +benchmarks like absolute trajectory error (ATE) +or relative pose error (RPE) [19, 59] are also +applicable to our data. +5.1.1 Evaluation Metrics +Similar to previous benchmarks, the main accu- +racy measure we are interested in is the RPE. In +general, the RPE is split up into a translational +and a rotational error. However, another compo- +nent we are interested in is the scale error. One +may argue that, especially for stereo approaches, +scale errors are marginal and therefore not rel- +evant. Nevertheless, our experience is different. +We observe that quite significant scale errors and +drift can occur when performing stereo VO and +SLAM in automotive environments. This can be +caused either by the miss-calibration of the cam- +eras, by the structure of the scene but also by +algorithm-specific design choices like the type of +keypoint detector, etc. Since the sensor setup has +a limited stereo baseline, parallaxes (i.e. pixel +disparities) for far object points are vanishing. +This means that, even for stereo approaches, the +scale becomes non-observable if no close static +objects are present in the scene. Increasing the +stereo baseline, however, could reduce the rigid- +ity of the sensor setup. We believe that it is very +valuable to conduct further research on stereo VO +and SLAM methods which explicitly consider the +depth uncertainties created by the length of the +stereo baseline. +Since in automotive use cases, the scale can +always be observed based on a reference system, +like wheel ticks, GNSS or a reference map, we +consider only relative errors (drifts) in scale, trans- +lation, and rotation in the proposed benchmark. +Therefore, before evaluation, a global scale align- +ment is performed for the entire trajectory with +respect to the reference trajectory. +For the proposed VO benchmark all evaluation +metrics are defined based on the estimated relative +pose Test +ij +∈ SE(3) between two frames i and j +and its corresponding reference pose Tref +ij ∈ SE(3) +with: +Tref +ij = +� +Tref +i +�−1 Tref +j +and +Test +ij = +� +Test +i +�−1 Test +j . +(3) +For a pair of frames (i, j) for which reference +poses are available, we calculate the relative trans- +lational error ϵt +ij, rotational error ϵr +ij, and scale +error ˜ϵs +ij as given in Equations (4) to (6). +ϵt +ij = ∥tref +ij − test +ij ∥2 +dij +(4) +ϵr +ij = arccos +� 1 +2 +� +trace ((Rref +ij )−1Rest +ij ) − 1 +�� +dij +(5) +˜ϵs +ij = +��test +ij +�� +2 +��tref +ij +�� +2 +(6) +From ˜ϵs +ij one obtains the final relative scale error +as ϵs +ij += +max[˜ϵs +ij, (˜ϵs +ij)−1]. The parameter dij +defines the reference path length between the two +poses Tref +i +and Tref +j . +Meaningful metrics are obtained by extracting +all possible sub-segments of length 100 m, 200 m, +400 m, 600 m, 800 m, and 1000 m from a trajec- +tory and calculating the relative poses between +the first and last frame of each sub-segment. Fur- +thermore, for trajectory segments where no GNSS +measurements are available for more than 1000 m +(e.g. in tunnels, garages, or urban canyons), also +the relative poses of such an entire stretch are +taken into account. This allows us to also consider +challenging scenarios like tunnels and the transi- +tion from bright to dark in the benchmark. Using +sub-segments of different lengths for evaluation is + +Wenzel et al. +11 +Fig. 7: Challenging scenes for global place +recognition. Top: two pictures share the same +location with different appearances. Bottom: two +pictures have a similar appearance but are taken +at different locations. +inspired by the KITTI benchmark [19] and allows +capturing both short and long-term accuracy of +VO algorithms. +To obtain single number metrics for every +sequence, we consider the visual VO successful +if the errors are within certain positional, rota- +tional, and scale bounds. We define three intervals +by varying the thresholds: high precision (0.5 %, +0.005 deg/m, 1.005 (multiplier)), medium pre- +cision (1 %, 0.01 deg/m, 1.01 (multiplier)), and +coarse precision (2 %, 0.02 deg/m, 1.02 (multi- +plier)). +While the translational error is the most mean- +ingful metric to evaluate VO algorithms, the +rotational error, and scale error still give valu- +able insight into the specific behavior of a certain +approach. +5.2 Global Place Recognition +Global place recognition refers to the task of +retrieving the most similar database image given +a query image [37]. To improve the searching +efficiency and the robustness against different +weather conditions, tremendous progress on global +descriptors [1, 2, 18, 29] has been seen. For +the localization pipeline, visual place recognition +serves as the initialization step to the downstream +local pose refinement by providing the most sim- +ilar database images as well as the corresponding +global poses. Due to the advent of deep neural +networks [24, 34, 56, 60], methods aggregating +deep image features are proposed and have shown +advantages over classical methods [3, 21, 44, 62]. +The proposed dataset is challenging for global +place recognition since it contains not only cross- +season images that have different appearances +with a similar geographical location but also intra- +season images which share similar appearances +but with different locations. This results in mainly +two different types: images taken at the same +place, but look different, or images taken at dif- +ferent places but look similar. Figure 7 depicts +example pairs of these scenarios. +5.2.1 Evaluation Metrics +We follow the standard metric widely used for +global place recognition [2, 3, 21, 51], namely the +recall at top N retrievals with a certain range +bound as the positive threshold. Specifically, one +query image is considered to be correctly local- +ized if at least one of the top N retrieved images +is within a certain translational (in meters) and a +certain rotational (in degrees) bound with respect +to the ground-truth location of the query image. +The translational error ϵt is measured as the +Euclidean distance: +ϵt = ∥tref − test∥2 +(7) +between the reference tref and estimated test cam- +era positions. The rotational error ϵr is measured +as an angle in degrees (following [23]) by calculat- +ing: +ϵr = arccos +�1 +2 +� +trace ((Rref)−1Rest) − 1 +�� +, (8) +where Rref, and Rest denote the reference and +estimated camera rotation matrices. In the evalu- +ation of global place recognition, we calculate the +recalls under different threshold settings: by fixing +N and changing the range bound, or by fixing the +range bound and changing N. We will describe the +specific settings in Section 6.2. +5.3 Map-Based Visual Localization +Map-based visual localization refers to the task +of locally refining the 6DoF pose between refer- +ence images and images from a query sequence. +In contrast to wide-baseline stereo matching, for +map-based visual localization, it is also possible to +utilize the sequential information of the sequence. +This allows estimating depth values by running +a standard VO method. Those depth estimates +can then be used to improve the tracking of the +individual localization candidates. + +12 +Wenzel et al. +In contrast to global place recognition which +only uses 2D images and no other information, +this task allows the use of a globally-consistent +3D reconstruction of the reference scene. In this +task, we assume to know the mapping between +reference and query samples and only focus on +the local pose refinement task. In practice, this +mapping can be found using image retrieval tech- +niques as described in Section 5.2 or by using +GNSS measurements as a coarse initialization if +available. +Accurately localizing in a pre-built map is +a challenging problem, especially if the visual +appearance of the query sequence significantly dif- +fers from the base map. This makes it extremely +difficult, especially for vision-based systems, since +the localization accuracy is often limited by the +discriminative power of feature descriptors. Our +proposed dataset allows evaluating visual localiza- +tion across multiple types of weather conditions +and diverse scenes, ranging from urban to country- +side driving. Furthermore, our up to centimeter- +accurate reference poses allow us to create more +strict evaluation settings with an increased level +of difficulty. This allows us to determine the lim- +itations and robustness of current state-of-the-art +methods. +5.3.1 Evaluation Metrics +For evaluation, we measure the translational and +rotational error of any method between the esti- +mated and the reference pose. Please refer to +Equation (7) and Equation (8) for the defini- +tions of the translational and rotational error, +respectively. +We consider the localization successful if a +query image is localized within certain positional +(in meters) and rotational (in degrees) bounds +with respect to their reference pose. We define +three localization intervals by varying the thresh- +olds: high precision (0.1 m, 1°), medium pre- +cision (0.25 m, 2°), and coarse precision (1 m, +5°). +6 Experimental Evaluation +In this section, we evaluate the current state-of- +the-art baseline methods for each of the three pro- +vided benchmarks (visual odometry, global place +recognition, and map-based visual localization) to +demonstrate the diversity and challenges of the +benchmark. We will establish an open leaderboard +for the benchmark to compare different methods +upon publication. This allows the reproduction of +the baseline results for every user. Furthermore, +we will set up a server for an automatic evaluation +of the results on the withheld test set. +6.1 Visual Odometry in Challenging +Conditions +We provide results for state-of-the-art baseline +stereo and stereo-inertial odometry and SLAM +approaches. The methods provided as baselines +are classical geometric approaches. Nevertheless, +we strongly encourage researchers to evaluate +learning-based methods on our benchmark as +well. In particular, we provide results for the +following stereo and stereo-inertial VO methods: +ORB-SLAM33 [9] and Basalt4 [66]. +The provided VO benchmark is divided into +two sets of evaluation sequences: unknown sce- +narios and known scenarios. Unknown scenarios +consist only of scenarios, for which no sequences at +all are provided in the training set. Namely, these +are the scenarios Highway and Maximilianeum. +Known scenarios are those scenarios for which are +also sequences provided in the training set. While +this is irrelevant for pure geometric approaches, +we believe that this separation will be impor- +tant to evaluate the generalization capabilities of +learning-based approaches. Table 2 shows the eval- +uation results on the individual sequences of the +benchmark for known scenarios. Figure 8 shows +the results across all sequences corresponding to +the known scenarios in cumulative error plots. +Table 3 shows the evaluation results on the indi- +vidual sequences of the benchmark for unknown +scenarios. Figure 9 shows the results across all +sequences corresponding to the unknown scenarios +in cumulative error plots. +From Table 2 and 3 one can observe that +all evaluated methods perform significantly worse +on the unknown scenarios. Nevertheless, this is +mainly due to the challenging conditions, which +are on one side highway sequences with high speed +and sudden lighting changes under bridges as well +as inner city night sequences. +3https://github.com/UZ-SLAMLab/ORB SLAM3 +4https://gitlab.com/VladyslavUsenko/basalt + +Wenzel et al. +13 +Table 2: Visual odometry results on known scenarios from the 4Seasons benchmark. This +table shows the evaluation results of state-of-the-art baseline methods on the VO benchmark. The best- +performing results are in bold. The results are shown in terms of the percentage of high / medium / +coarse precision. +Method +office loop 2 test office loop 3 test neighborhood 2 test business campus 1 test countryside 2 test city loop 2 test old town 2 test old town 3 test parking garage 1 test +Average +Basalt [66] (stereo) +9.1 / 65.7 / 96.7 +6.3 / 53.0 / 94.4 +4.2 / 21.5 / 70.1 +2.3 / 28.5 / 71.5 +7.7 / 38.3 / 77.6 +11.4 / 43.8 / 72.3 +5.6 / 31.1 / 78.0 +1.3 / 9.0 / 37.4 +0.0 / 0.0 / 33.3 +5.3 / 32.3 / 70.2 +Basalt [66] (stereo-inertial) +3.3 / 35.0 / 92.0 +2.1 / 20.9 / 80.8 +3.5 / 23.6 / 72.9 +11.3 / 59.0 / 95.7 +16.4 / 48.5 / 88.8 +23.1 / 59.2 / 88.2 +0.0 / 0.0 / 0.0 +1.5 / 15.4 / 42.6 +0.0 / 11.1 / 55.6 +6.8 / 30.3 / 68.5 +ORB-SLAM3 [9] (stereo) +16.8 / 65.3 / 94.9 +1.4 / 24.0 / 82.2 +4.9 / 55.6 / 95.8 +3.9 / 42.2 / 82.8 +5.8 / 41.4 / 76.6 +1.2 / 12.6 / 49.3 +0.8 / 17.7 / 57.0 +0.3 / 1.0 / 2.8 +0.0 / 22.2 / 77.8 +3.9 / 31.3 / 68.8 +ORB-SLAM3 [9] (stereo-inertial) +7.3 / 33.6 / 84.7 +2.1 / 15.7 / 50.2 +13.2 / 44.4 / 84.0 +19.9 / 64.8 / 91.0 +2.9 / 11.4 / 42.6 +26.1 / 59.2 / 77.9 12.9 / 43.3 / 87.1 +0.0 / 0.0 / 0.0 +0.0 / 11.1 / 44.4 +9.4 / 31.5 / 62.4 +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +Fig. 8: Performance of state-of-the-art baseline visual odometry methods on known scenar- +ios from the 4Seasons benchmark. The figure shows the translational error (in %), rotational error +(in mdeg/m), and scale error (multiplier). +Table +3: +Visual +odometry +results +on +unknown +scenarios +from +the +4Seasons +benchmark. This table shows the evaluation +results of state-of-the-art baseline methods on the +VO benchmark. The best-performing results are +in bold. The results are shown in terms of the +percentage of high / medium / coarse precision. +Method +highway 1 test +highway 2 test +maximilianeum 1 test maximilianeum 2 test +Average +Basalt [66] (stereo) +9.4 / 32.0 / 63.1 +10.3 / 29.5 / 52.7 +35.1 / 75.3 / 91.8 +1.2 / 9.8 / 38.2 +14.0 / 36.6 / 61.4 +Basalt [66] (stereo-inertial) +32.3 / 68.6 / 85.8 21.3 / 49.2 / 71.2 +34.0 / 69.6 / 94.3 +0.0 / 6.9 / 27.2 +21.9 / 48.6 / 69.6 +ORB-SLAM3 [9] (stereo) +0.6 / 3.9 / 22.1 +0.0 / 0.0 / 0.0 +0.0 / 19.6 / 56.7 +0.0 / 0.0 / 0.0 +0.2 / 5.9 / 19.7 +ORB-SLAM3 [9] (stereo-inertial) 10.3 / 29.6 / 48.6 12.9 / 25.1 / 68.3 +26.3 / 60.8 / 79.4 +10.4 / 39.3 / 56.1 +15.0 / 38.7 / 63.1 +While the results above provide average num- +bers across all sequences of the benchmark, we +provide in Figure 10 and 11 side-by-side the results +for identical scenarios but for different conditions, +respectively. +Figure 10 provides VO results on the Max- +imilianeum +scenario +in +the +afternoon +(max- +imilianeum 1 test) +and +at +night +(maximilia- +neum 2 test), respectively. As one could expect, +there is a significant drop in performance when +going from day to night due to less visible land- +marks. Nevertheless, it is interesting to observe +that ORB-SLAM3 (with IMU) can perform bet- +ter during the night than Basalt (with IMU). A +reason might be that ORB-SLAM3 is using fea- +ture matching to find point correspondences, while +Basalt is relying on optical flow. This correla- +tion cannot be observed when running without +IMU, where ORB-SLAM3 is failing. However, +especially during the night and without IMU, the +task becomes inordinately more difficult. +Figure 11 provides performance comparisons +between a sunny (office loop2 test) and a cloudy +(office loop3 test) condition on the Office Loop +scenario. Across all algorithms, one can observe +improved performance during sunny weather con- +ditions. A likely reason for this is the presence +of more static feature points caused by shadows, +especially on the road. This can be seen on the +right side images in Figure 11, where much more +texture is on the road for the sunny than for the +cloudy conditions. +While the evaluated methods show all-in-all +good performance in good weather and light- +ing conditions, we believe that our dataset and +benchmark will contribute to improving the per- +formance in conditions with fewer and unreliable +feature points. The results show that the pro- +posed benchmark is highly challenging and still +provides room for improving state-of-the-art VO +algorithms. + +14 +Wenzel et al. +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +Fig. 9: Performance of state-of-the-art visual odometry methods on unknown scenarios +from the 4Seasons benchmark. The figure shows the translational error (in %), rotational error (in +mdeg/m), and scale error (multiplier). +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +(a) Afternoon. +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +(b) Night. +Fig. 10: Comparison of visual odometry performance for afternoon and night. The figure shows +the performance for different state-of-the-art baseline VO algorithms on the same route for afternoon and +night conditions. One can observe a significant drop in performance when going from day to night due +to less visible landmarks. +6.2 Global Place Recognition +We evaluate the current state-of-the-art base- +line deep image descriptors methods including +NetVLAD5 [3] pretrained on Pittsburgh30k [63], +Deep Image Retrieval (DIR)6 [22, 46] (aka AP- +GeM) trained on the Landmarks dataset [4], and +CNN Image Retrieval (CIR)7 [43, 44] trained on +the dataset derived from [54]. For each scenario +of the 4Seasons benchmark, we use a predefined +recording as the reference map and a predefined +recording as the query map. Note that we leave +5https://github.com/cvg/Hierarchical-Localization +6https://github.com/naver/deep-image-retrieval +7https://github.com/filipradenovic/ +cnnimageretrieval-pytorch +out the Business Campus scenario for global place +recognition. +As shown in Figure 12, we first plot two differ- +ent recall curves: (1) Recall[%] – Threshold [m] @ +Top1: the recalls of different methods when chang- +ing the distance threshold in the range 1 m–20 m +using only the top 1 retrieved images, and (2) +Recall[%] – Top N @ 1 m: the recalls of different +methods when changing the number of candidate +retrievals N ∈ {1, 2, 3, . . . , 20} using the fixed +range bound 1 m. We also show the optimal recall +by using the closest database images with respect +to the ground-truth query image location as the +candidates. This shows the upper bound of the +global place recognition accuracy. + +Wenzel et al. +15 +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +(a) Sunny. +0 +1 +2 +3 +Translational error [%] +0 +20 +40 +60 +80 +100 +Occurrence [%] +Basalt (stereo) +Basalt (stereo-inertial) +ORB-SLAM3 (stereo) +ORB-SLAM3 (stereo-inertial) +0 +5 +10 +15 +20 +25 +Rotational error [mdeg/m] +0 +20 +40 +60 +80 +100 +1.000 +1.005 +1.010 +1.015 +1.020 +Scale error (multiplier) +0 +20 +40 +60 +80 +100 +(b) Cloudy. +Fig. 11: Comparison of visual odometry performance for sunny and cloudy. The figure shows +the performance for different state-of-the-art baseline VO algorithms on the same route for sunny and +cloudy conditions. Across all algorithms, one can observe improved performance during sunny weather +conditions. A likely reason for this is the presence of more static feature points caused by shadows, +especially on the road. This can be seen on the right side images, where much more texture is on the +road for the sunny than for the cloudy conditions. +From the results, we can see that NetVLAD +still outperforms the other recent methods by a +notable margin. One reason for NetVLAD’s supe- +rior performance could be the introduction of the +inductive bias into the network design, based on +the established principle of classical VLAD [2, +29]. However, one shall also admit that the gap +between the state-of-the-art methods and the opti- +mal performance is still quite large, and more +research on global place recognition still needs to +be conducted. +We show the localization accuracy of the global +place recognition methods without using local +pose refinement. Note that for these methods we +use the top 20 candidates and loosened range +bounds, namely, (1 m, 5°) for high precision, +(5 m, 10°) for medium precision, and (10 m, +20°) for coarse precision. The last three rows of +Table 4 show the individual global place recogni- +tion (GPR) performance on each of the evaluated +scenarios from the 4Seasons benchmark. +6.3 Map-Based Visual Localization +For the evaluation of map-based visual localiza- +tion, we use the following processing pipeline: we +first build a SfM model from the reference scene +0 +5 +10 +15 +20 +Threshold [m] @ Top1 +0 +20 +40 +60 +80 +100 +Recall [%] +NetVLAD +CIR +DIR +optimal +0 +5 +10 +15 +20 +Top N @ 1.0m +0 +20 +40 +60 +80 +100 +Fig. +12: Performance +of +state-of-the-art +baseline global place recognition methods +on the 4Seasons benchmark. The gray line +indicates the upper bound for the global place +recognition accuracy. +that provides correspondences between local fea- +tures and 3D points in the reconstructed map. +This is followed by 2D-3D matching between the +query images and the database images. As the +last step, those 2D-3D matches are used for cam- +era pose estimation via Perspective-n-Point (PnP) +and random sample consensus (RANSAC) [17]. In +particular, we evaluate the current state-of-the-art +coarse-to-fine hierarchical localization method [48] +based on the following learned deep local fea- +ture descriptors: SuperPoint [11], D2-Net [12], and +R2D2 [47]. Additionally, we use the classic scale- +invariant feature transform (SIFT) [36] algorithm. +Hloc [48] simultaneously predicts local features + +16 +Wenzel et al. +Table 4: Visual localization results on the 4Seasons benchmark. We report the percentage of +images localized within 0.1 m and 1°, 0.25 m and 2°, 1 m and 5° of the reference poses for map-based visual +localization (MBVL) pipelines. For global place recognition (GPR) methods, we report the percentage of +images localized within 1 m and 5°, 5 m and 10°, 10 m and 20° of the reference poses. The best-performing +results for both MBVL and GPR pipelines are in bold. +Method +Office Loop +Neighborhood +Business Campus +Countryside +City Loop +Old Town +Parking Garage +Average +MBVL +hloc [48] (SuperPoint [11] + SuperGlue [49]) 68.6 / 85.1 / 89.2 56.0 / 73.3 / 86.8 +38.2 / 74.5 / 89.0 +8.0 / 29.2 / 64.7 +37.0 / 72.6 / 83.2 27.1 / 43.2 / 58.4 46.9 / 63.7 / 76.1 40.3 / 63.1 / 78.2 +hloc [48] (D2-Net [12] + NN) +43.3 / 71.5 / 88.3 28.7 / 54.6 / 85.1 +22.4 / 62.2 / 85.5 +3.4 / 18.6 / 63.4 +14.9 / 50.0 / 86.2 +9.7 / 26.3 / 47.9 +28.3 / 52.2 / 76.1 +21.5 / 47.9 / 76.1 +hloc [48] (SIFT [36] + NN) +24.6 / 39.3 / 52.6 33.6 / 50.6 / 66.1 +3.5 / 9.9 / 18.9 +0.1 / 0.6 / 2.4 +9.0 / 22.3 / 40.3 +0.0 / 0.3 / 4.0 +9.7 / 16.8 / 27.4 +11.5 / 20.0 / 30.2 +hloc [48] (R2D2 [47] + NN) +66.4 / 83.1 / 88.3 59.8 / 81.9 / 96.0 +36.9 / 68.6 / 82.9 +3.8 / 18.1 / 53.6 +36.1 / 77.2 / 92.0 16.5 / 27.8 / 40.0 41.6 / 66.4 / 78.8 +37.3 / 60.4 / 75.9 +GPR +NetVLAD [3] +55.4 / 90.2 / 93.5 49.5 / 80.6 / 83.5 +– +10.6 / 29.9 / 34.1 24.6 / 54.0 / 62.3 30.8 / 64.3 / 79.2 37.9 / 79.3 / 86.2 34.8 / 66.4 / 73.1 +CNN Image Retrieval (CIR) [44] +33.7 / 66.3 / 71.7 43.7 / 71.8 / 78.6 +– +6.8 / 22.0 / 28.8 +6.7 / 25.0 / 31.3 +14.9 / 45.7 / 62.0 27.6 / 55.2 / 65.5 +22.2 / 47.7 / 56.3 +Deep Image Retrieval (DIR) [46] +26.1 / 58.7 / 68.5 41.7 / 72.8 / 79.6 +– +10.2 / 23.9 / 29.5 +7.9 / 24.2 / 30.6 +19.0 / 43.0 / 63.3 27.6 / 79.3 / 89.7 +22.1 / 50.3 / 60.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Distance threshold [m] +0 +20 +40 +60 +80 +100 +Correctly classified queries [%] +SP+SG +D2-Net+NN +SIFT+NN +R2D2+NN +0 +1 +2 +3 +4 +5 +Orientation threshold [deg] +0 +20 +40 +60 +80 +100 +Fig. +13: Performance +of +state-of-the-art +baseline +map-based +visual +localization +approaches on the 4Seasons benchmark. +The figure shows the cumulative localization +accuracy against the translational, and rotational +error, respectively. +and global descriptors for accurate 6DoF local- +ization. This approach first performs global place +recognition to obtain location candidates, and +afterward matches the local features only within +those places. The extracted local image features +are used to establish 2D-3D matches within a pre- +built SfM model. Pose estimation is performed +using COLMAP [53]. Therefore, this pipeline can +be seen as a pose refinement strategy. For each +scenario of the 4Seasons benchmark, we use a +predefined recording as the reference map and a +predefined recording as the query. +Figure 13 shows the percentage of correctly +classified queries when changing the distance and +orientation thresholds, respectively. This figure +shows the average performance for the differ- +ent state-of-the-art map-based visual localization +(MBVL) approaches across all evaluated scenar- +ios. The first four rows of Table 4 show the indi- +vidual hierarchical localization performance on +each of the evaluated scenarios from the 4Seasons +benchmark. +From the results, we can see that the clas- +sic SIFT+nearest neighbor (NN) approach shows +a bad performance estimating the 6DoF pose +between the reference and query candidates. The +results also suggest that deep learning-based +algorithms dramatically outperform the classical +method. This is due to the challenging nature +of the benchmark since it deals with dras- +tic lighting and illumination changes, occlusions, +and changing environmental/weather conditions. +Those results provide valuable insights into the +limitations and failure cases of the different meth- +ods. We observe that learned feature descriptors +significantly outperform classic methods under +challenging conditions contained in the 4Seasons +benchmark, with SuperPoint+SuperGlue yielding +the best results overall. Nevertheless, the results +from Table 4 show that the long-term localization +problem is still far from being solved, especially +for highly dynamic environments (e.g. Old Town) +and scenes that exhibit very similar structure (e.g. +Countryside). +Our benchmark provides the basis to enable +more research advances that are needed to close +this performance gap. +7 Conclusion +Current benchmarks either focus mainly on evalu- +ating the performance of simultaneous localization +and mapping algorithms or visual localization +in isolation. To close this gap, we introduce a +benchmark that by providing a holistic way for +jointly benchmarking long-term visual SLAM and +localization. +In this paper, we have introduced a compre- +hensive benchmark suite for visual SLAM and + +Wenzel et al. +17 +visual localization for autonomous driving under +challenging conditions. The benchmark covers a +huge variety of environmental conditions, along +with short-term and long-term weather and illu- +mination changes. Moreover, we have reviewed +and evaluated the current state-of-the-art baseline +approaches for visual SLAM and visual localiza- +tion. We have observed large performance gaps +and see huge potential in future work to close +those gaps. +The benchmark dataset, evaluation server, +and leaderboard will be available upon accep- +tance via the benchmark’s website, https://www. +4seasons-dataset.com. +Acknowledgments. +We express our apprecia- +tion to our colleagues at Artisense for their help +with setting up the recording setup and sensor +design. +Declarations +Competing +Interests. +The authors declare +that they have no conflict of interest. +References +[1] Angeli A, Filliat D, Doncieux S, et al +(2008) Fast and incremental method for +loop-closure detection using bags of visual +words. 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International Journal of +Computer Vision (IJCV) 103(3):267–305 + +elBogen +Metro +品 +Harnierplatz +Alois-Woifmuller-StraBe +EsVFreimann +Werk +01.20 +Campus +Freimann +Freimann +DBNetz7 +A9 +DBSystemtechnik +Munchen-FraahurterRir +Guterbahnhof +Edisonstrage +Munchen +Bahn-Landwirischot +Frelmonn +Manchen- +Freirmann +Leaflet Data by OpenStreetMap, under ODblRodelhuge +Garching-5ud +merkona! +Hochbruck +490m +Schweizerholz +GOrCH +4-502 +Frottmaninger +B388 +st2053 +Heide +Kreuz.Mop +nen-Mord +Munchen-Weuherbergl +120 +Monchen-Froumonng-Word +120 +Norurschutroebiet +Ismaning +Ponzerwese +Ona Hartelholz +Sudliche +Frottmaninger +Ponzerwese +Kreuz Munchen-Word +und Hartelhol2 +Heide +st2053 +13 +B471 +Kreuz Mur +hen-Nord +B13 +Munchen-Fro +maning-Sud +Neuherberg +A99 +Monchen +Am.Har +Neufreimann +Monchen +Unterfohring +Aschheim/smoning +Euro-dstreork +14 +Freimann +皖A5G5H +Munchen-Frankfurter RingMunchen-reiman +74 +Mceschule +QP·Q·27 +S/tustrope +口 +Spengelh +Schuisoortp/az +an der +Kulturheimes +tetmer +Situlistrose +Tenrs +Park St. +St 2350 +Florion +Freimal +Smuhistrabe +Freimann +毛 +A9 +P +LeafletIDataby OpenStreetMap, underODblParkrinn +ACoS +Balpark +Business +CQMDUS +BMW +Sportpark +Group +VIPFlottenservice +Presse +Testfahrzeuge +SWiss Le +Deutschland +StraBe +Hardwe +P+R Garching +Hochbruck +Leaflet I Data by @ OpenStreetMaprunder oDblWaturschutzgebiet +Gorchnger +Heide +FS 20 +Eehnoer +Zettelhof +Kieswerk +Eching +Sud +Naturschutzqebiet +Landkreis Freising +Mallertshofer. +Holz +Landkreis Munchen +Garching-Nord +UAdFoAm.Hart +Neufreimann +Monichen-Frermonn +Jnterfohrind +ASCHRem +Euro-Industrepark +14 +Freimann +Mupchen-FanhuerRe +Schwabing +Nord +Riesenfeld +Milbertshofen +te Heide +Hirschau +Oberfohrin +Schwabing +Schwabing +Johanneskirchen +rg +West +Ennsche +18.8 +GOren +Alt-Schwabinig +Herzogpar +Englschaking +Bogenhausen +Maxvorstadt +Denning +Daglfinig +Altbogenhausen +Parkstadt Bogenhausen +Zamdorf +Riem +MupchepstephousenKonigsplatz +GaleriestraBe +Hinbse +Odeonsplatz +Arcostrae +Odeonsplatz +nsoldstrane +Liebinstrale +ptbahnho +ntiet +Karlsplatz +Hauptbahnhof +(stachus) +Kreuzviertel +BayerstraBe +Lehel +KE +Isplatz +otbahnhot +achus) +LGraggenauervierte +Lehel +Altstadt +SchwanthalerstralSe +Maximians +Marienplatz +Praterinsel +Landwehrstrac +Hackenviertel ++ ++ +Angerviertel +Isartor +Fo +ser +linger +Gartnerplatz +fugsburgerstrane ++IstraBe +Odeonsplatz +Odeonsplatz +soldstrane +Maximiiansanlogen +Kreuzvierte +Lehel +Graoo +Lee +Altstadt +Maxin +Marienplatz +Praterinsel +Max- +Webel +Isartor +Haidhausen +Gartnerplat +Museumsinsel +Maidhausen +Isarvorstadt +Leaflet | Data by @ OpenStreetMap. under ODbl170 +&Warenannahme +Bauhaus +d +Fu +u +Leaflet I Data by @ OpenStreetMap, under ODbl20 +Wenzel et al. +Table 5: Statistics of the 4Seasons dataset. This table shows the different scenarios and recordings +along with the weather condition, seasons, and time of the day from our benchmark. We provide a variety +of scenarios and short-term to long-term changes. These recordings are all released with ground truth +(GNSS/IMU, point clouds, and reference poses) and can therefore be used for training learning-based +techniques. +Scenario +Recording +Weather +(cloudy, rainy, snowy, sunny) +Season +(winter, spring, summer, fall) +Daytime +(morning, afternoon, evening, night) +Map Accuracy +Horizontal RMSE +(GNSS-Ref. Pose) +Map Accuracy +% of Accurate Poses +office loop 1 train +2020-03-24 17-36-22 sunny +spring +afternoon +6.84 cm +85.93 % +office loop 2 train +2020-03-24 17-45-31 sunny +spring +afternoon +6.34 cm +86.92 % +office loop 3 train +2020-04-07 10-20-32 sunny +spring +morning +5.44 cm +77.72 % +office loop 4 train +2020-06-12 10-10-57 sunny +summer +morning +2.74 cm +54.01 % +office loop 5 train +2021-01-07 12-04-03 cloudy/snowy +winter +afternoon +3.79 cm +96.00 % +office loop 6 train +2021-02-25 13-51-57 sunny +winter +afternoon +2.90 cm +91.45 % +neighborhood 1 train +2020-03-26 13-32-55 cloudy +spring +afternoon +4.13 cm +56.71 % +neighborhood 2 train +2020-10-07 14-47-51 cloudy +fall +afternoon +1.19 cm +85.00 % +neighborhood 3 train +2020-10-07 14-53-52 rainy +fall +afternoon +2.00 cm +84.12 % +neighborhood 4 train +2020-12-22 11-54-24 cloudy +winter +morning +3.47 cm +87.92 % +neighborhood 5 train +2021-02-25 13-25-15 sunny +winter +afternoon +2.45 cm +86.23 % +neighborhood 6 train +2021-05-10 18-02-12 cloudy +spring +evening +1.74 cm +69.43 % +neighborhood 7 train +2021-05-10 18-32-32 cloudy +spring +evening +1.44 cm +85.45 % +business campus 1 train 2020-10-08 09-30-57 sunny +fall +morning +5.49 cm +83.08 % +business campus 2 train 2021-01-07 13-12-23 cloudy/snowy +winter +afternoon +1.77 cm +99.13 % +business campus 3 train 2021-02-25 14-16-43 sunny +winter +afternoon +7.33 cm +66.86 % +countryside 1 train +2020-04-07 11-33-45 sunny +spring +morning +3.96 cm +90.89 % +countryside 2 train +2020-06-12 11-26-43 sunny +summer +morning +2.54 cm +87.00 % +countryside 3 train +2020-10-08 09-57-28 sunny +fall +morning +1.94 cm +89.37 % +countryside 4 train +2021-01-07 13-30-07 cloudy/snowy +winter +afternoon +5.42 cm +92.02 % +city loop 1 train +2020-12-22 11-33-15 rainy +winter +morning +6.85 cm +83.08 % +city loop 2 train +2021-01-07 14-36-17 snowy/sunny +winter +afternoon +4.76 cm +84.27 % +city loop 3 train +2021-02-25 11-09-49 sunny +winter +morning +3.41 cm +85.14 % +old town 1 train +2020-10-08 11-53-41 cloudy +fall +morning +2.90 cm +93.76 % +old town 2 train +2021-01-07 10-49-45 cloudy/snowy/sunny +winter +morning +1.80 cm +93.16 % +old town 3 train +2021-02-25 12-34-08 sunny +winter +afternoon +1.43 cm +83.12 % +old town 4 train +2021-05-10 21-32-00 cloudy +spring +night +13.45 cm +95.81 % +parking garage 1 train +2020-12-22 12-04-35 cloudy +winter +afternoon +1.43 cm +33.22 % +parking garage 2 train +2021-02-25 13-39-06 sunny +winter +afternoon +2.52 cm +40.54 % +parking garage 3 train +2021-05-10 19-15-19 cloudy +spring +evening +3.41 cm +34.15 % +[24] He K, Zhang X, Ren S, et al (2016) Deep +residual learning for image recognition. 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In: Proceedings of the +European Conference on Computer Vision +(ECCV), pp 817–833 + diff --git a/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/load_file.txt b/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..639dff772a4922119afd4be0f87bd8933e886147 --- /dev/null +++ b/j9AzT4oBgHgl3EQfNPvf/content/tmp_files/load_file.txt @@ -0,0 +1,1223 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf,len=1222 +page_content='4Seasons: Benchmarking Visual SLAM and Long-Term Localization for Autonomous Driving in Challenging Conditions Patrick Wenzel1*, Nan Yang2†, Rui Wang3†, Niclas Zeller4† and Daniel Cremers1 1Department of Computer Science, Technical University of Munich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2Reality Labs at Meta, Redmond, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3Microsoft Mixed Reality & AI Lab, Zurich, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4Karlsruhe University of Applied Sciences, Karlsruhe, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' E-mail(s): patrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='wenzel@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' †Work done at Technical University of Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Abstract In this paper, we present a novel visual SLAM and long-term localization benchmark for autonomous driving in challenging conditions based on the large-scale 4Seasons dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The proposed bench- mark provides drastic appearance variations caused by seasonal changes and diverse weather and illumination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While significant progress has been made in advancing visual SLAM on small-scale datasets with similar conditions, there is still a lack of unified benchmarks represen- tative of real-world scenarios for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We introduce a new unified benchmark for jointly evaluating visual odometry, global place recognition, and map-based visual localiza- tion performance which is crucial to successfully enable autonomous driving in any condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The data has been collected for more than one year, resulting in more than 300 km of record- ings in nine different environments ranging from a multi-level parking garage to urban (including tunnels) to countryside and highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide globally consistent reference poses with up to centimeter-level accuracy obtained from the fusion of direct stereo-inertial odometry with RTK GNSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We evaluate the performance of several state-of-the-art visual odometry and visual localiza- tion baseline approaches on the benchmark and analyze their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The experimental results provide new insights into current approaches and show promising potential for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Our benchmark and evaluation protocols will be available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4seasons-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Keywords: Autonomous Driving, Benchmark, Long-Term Visual Localization, SLAM, Visual Odometry, Camera Pose Estimation 1 Introduction During the last decade, research on visual odome- try (VO) and simultaneous localization and map- ping (SLAM) has made tremendous strides [13, 16, 40, 41], particularly in the context of autonomous driving [14, 39, 69, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' One reason for this progress has been the publication of large-scale datasets tailored for benchmarking these meth- ods [8, 10, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nonetheless, existing algorithms have significant limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Most approaches are 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='01147v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='CV] 31 Dec 2022 2 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 1: 4Seasons benchmark dataset overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Top: overlaid maps recorded at differ- ent times and environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The 3D points from the reference map (black) align well with the 3D points from the query map (blue), indicating that the reference poses are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Bottom: sample images demonstrating the diver- sity of our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The first row shows a collection from the same scene across different weather and lighting conditions: snowy, cloudy, sunny, and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The second row depicts the variety of scenarios within the benchmark: inner city, suburban, countryside, and a parking garage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' tailored to work well on small-scale datasets which exhibit limited challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, the next logical step towards pro- gressing research in the direction of visual SLAM is to make it robust under dynamically chang- ing and challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This includes VO, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' at night or rain, as well as long-term place recognition and localization against a pre-built map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In this regard, the advent of deep learning has exhibited itself to be a promising poten- tial in complementing the performance of visual SLAM [12, 28, 30, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, it has become all the more important to have datasets that are commensurate with handling the challenges of any real-world environment while also being capable of discerning the performance of state-of-the-art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To accommodate this demand, we present a cross-season and multi-weather benchmark, par- ticularly focusing on visual SLAM and long-term localization for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This bench- mark is based on the versatile large-scale 4Seasons dataset [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To the best of our knowledge, we pro- vide the first large-scale cross-season benchmark dataset comprising stereo images, corresponding high frame-rate inertial measurement unit (IMU), and accurate RTK GNSS measurements to evalu- ate sequential localization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' By traversing the same route under different conditions and over a long-term time horizon, we capture variety in illumination and weather, as well as in the appear- ance of the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For each scenario, we provide multiple traversals exhibiting different environ- mental conditions, as described in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The recordings show vastly different variations in the scene geometry including dynamic objects, road- works, construction sites, and seasonal changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To acquire accurate reference poses of large-scale scenes, we use a custom stereo-inertial sensor together with a RTK GNSS system to obtain up to centimeter-accurate poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 1 visualizes two overlaid 3D reconstructions of the same scene recorded at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, the figure depicts sample images of the dataset used to eval- uate six degrees of freedom (6DoF) localization against a prior map using query images taken from a variety of challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We pro- vide reference poses for a subset of the recordings, and withhold the remaining for an online evalua- tion benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We design a benchmark to measure the impact of long-term environmental changes on the performance of visual SLAM and localization for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The main contributions of this paper are the extensive benchmark suite for evaluating the long- term visual localization problem for autonomous driving, the evaluation of state-of-the-art baseline SLAM and visual localization algorithms, and the interpretation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This work extends our paper published at GCPR 2020 [72] through the following additional contributions: We propose a large-scale cross-season and multi- weather benchmark suite for long-term visual SLAM in automotive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' It allows the joint evaluation of visual odometry, global place recognition, and map-based visual localization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We release plenty of additional sequences cover- ing nine different types of environments, ranging from a multi-level parking garage to urban (including tunnels) to countryside and highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3 We provide an extensive evaluation of state- of-the-art baseline approaches for visual SLAM and visual localization on the presented bench- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To foster research, our benchmark and evalu- ation protocols will be available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4seasons-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2 Related Work There exists a variety of benchmarks and datasets focusing on VO and SLAM for autonomous driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Here, we divide these datasets into the ones which focus only on VO as well as those covering different weather conditions and therefore aiming towards long-term SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Visual Odometry Datasets & Benchmarks The most popular benchmark for autonomous driving probably is KITTI [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This multi-sensor dataset covers a wide range of tasks including not only VO, but also 3D object detection and tracking, scene flow estimation as well as seman- tic scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The dataset contains diverse scenarios ranging from urban to country- side to highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, all scenarios are only recorded once and under similar weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Ground truth is obtained based on a high-end inertial navigation system (INS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Another dataset containing LiDAR, IMU, and image data at a large scale is the M´alaga Urban dataset [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, in contrast to KITTI, no accurate 6DoF ground truth is provided, and therefore it does not allow for an appropriate quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, only a few places are visited multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Other popular datasets for the evaluation of VO and visual-inertial odometry (VIO) algo- rithms that are not related to autonomous driv- ing include [59] (handheld RGB-D), [7] (UAV stereo-inertial), [15] (handheld mono), and [55] (handheld stereo-inertial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Long-Term SLAM Datasets & Benchmarks More related to our work are datasets containing multiple traversals of the same environment over a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Concerning SLAM for autonomous driving, the Oxford RobotCar Dataset [38] repre- sents a kind of pioneer work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This dataset consists of large-scale sequences recorded multiple times in the same environment for one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Hence, it covers large variations in the appearance and structure of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, the diversity of the scenarios is only limited to an urban envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Also, the ground truth provided for the dataset is not accurate up to centimeter-level [38, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Other existing datasets are lacking sequen- tial structure [33], only provide a certain adverse condition [42], or focus on AR scenarios [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The work by [52] proposes three comple- mentary benchmark datasets based on exist- ing datasets, namely RobotCar Seasons (based on [38]), Aachen Day-Night (based on [51]), and CMU Seasons (based on [5]) that have been used for benchmarking visual localization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The ground truth of the RobotCar Seasons [52] dataset is obtained based on structure from motion (SfM) and LiDAR point cloud align- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, due to inaccurate GNSS mea- surements [38], a globally consistent ground truth up to centimeter-level accuracy can not be guar- anteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, this dataset only provides one reference traversal in the overcast condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In contrast, we provide globally consistent reference models for all training traversals cov- ering a wide variety of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Hence, every traversal can be used as a reference model that allows further research on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' analyzing suitable reference-query pairs for long-term localization and mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Global place recognition datasets such as Pittsburgh [63], Tokyo 24/7 [64], and Mapil- lary Street-Level Sequences [71] provide only coarse-scale location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Other related localization datasets include 12-Scenes [67], InLoc [61], Cambridge Landmarks [32], and CrowdDriven [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Other Datasets Examples of further multipurpose autonomous driving datasets that also can be used for VO are [8, 10, 26, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' As stated in Section 1, our proposed bench- mark dataset differentiates from previous related work in terms of being both large-scale (similar to [20]) and having high variations in appear- ance and conditions (similar to [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, 4 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' accurate reference poses based on the fusion of direct stereo VIO and RTK GNSS are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To the best of our knowledge, we are the first to introduce a public, modular benchmark for evalu- ating visual SLAM, global place recognition, and map-based visual localization approaches under challenging conditions for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3 System Overview This section presents the sensor setup which is used for data recording (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Further- more, we describe the calibration of the entire sensor suite (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2) as well as our approach to obtain up to centimeter-accurate global 6DoF reference poses (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Sensor Setup The hardware setup consists of a custom stereo- inertial sensor for 6DoF pose estimation, as well as a high-end RTK GNSS receiver for global posi- tioning and global pose refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 2 shows our test vehicle equipped with the sensor system used for data acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Stereo-Inertial Sensor The core of the sensor system is our custom stereo- inertial sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This sensor consists of a pair of monochrome industrial-grade global shutter cam- eras (Basler acA2040-35gm) and lenses with a fixed focal length of f = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 mm (Stemmer Imag- ing CVO GMTHR23514MCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The cameras are mounted on a highly-rigid aluminum rail with a stereo baseline of 30 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' On the same rail, a Pre- cision MEMS IMU (Analog Devices ADIS16465) is mounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The cameras and the IMU are trig- gered over an external clock generated by an FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Here, the trigger accounts for exposure compensations, meaning that the time between the centers of the exposure interval for two con- secutive images is always kept constant (1/[frame rate]) independent of the exposure time itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, based on the FPGA, the IMU is properly synchronized with the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In the dataset, we record stereo sequences with a frame rate of 30 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We perform pixel binning with a factor of two and crop the image to a resolution of 800 × 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This results in a field of view of approx- imately 77° horizontally and 43° vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The IMU is recorded at a frequency of 2000 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' During recording, we guarantee an equal exposure time for the left and the right image of each stereo pair as well as a smooth exposure transition in highly dynamic lighting conditions, as it is favorable to visual SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide those exposure times for each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 GNSS Receiver For global positioning and to compensate drift in the VIO system, we utilize an RTK GNSS receiver (mosaic-X5) from Septentrio in combination with an Antcom Active G8 GNSS antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The GNSS receiver provides a horizontal position accuracy of up to 6 mm by utilizing RTK correction sig- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While the high-end GNSS receiver is used for accurate positioning, we use a second receiver connected to the time-synchronization FPGA to obtain GNSS timestamps for the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Calibration 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Aperture and Focus Adjustment The lenses used in the stereo system have both adjustable aperture and focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, before performing the geometric calibration of all sensors, we manually adjust both cameras for a matching average brightness and a minimum focus blur [25], across a structured planar target in 10 m distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Stereo Camera and IMU For the intrinsic and extrinsic calibration of the stereo cameras, as well as the extrinsic calibra- tion and time-synchronization of the IMU, we use Kalibr1 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The stereo cameras are mod- eled using the Kannala-Brandt model [31], a generic camera model consisting of a total of eight parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We validated the calibration accuracy of each recording by performing a feature-based epipolar-line consistency check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 GNSS Antenna Since the GNSS antenna does not have any orien- tation but has an isotropic reception pattern, only the 3D translation vector between one of the cam- eras and the antenna within the camera frame has to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This vector was measured manually for our sensor setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/ethz-asl/kalibr Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5 (a) Test vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (b) Sensor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2: Recording setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Test vehicle and sensor system used for dataset recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The sensor system consists of a custom stereo-inertial sensor with a stereo baseline of 30 cm and a high-end RTK GNSS receiver from Septentrio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Ground Truth Generation Reference poses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' ground truth) for VO and SLAM should provide high accuracy in both local relative 6DoF transformations and global posi- tioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To fulfill the first requirement, we extend the state-of-the-art stereo direct sparse VO [69] by integrating IMU measurements [68], achieving a stereo-inertial SLAM system offering average tracking drift around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 % of the traveled dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To fulfill the second requirement, the poses estimated by our stereo-inertial system are fused with the RTK GNSS measurements using a global pose graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We first estimate a Sim(3) transfor- mation to globally align the camera positions in the VIO coordinate system to those in the GNSS coordinate system using the Kabsch–Umeyama algorithm [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A transformation in Sim(3) is esti- mated instead of in SE(3) to account for the global scale drift in the VIO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Denoting the Lie- algebra of SE(3) as se(3), each aligned camera pose ξVIO wi ∈ se(3) is added to the pose graph as a se(3) node, where ξwi defines a transformation from the i-th camera coordinate system to the world coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The camera connections from the VIO sliding window (one connection cor- responds to two cameras co-observing a part of the scene) are added as se(3) − se(3) edges, with the relative poses ξVIO ji as the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' If a camera pose has a valid corresponding GNSS pose, that is, the GNSS pose is available and the observed standard deviation of the position is smaller than a predefined threshold, the GNSS pose ti ∈ R3 is added to the pose graph as a fixed R3 node and an se(3) − R3 edge is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The energy function defined for the pose graph optimization is thus defined as: E(ξwi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' , ξwn) = � ξVIO ji ∈ε (ξVIO ji ξ−1 wi ◦ ξwj)⊤Σ−1 ji (ξVIO ji ξ−1 wi ◦ ξwj)+ ω � ti∈ν (ti − (ξwi ◦ ξcg)[t])⊤Σ−1 i (ti − (ξwi ◦ ξcg)[t]), (1) where ε is the set of VIO camera connections, ν is the set of valid RTK GNSS poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Σji ∈ R6×6 and Σi ∈ R3×3 are the covariance matrices from the VIO and GNSS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' ξcg denotes the extrin- sic calibration between the camera and the GNSS antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A scale term ω is added to balance the two different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The ◦-operator defines the concatenation of poses defined as se(3) and therefore is defined as follows: ξi ◦ ξj := log(exp(ξi) · exp(ξj)), (2) where log(·) defines the logarithm and exp(·) the exponential map of the SE(3) Lie-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' ξ[t] denotes the translation part in se(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The energy function is optimized using the Leven- berg–Marquardt algorithm in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' GNsS antenna stereo-inertial sensor6 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' One crucial aspect of the dataset is that the reference poses that we provide are accu- rate enough, even though some recorded sequences contain challenging conditions in partially GNSS- denied environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Although the stereo-inertial sensor system has an average drift around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 %, this cannot be guaranteed for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Hence, for the reference poses in our dataset, we report whether a pose can be considered to be reliable by measuring the distance to the corresponding RTK GNSS measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For all poses, with- out corresponding RTK GNSS measurement we do not guarantee a certain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, due to the highly accurate stereo-inertial odom- etry system, these poses can be considered accu- rate in most cases, even in environments without GNSS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' tunnels, or areas with tall buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide details about the pose accuracy in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4 Benchmark Setup To overcome the shortcomings of existing bench- marks and datasets for autonomous driving, as discussed in Section 2, we define the following requirements for an appropriate benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Accuracy: we provide up to centimeter-accurate 6DoF poses obtained by fusing VIO measure- ments with RTK GNSS correction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Large-scale: we provide large-scale sequences (trajectories longer than 10 km) to allow for extensive evaluations of SLAM and visual local- ization under challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Diversity: besides large-scale, we also provide both short-term and long-term changes within the recorded scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This is important to eval- uate the generalization capabilities of recent learning-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Multitask: the benchmark can be used to eval- uate visual odometry, global place recognition, and map-based visual localization under chal- lenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Based on these properties, we propose a novel large-scale dataset that is used as an extensive benchmark suite for evaluating multitasking chal- lenges related to autonomous driving under chang- ing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The sequences have been collected in the metropolitan area of Munich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The different scenes are described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3: Data collection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This figure shows the map of the covered area of our benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide sequences at a large scale and a huge variety of different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A detailed visualization of each scenario’s trajectory is shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Scenarios This section describes the different sequences we have collected for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The sequences involve different scenarios – ranging from urban driving to a parking garage and rural areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide complex trajectories, which include partially overlapping routes, and multiple loops within a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For each scenario, we have col- lected multiple traversals covering a large range of variations in the structure and environmental appearance due to weather, illumination, dynamic objects, and seasonal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In total, our bench- mark dataset consists of nine different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 3 shows the covered area, including highlighted traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Each scenario is visualized in a separate color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We now describe each scene in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Office Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A loop around an industrial area of the city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A drive along the A9 three-lane highway in the northern part of Munich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' St 2053 FS 20 ohho 5 A9 Landkreis Fr Unterschleisheim Naturschutzgebiet Malertshofer Holz hing-ord B13 Garching-Word und Fo Berglhoiz 3heim 4 Garching bei B 471 Hochbruck Munchen Hochbruck Garching-Sud上 Schweizerholz, eigher 5t 2053 Frottmaninger Heide Kreuz Mun nen-Nord Munchen Weuherberg Monchen-Frotmaning-Nord Naturschutzgebiet Sudliche Frottmaninger und Hartelholz Heide chen-Nord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' St 2053 B13 euzMu Munchen-Frd maning-Sug Harthof: 99 9 Freim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='n AmHart Muniche Euro-Industriepar!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3 Unterfohring C kfurter Ring Ring .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='Schwabing esenfeld Milbertshofen Nord M3 Oberfohrir Schwabing johanneskirchen 6 Herzogpar Englschalk laxvorstadt Dennin Daglfing Bogenhausen Zamdorf lehel Munchen-Steinhousen Munchen-Zamdo Munchen-Dog!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Munchen Haidhausen rstadWenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 7 Table 1: Statistics of the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This table shows the different scenarios and record- ings along with the weather condition, seasons, and time of the day from our benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide a variety of scenarios and short-term to long-term changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The recordings in this table are used for the benchmark evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The ground truth (GNSS/IMU, point clouds, and reference poses) is withheld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Benchmark type (VO = visual odometry, GPR = global place recognition, MBVL = map-based visual localization) defines the benchmark a sequence is used for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' All recordings with ground truth are shown in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Scenario Recording Weather (cloudy, rainy, snowy, sunny) Season (winter, spring, summer, fall) Daytime (morning, afternoon, evening, night) Benchmark Type Map Accuracy Horizontal RMSE (GNSS-Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Pose) Map Accuracy % of Accurate Poses office loop 1 test 2020-03-03 12-12-32 cloudy spring afternoon GPR 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='29 cm 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='91 % office loop 2 test 2020-03-26 15-03-02 cloudy/sunny spring afternoon VO 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='14 cm 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='22 % office loop 3 test 2021-05-10 19-25-54 cloudy spring evening VO + MBVL 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='78 cm 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='06 % highway 1 test 2020-10-08 10-19-46 sunny fall morning VO 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='04 cm 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='65 % highway 2 test 2021-02-25 13-11-30 sunny winter afternoon VO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='80 cm 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='31 % neighborhood 1 test 2020-03-26 14-54-05 cloudy spring afternoon GPR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='20 cm 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='38 % neighborhood 2 test 2021-05-10 18-26-26 cloudy spring evening VO + MBVL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='51 cm 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='42 % business campus 1 test 2021-01-07 13-03-56 cloudy/snowy winter afternoon VO + MBVL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='39 cm 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='36 % countryside 1 test 2020-03-26 14-30-52 cloudy spring afternoon GPR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='53 cm 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='75 % countryside 2 test 2021-01-07 14-03-57 cloudy/snowy winter afternoon VO + MBVL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='36 cm 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='21 % city loop 1 test 2020-03-03 12-28-45 cloudy spring afternoon GPR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='36 cm 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='62 % city loop 2 test 2021-02-25 11-27-40 sunny winter morning VO + MBVL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='36 cm 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='40 % old town 1 test 2020-10-08 12-11-19 cloudy fall afternoon GPR 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='19 cm 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='26 % old town 2 test 2021-05-10 19-51-14 cloudy spring evening VO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='84 cm 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='04 % old town 3 test 2021-05-10 21-18-00 cloudy spring night VO + MBVL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='94 cm 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='07 % maximilianeum 1 test 2021-02-25 12-16-32 sunny winter afternoon VO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='90 cm 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='13 % maximilianeum 2 test 2021-05-10 20-59-00 cloudy spring night VO 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='46 cm 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='46 % parking garage 1 test 2020-06-12 10-29-20 sunny summer morning VO + MBVL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='75 cm 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='06 % parking garage 2 test 2021-05-10 19-18-36 cloudy spring evening GPR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='54 cm 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='75 % 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Traversal through a neigh- borhood at the outskirts of the city, covering detached houses with gardens and trees in the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Business Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Several loops around a campus in a business area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Countryside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Rural area around agricultural fields that exhibits very homogeneous and repetitive structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' City Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A large-scale loop at a ring road within the city of Munich, including a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Old Town.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Loop around the urban city center with tall buildings, much traffic, and dynamic objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Maximilianeum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The Maximilianeum is a famous palatial building in Munich which is located at the eastern end of a royal avenue with paving stones and a tram route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Parking Garage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A three-level parking garage to benchmark combined indoor and outdoor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The VIO traces for each scenario are shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide reference poses and 3D models as sparse point clouds generated by our ground truth generation pipeline (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 4) along with the corresponding raw image frames and raw IMU measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 5 shows an example of the optimized trajectory, which depicts the accuracy of the provided reference poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Table 1 shows all the sequences with withheld ground truth used for benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The benchmark dataset presents a challenge to current approaches to visual SLAM and long-term localization because it contains data from different seasons and weather conditions, as well as from different times of day, as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Reference Pose Validation The top part of Figure 1 shows two overlaid point clouds from different runs across the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Note that despite the weather and sea- sonal differences, the point clouds align very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This shows that our reference poses are suffi- ciently accurate for benchmarking long-term local- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, a qualitative assessment of the point-to-point correspondences is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows a subset of very accu- rate pixel-wise correspondences across different seasons (fall/winter) in the top and different illu- mination conditions (sunny/night) in the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 8 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4: 3D models of different scenarios contained in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows an office loop around an industrial area (left), multiple loops around a business campus with high buildings (middle), and a stretch recorded in a multi-level parking garage (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The green lines encode the GNSS trajec- tories, and the red lines encode the VIO trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Top: shows the trajectories before the fusion using pose graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Bottom: shows the results after the pose graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Note that after the pose graph optimization, the reference trajectories are well aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' These point-to-point correspondences are a result of our up to centimeter-accurate global reference poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This makes them suitable as training pairs for learning-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Recently, there has been an increasing demand for pixel-wise cross- season correspondences, which are needed to learn dense feature descriptors [12, 47, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, there is still a lack of datasets to satisfy this demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The KITTI [20] dataset does not pro- vide cross-season data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The Oxford RobotCar Dataset [38] provides cross-season data, however, since the ground truth is not accurate enough, the paper does not recommend benchmarking localization and mapping approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Recently, RobotCar Seasons [52] was proposed to overcome the inaccuracy of the provided ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, similar to the authors of [57], we found that it is still challenging to obtain accurate cross-season pixel-wise matches due to pose incon- sistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, this dataset only provides images captured from three synchronized cameras mounted on a car, pointing to the rear-left, rear, and rear-right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, another limitation of the dataset is that it only provides relatively small segments and no long trajecto- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, a significant portion of it suffers from strong motion blur and low image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Pose Accuracy One potential limitation of our benchmark dataset is that we can only guarantee a certain pose accu- racy when GNSS is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Naturally, GNSS is unreliable in urban canyons or tunnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, for the benchmark evaluation, we only consider poses as reference poses if GNSS is available and the observed standard deviation of the position is less than 5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Please note that we only require accurate reference poses for the evaluation of visual localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The evaluation of VO is based on the accumulated drift over time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' it is only required that the start and end positions for each segment of a sequence are accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, we provide quantitative measures of the quality of the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We report the percentage of accurate reference poses for each trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, we 11Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5: Reference poses validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This figure shows two additional 3D models of the scenar- ios collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Note that these two sequences are quite large (more than 10 km and 6 km, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Top: before the fusion using pose graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Bottom: results after optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The green lines encode the GNSS trajectories, the red lines show the VIO trajectories (before fusion) and the fused trajectories (after fusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The left part of the figure shows a zoomed-in view of a tun- nel, where the GNSS signal becomes very noisy, as highlighted in the red boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Besides, due to the large size of the sequence, the accumulated track- ing error leads to a significant deviation of the VIO trajectory from the GNSS recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Our pose graph optimization, by depending globally on GNSS positions and locally on VIO relative poses, successfully eliminates global VIO drifts and local GNSS positioning flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' report the overall map accuracy in terms of hor- izontal RMSE between the GNSS poses and the refined poses after pose graph optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The percentage of accurate poses for each test sequence can be seen in Table 1 and Table 5 for the training sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For a qualitative visual anal- ysis, we show accurate pixel-wise correspondences in Figure 6, indicating that the reference poses are sufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We do not claim that our poses are consistently centimeter-accurate, how- ever, by analyzing the map accuracy we can assure the quality of the poses used for benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Data Source We release (distorted & undistorted) 8-bit grayscale images, IMU measurements, and sensor calibration, including the calibration sequences, for all sequences (training and testing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In addi- tion, RTK GNSS measurements, in NMEA for- mat, VO point clouds, and reference poses are released only for training sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For the test- ing sequences, such data is withheld for evalua- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, we specify the distance between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6: Accurate pixel-wise correspon- dences, making cross-season training pos- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Qualitative assessment of the accuracy of our data collection and geometric reconstruction method for a sample of four different conditions (from top left in clockwise order: cloudy, snowy, night, sunny) across the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Each same colored point in the four images corresponds to the same geometric point in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The cameras corresponding to these images have different poses in the global frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Please note that the points are not matched, but rather a result of our accurate reference poses and geometric reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' the refined reference poses and the raw RTK GNSS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5 Benchmark Tasks In this section, we define the benchmark evalua- tion metrics, tasks, and their evaluation protocols for visual odometry, global place recognition, and map-based visual localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Visual localization consists of retrieving the 6DoF pose of a query within an existing 3D model and can be inter- preted as a two-step approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' First, global image retrieval is performed to obtain a rough esti- mate of the query pose w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Second, local feature matching is used to refine the pose estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For the evaluation, in each task, we consider a set of estimated 6DoF poses Test i ∈ SE(3), as well as a set of reference, poses Tref i ∈ SE(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While the reference poses are always defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' a global world frame, the estimated poses are defined either w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' the same global world frame (for global place recognition and map-based visual localization) or to a selected local frame2 (visual odometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 2Can be for instance the camera frame of the first recorded left camera image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 10 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Visual Odometry in Challenging Conditions Visual odometry aims to accurately estimate the relative 6DoF camera pose based on recorded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To benchmark the task of VO there already exists various datasets [15, 19, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' All of these existing datasets consist of sequences recorded at rather homogeneous con- ditions (indoors, or sunny/overcast outdoor con- ditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, methods specially developed for autonomous driving use cases must perform robustly under almost any condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We believe that the proposed benchmark will contribute to improving the performance of VO under diverse weather and lighting conditions in an automotive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, instead of replacing exist- ing benchmarks and datasets, we aim to provide an extension that is more focused on challenging conditions in autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' As we pro- vide frame-wise accurate poses for large portions of the sequences, metrics well known from other benchmarks like absolute trajectory error (ATE) or relative pose error (RPE) [19, 59] are also applicable to our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Evaluation Metrics Similar to previous benchmarks, the main accu- racy measure we are interested in is the RPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In general, the RPE is split up into a translational and a rotational error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, another compo- nent we are interested in is the scale error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' One may argue that, especially for stereo approaches, scale errors are marginal and therefore not rel- evant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, our experience is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We observe that quite significant scale errors and drift can occur when performing stereo VO and SLAM in automotive environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This can be caused either by the miss-calibration of the cam- eras, by the structure of the scene but also by algorithm-specific design choices like the type of keypoint detector, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Since the sensor setup has a limited stereo baseline, parallaxes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' pixel disparities) for far object points are vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This means that, even for stereo approaches, the scale becomes non-observable if no close static objects are present in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Increasing the stereo baseline, however, could reduce the rigid- ity of the sensor setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We believe that it is very valuable to conduct further research on stereo VO and SLAM methods which explicitly consider the depth uncertainties created by the length of the stereo baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Since in automotive use cases, the scale can always be observed based on a reference system, like wheel ticks, GNSS or a reference map, we consider only relative errors (drifts) in scale, trans- lation, and rotation in the proposed benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, before evaluation, a global scale align- ment is performed for the entire trajectory with respect to the reference trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For the proposed VO benchmark all evaluation metrics are defined based on the estimated relative pose Test ij ∈ SE(3) between two frames i and j and its corresponding reference pose Tref ij ∈ SE(3) with: Tref ij = � Tref i �−1 Tref j and Test ij = � Test i �−1 Test j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (3) For a pair of frames (i, j) for which reference poses are available, we calculate the relative trans- lational error ϵt ij, rotational error ϵr ij, and scale error ˜ϵs ij as given in Equations (4) to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' ϵt ij = ∥tref ij − test ij ∥2 dij (4) ϵr ij = arccos � 1 2 � trace ((Rref ij )−1Rest ij ) − 1 �� dij (5) ˜ϵs ij = ��test ij �� 2 ��tref ij �� 2 (6) From ˜ϵs ij one obtains the final relative scale error as ϵs ij = max[˜ϵs ij, (˜ϵs ij)−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The parameter dij defines the reference path length between the two poses Tref i and Tref j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Meaningful metrics are obtained by extracting all possible sub-segments of length 100 m, 200 m, 400 m, 600 m, 800 m, and 1000 m from a trajec- tory and calculating the relative poses between the first and last frame of each sub-segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fur- thermore, for trajectory segments where no GNSS measurements are available for more than 1000 m (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' in tunnels, garages, or urban canyons), also the relative poses of such an entire stretch are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This allows us to also consider challenging scenarios like tunnels and the transi- tion from bright to dark in the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Using sub-segments of different lengths for evaluation is Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 7: Challenging scenes for global place recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Top: two pictures share the same location with different appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Bottom: two pictures have a similar appearance but are taken at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' inspired by the KITTI benchmark [19] and allows capturing both short and long-term accuracy of VO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To obtain single number metrics for every sequence, we consider the visual VO successful if the errors are within certain positional, rota- tional, and scale bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We define three intervals by varying the thresholds: high precision (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 deg/m, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 (multiplier)), medium pre- cision (1 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='01 deg/m, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='01 (multiplier)), and coarse precision (2 %, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='02 deg/m, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='02 (multi- plier)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While the translational error is the most mean- ingful metric to evaluate VO algorithms, the rotational error, and scale error still give valu- able insight into the specific behavior of a certain approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Global Place Recognition Global place recognition refers to the task of retrieving the most similar database image given a query image [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To improve the searching efficiency and the robustness against different weather conditions, tremendous progress on global descriptors [1, 2, 18, 29] has been seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For the localization pipeline, visual place recognition serves as the initialization step to the downstream local pose refinement by providing the most sim- ilar database images as well as the corresponding global poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Due to the advent of deep neural networks [24, 34, 56, 60], methods aggregating deep image features are proposed and have shown advantages over classical methods [3, 21, 44, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The proposed dataset is challenging for global place recognition since it contains not only cross- season images that have different appearances with a similar geographical location but also intra- season images which share similar appearances but with different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This results in mainly two different types: images taken at the same place, but look different, or images taken at dif- ferent places but look similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 7 depicts example pairs of these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Evaluation Metrics We follow the standard metric widely used for global place recognition [2, 3, 21, 51], namely the recall at top N retrievals with a certain range bound as the positive threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Specifically, one query image is considered to be correctly local- ized if at least one of the top N retrieved images is within a certain translational (in meters) and a certain rotational (in degrees) bound with respect to the ground-truth location of the query image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The translational error ϵt is measured as the Euclidean distance: ϵt = ∥tref − test∥2 (7) between the reference tref and estimated test cam- era positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The rotational error ϵr is measured as an angle in degrees (following [23]) by calculat- ing: ϵr = arccos �1 2 � trace ((Rref)−1Rest) − 1 �� , (8) where Rref, and Rest denote the reference and estimated camera rotation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In the evalu- ation of global place recognition, we calculate the recalls under different threshold settings: by fixing N and changing the range bound, or by fixing the range bound and changing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We will describe the specific settings in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Map-Based Visual Localization Map-based visual localization refers to the task of locally refining the 6DoF pose between refer- ence images and images from a query sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In contrast to wide-baseline stereo matching, for map-based visual localization, it is also possible to utilize the sequential information of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This allows estimating depth values by running a standard VO method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Those depth estimates can then be used to improve the tracking of the individual localization candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 12 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In contrast to global place recognition which only uses 2D images and no other information, this task allows the use of a globally-consistent 3D reconstruction of the reference scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In this task, we assume to know the mapping between reference and query samples and only focus on the local pose refinement task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In practice, this mapping can be found using image retrieval tech- niques as described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 or by using GNSS measurements as a coarse initialization if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Accurately localizing in a pre-built map is a challenging problem, especially if the visual appearance of the query sequence significantly dif- fers from the base map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This makes it extremely difficult, especially for vision-based systems, since the localization accuracy is often limited by the discriminative power of feature descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Our proposed dataset allows evaluating visual localiza- tion across multiple types of weather conditions and diverse scenes, ranging from urban to country- side driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, our up to centimeter- accurate reference poses allow us to create more strict evaluation settings with an increased level of difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This allows us to determine the lim- itations and robustness of current state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Evaluation Metrics For evaluation, we measure the translational and rotational error of any method between the esti- mated and the reference pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Please refer to Equation (7) and Equation (8) for the defini- tions of the translational and rotational error, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We consider the localization successful if a query image is localized within certain positional (in meters) and rotational (in degrees) bounds with respect to their reference pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We define three localization intervals by varying the thresh- olds: high precision (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 m, 1°), medium pre- cision (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='25 m, 2°), and coarse precision (1 m, 5°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6 Experimental Evaluation In this section, we evaluate the current state-of- the-art baseline methods for each of the three pro- vided benchmarks (visual odometry, global place recognition, and map-based visual localization) to demonstrate the diversity and challenges of the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We will establish an open leaderboard for the benchmark to compare different methods upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This allows the reproduction of the baseline results for every user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Furthermore, we will set up a server for an automatic evaluation of the results on the withheld test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 Visual Odometry in Challenging Conditions We provide results for state-of-the-art baseline stereo and stereo-inertial odometry and SLAM approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The methods provided as baselines are classical geometric approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, we strongly encourage researchers to evaluate learning-based methods on our benchmark as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In particular, we provide results for the following stereo and stereo-inertial VO methods: ORB-SLAM33 [9] and Basalt4 [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The provided VO benchmark is divided into two sets of evaluation sequences: unknown sce- narios and known scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Unknown scenarios consist only of scenarios, for which no sequences at all are provided in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Namely, these are the scenarios Highway and Maximilianeum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Known scenarios are those scenarios for which are also sequences provided in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While this is irrelevant for pure geometric approaches, we believe that this separation will be impor- tant to evaluate the generalization capabilities of learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Table 2 shows the eval- uation results on the individual sequences of the benchmark for known scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 8 shows the results across all sequences corresponding to the known scenarios in cumulative error plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Table 3 shows the evaluation results on the indi- vidual sequences of the benchmark for unknown scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 9 shows the results across all sequences corresponding to the unknown scenarios in cumulative error plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' From Table 2 and 3 one can observe that all evaluated methods perform significantly worse on the unknown scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, this is mainly due to the challenging conditions, which are on one side highway sequences with high speed and sudden lighting changes under bridges as well as inner city night sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/UZ-SLAMLab/ORB SLAM3 4https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/VladyslavUsenko/basalt Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 13 Table 2: Visual odometry results on known scenarios from the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This table shows the evaluation results of state-of-the-art baseline methods on the VO benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The best- performing results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The results are shown in terms of the percentage of high / medium / coarse precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Method office loop 2 test office loop 3 test neighborhood 2 test business campus 1 test countryside 2 test city loop 2 test old town 2 test old town 3 test parking garage 1 test Average Basalt [66] (stereo) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 53.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Basalt [66] (stereo-inertial) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 92.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 / 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='8 / 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 ORB-SLAM3 [9] (stereo) 16.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 / 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': 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translational error (in %), rotational error (in mdeg/m), and scale error (multiplier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Table 3: Visual odometry results on unknown scenarios from the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This table shows the evaluation results of state-of-the-art baseline methods on the VO benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The best-performing results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The results are shown in terms of the percentage of high / medium / coarse precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Method highway 1 test highway 2 test maximilianeum 1 test maximilianeum 2 test Average Basalt [66] (stereo) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 / 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 / 52.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='9 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='8 / 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 / 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 While the results above provide average num- bers across all sequences of the benchmark, we provide in Figure 10 and 11 side-by-side the results for identical scenarios but for different conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 10 provides VO results on the Max- imilianeum scenario in the afternoon (max- imilianeum 1 test) and at night (maximilia- neum 2 test), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' As one could expect, there is a significant drop in performance when going from day to night due to less visible land- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, it is interesting to observe that ORB-SLAM3 (with IMU) can perform bet- ter during the night than Basalt (with IMU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A reason might be that ORB-SLAM3 is using fea- ture matching to find point correspondences, while Basalt is relying on optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This correla- tion cannot be observed when running without IMU, where ORB-SLAM3 is failing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, especially during the night and without IMU, the task becomes inordinately more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 11 provides performance comparisons between a sunny (office loop2 test) and a cloudy (office loop3 test) condition on the Office Loop scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Across all algorithms, one can observe improved performance during sunny weather con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A likely reason for this is the presence of more static feature points caused by shadows, especially on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This can be seen on the right side images in Figure 11, where much more texture is on the road for the sunny than for the cloudy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' While the evaluated methods show all-in-all good performance in good weather and light- ing conditions, we believe that our dataset and benchmark will contribute to improving the per- formance in conditions with fewer and unreliable feature points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The results show that the pro- posed benchmark is highly challenging and still provides room for improving state-of-the-art VO algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 14 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='020 Scale error (multiplier) 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 9: Performance of state-of-the-art visual odometry methods on unknown scenarios from the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows the translational error (in %), rotational error (in mdeg/m), and scale error (multiplier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='020 Scale error (multiplier) 0 20 40 60 80 100 (a) Afternoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='020 Scale error (multiplier) 0 20 40 60 80 100 (b) Night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 10: Comparison of visual odometry performance for afternoon and night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows the performance for different state-of-the-art baseline VO algorithms on the same route for afternoon and night conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' One can observe a significant drop in performance when going from day to night due to less visible landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 Global Place Recognition We evaluate the current state-of-the-art base- line deep image descriptors methods including NetVLAD5 [3] pretrained on Pittsburgh30k [63], Deep Image Retrieval (DIR)6 [22, 46] (aka AP- GeM) trained on the Landmarks dataset [4], and CNN Image Retrieval (CIR)7 [43, 44] trained on the dataset derived from [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For each scenario of the 4Seasons benchmark, we use a predefined recording as the reference map and a predefined recording as the query map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Note that we leave 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/cvg/Hierarchical-Localization 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/naver/deep-image-retrieval 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com/filipradenovic/ cnnimageretrieval-pytorch out the Business Campus scenario for global place recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' As shown in Figure 12, we first plot two differ- ent recall curves: (1) Recall[%] – Threshold [m] @ Top1: the recalls of different methods when chang- ing the distance threshold in the range 1 m–20 m using only the top 1 retrieved images, and (2) Recall[%] – Top N @ 1 m: the recalls of different methods when changing the number of candidate retrievals N ∈ {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' , 20} using the fixed range bound 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We also show the optimal recall by using the closest database images with respect to the ground-truth query image location as the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This shows the upper bound of the global place recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 15 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='020 Scale error (multiplier) 0 20 40 60 80 100 (a) Sunny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 0 1 2 3 Translational error [%] 0 20 40 60 80 100 Occurrence [%] Basalt (stereo) Basalt (stereo-inertial) ORB-SLAM3 (stereo) ORB-SLAM3 (stereo-inertial) 0 5 10 15 20 25 Rotational error [mdeg/m] 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='005 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='020 Scale error (multiplier) 0 20 40 60 80 100 (b) Cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 11: Comparison of visual odometry performance for sunny and cloudy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows the performance for different state-of-the-art baseline VO algorithms on the same route for sunny and cloudy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Across all algorithms, one can observe improved performance during sunny weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' A likely reason for this is the presence of more static feature points caused by shadows, especially on the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This can be seen on the right side images, where much more texture is on the road for the sunny than for the cloudy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' From the results, we can see that NetVLAD still outperforms the other recent methods by a notable margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' One reason for NetVLAD’s supe- rior performance could be the introduction of the inductive bias into the network design, based on the established principle of classical VLAD [2, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' However, one shall also admit that the gap between the state-of-the-art methods and the opti- mal performance is still quite large, and more research on global place recognition still needs to be conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We show the localization accuracy of the global place recognition methods without using local pose refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Note that for these methods we use the top 20 candidates and loosened range bounds, namely, (1 m, 5°) for high precision, (5 m, 10°) for medium precision, and (10 m, 20°) for coarse precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The last three rows of Table 4 show the individual global place recogni- tion (GPR) performance on each of the evaluated scenarios from the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Map-Based Visual Localization For the evaluation of map-based visual localiza- tion, we use the following processing pipeline: we first build a SfM model from the reference scene 0 5 10 15 20 Threshold [m] @ Top1 0 20 40 60 80 100 Recall [%] NetVLAD CIR DIR optimal 0 5 10 15 20 Top N @ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0m 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 12: Performance of state-of-the-art baseline global place recognition methods on the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The gray line indicates the upper bound for the global place recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' that provides correspondences between local fea- tures and 3D points in the reconstructed map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This is followed by 2D-3D matching between the query images and the database images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' As the last step, those 2D-3D matches are used for cam- era pose estimation via Perspective-n-Point (PnP) and random sample consensus (RANSAC) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In particular, we evaluate the current state-of-the-art coarse-to-fine hierarchical localization method [48] based on the following learned deep local fea- ture descriptors: SuperPoint [11], D2-Net [12], and R2D2 [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Additionally, we use the classic scale- invariant feature transform (SIFT) [36] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Hloc [48] simultaneously predicts local features 16 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Table 4: Visual localization results on the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We report the percentage of images localized within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 m and 1°, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='25 m and 2°, 1 m and 5° of the reference poses for map-based visual localization (MBVL) pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For global place recognition (GPR) methods, we report the percentage of images localized within 1 m and 5°, 5 m and 10°, 10 m and 20° of the reference poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The best-performing results for both MBVL and GPR pipelines are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Method Office Loop Neighborhood Business Campus Countryside City Loop Old Town Parking Garage Average MBVL hloc [48] (SuperPoint [11] + SuperGlue [49]) 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 / 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 / 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 / 73.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 CNN Image Retrieval (CIR) [44] 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='8 / 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 – 6.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 / 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 / 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 / 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='3 Deep Image Retrieval (DIR) [46] 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='1 / 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='7 / 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='8 / 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='0 Distance threshold [m] 0 20 40 60 80 100 Correctly classified queries [%] SP+SG D2-Net+NN SIFT+NN R2D2+NN 0 1 2 3 4 5 Orientation threshold [deg] 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 13: Performance of state-of-the-art baseline map-based visual localization approaches on the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure shows the cumulative localization accuracy against the translational, and rotational error, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' and global descriptors for accurate 6DoF local- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This approach first performs global place recognition to obtain location candidates, and afterward matches the local features only within those places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The extracted local image features are used to establish 2D-3D matches within a pre- built SfM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Pose estimation is performed using COLMAP [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Therefore, this pipeline can be seen as a pose refinement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' For each scenario of the 4Seasons benchmark, we use a predefined recording as the reference map and a predefined recording as the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Figure 13 shows the percentage of correctly classified queries when changing the distance and orientation thresholds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This figure shows the average performance for the differ- ent state-of-the-art map-based visual localization (MBVL) approaches across all evaluated scenar- ios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The first four rows of Table 4 show the indi- vidual hierarchical localization performance on each of the evaluated scenarios from the 4Seasons benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' From the results, we can see that the clas- sic SIFT+nearest neighbor (NN) approach shows a bad performance estimating the 6DoF pose between the reference and query candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The results also suggest that deep learning-based algorithms dramatically outperform the classical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This is due to the challenging nature of the benchmark since it deals with dras- tic lighting and illumination changes, occlusions, and changing environmental/weather conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Those results provide valuable insights into the limitations and failure cases of the different meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We observe that learned feature descriptors significantly outperform classic methods under challenging conditions contained in the 4Seasons benchmark, with SuperPoint+SuperGlue yielding the best results overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Nevertheless, the results from Table 4 show that the long-term localization problem is still far from being solved, especially for highly dynamic environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Old Town) and scenes that exhibit very similar structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Countryside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Our benchmark provides the basis to enable more research advances that are needed to close this performance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 7 Conclusion Current benchmarks either focus mainly on evalu- ating the performance of simultaneous localization and mapping algorithms or visual localization in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' To close this gap, we introduce a benchmark that by providing a holistic way for jointly benchmarking long-term visual SLAM and localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In this paper, we have introduced a compre- hensive benchmark suite for visual SLAM and Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 17 visual localization for autonomous driving under challenging conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The benchmark covers a huge variety of environmental conditions, along with short-term and long-term weather and illu- mination changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Moreover, we have reviewed and evaluated the current state-of-the-art baseline approaches for visual SLAM and visual localiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We have observed large performance gaps and see huge potential in future work to close those gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The benchmark dataset, evaluation server, and leaderboard will be available upon accep- tance via the benchmark’s website, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 4seasons-dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We express our apprecia- tion to our colleagues at Artisense for their help with setting up the recording setup and sensor design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Declarations Competing Interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' References [1] Angeli A, Filliat D, Doncieux S, et al (2008) Fast and incremental method for loop-closure detection using bags of visual words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' IEEE Transactions on Robotics (T- RO) 24(5):1027–1037 [2] Arandjelovic R, Zisserman A (2013) All about VLAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 1578–1585 [3] Arandjelovic R, Gronat P, Torii A, et al (2016) NetVLAD: CNN architecture for weakly supervised place recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 5297–5307 [4] Babenko A, Slesarev A, Chigorin A, et al (2014) Neural codes for image retrieval.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Interna- tional Journal of Robotics Research (IJRR) 35(10):1157–1163 [8] Caesar H, Bankiti V, Lang AH, et al (2020) nuScenes: A multimodal dataset for autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 11,621– 11,631 [9] Campos C, Elvira R, G´omez JJ, et al (2020) ORB-SLAM3: An accurate open- source library for visual, visual-inertial and multi-map SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: arXiv preprint arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='11898 [10] Cordts M, Omran M, Ramos S, et al (2016) The cityscapes dataset for semantic urban scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 3213–3223 [11] DeTone D, Malisiewicz T, Rabinovich A (2018) SuperPoint: Self-supervised interest point detection and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pp 224–236 [12] Dusmanu M, Rocco I, Pajdla T, et al (2019) D2-Net: A trainable CNN for joint detection and description of local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceed- ings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 8092–8101 [13] Engel J, Sch¨ops T, Cremers D (2014) LSD- SLAM: Large-scale direct monocular SLAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the European Conference on Computer Vision (ECCV), pp 834–849 18 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 14: Dataset overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Example images from our benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' First row: office loop, second row: highway, third row: neighborhood, fourth row: business campus, fifth row: countryside, sixth row: city loop, seventh row: old town, eighth row: maximilianeum, ninth row: parking garage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' The figure illustrates the large appearance changes, occlusions, seasonal, and structural changes present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' [14] Engel J, St¨uckler J, Cremers D (2015) Large- scale direct SLAM with stereo cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE/RSJ Conference on Intelligent Robots and Systems (IROS), pp 1935–1942 [15] Engel J, Usenko V, Cremers D (2016) A photometrically calibrated benchmark for monocular visual odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: arXiv preprint arXiv:1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='02555 [16] Engel J, Koltun V, Cremers D (2017) Direct sparse odometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI) 40(3):611–625 [17] Fischler MA, Bolles RC (1981) Random sam- ple consensus: a paradigm for model fitting with applications to image analysis and auto- mated cartography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Communications of the ACM 24(6):381–395 Wenzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 19 (1) Office Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (2) Highway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (3) Neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (4) Business Campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (5) Countryside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (6) City Loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (7) Old Town.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (8) Maximilianeum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' (9) Parking Garage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' 15: Scenarios overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' This figure shows all the covered scenarios of our benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' We provide vastly different environments in and around the city of Munich, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' [18] G´alvez-L´opez D, Tardos JD (2012) Bags of binary words for fast place recognition in image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' IEEE Transactions on Robotics (T-RO) 28(5):1188–1197 [19] Geiger A, Lenz P, Urtasun R (2012) Are we ready for autonomous driving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' the KITTI vision benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp 3354– 3361 [20] Geiger A, Lenz P, Stiller C, et al (2013) Vision meets robotics: The KITTI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' International Journal of Robotics Research (IJRR) 32(11):1231–1237 [21] Gordo A, Almaz´an J, Revaud J, et al (2016) Deep image retrieval: Learning global repre- sentations for image search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' In: Proceedings of the European Conference on Computer Vision (ECCV), pp 241–257 [22] Gordo A, Almazan J, Revaud J, et al (2017) End-to-end learning of deep visual representations for image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Interna- tional Journal of Computer Vision (IJCV) 124(2):237–254 [23] Hartley R, Trumpf J, Dai Y, et al (2013) Rotation averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' International Journal of Computer Vision (IJCV) 103(3):267–305 elBogen Metro 品 Harnierplatz Alois-Woifmuller-StraBe EsVFreimann Werk 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content='20 Campus Freimann Freimann DBNetz7 A9 DBSystemtechnik Munchen-FraahurterRir Guterbahnhof Edisonstrage Munchen Bahn-Landwirischot Frelmonn Manchen- Freirmann Leaflet Data by OpenStreetMap, under ODblRodelhuge Garching-5ud merkona!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9AzT4oBgHgl3EQfNPvf/content/2301.01147v1.pdf'} +page_content=' Hochbruck 490m Schweizerholz GOrCH 4-502 Frottmaninger B388 st2053 Heide Kreuz.' 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+1,672 @@ +Sub-shot-noise sensitivity in a ring laser gyroscope +Angela D. V. Di Virgilio1, Francesco Bajardi2,3, Andrea Basti1,4, Nicol`o Beverini4, Giorgio +Carelli1,4, Donatella Ciampini1,4, Giuseppe Di Somma1,4, Francesco Fuso1,4, Enrico +Maccioni1,4, Paolo Marsili1,4, Antonello Ortolan5, Alberto Porzio2,6,∗ and David Vitali7,8 +1INFN Sez. +di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy +2INFN, Sez. di Napoli, Compl. Univ. Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy +3Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138, Napoli, Italy +4Dipartimento di Fisica, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy +5INFN - LNL , Viale dell’Universit‘a 2, 35020 Legnaro (PD), Italy +6Department of Civil and Mechanical Engineering – DICEM, +Universit´a di Cassino e Lazio Meridionale, 03043, Cassino, Italy +7Physics Division, School of Science and Technology, +Universit`a di Camerino Via Madonna delle Carceri 9, I-62032 Camerino (MC), Italy and +8INFN Sez. +di Perugia, Via A. Pascoli, 06123 Perugia, Italy +(Dated: January 5, 2023) +Absolute angular rotation rate measurements with sensitivity better than prad/sec would be +beneficial for fundamental science investigations. On this regard, large frame Earth based ring laser +gyroscopes are top instrumentation as far as bandwidth, long term operation and sensitivity are +concerned. +Their classical sensitivity limit is given by the shot–noise of the two beams counter +propagating inside the cavity usually considered as two independent propagating modes. Thus, it is +given by the sum of the shot–noise associated to each beam. Here we prove that the GINGERINO +active ring laser prototype upper limiting noise allows an unprecedented sensitivity close to 10−15 +rad/sec. This is more than a factor 10 better than the theoretical prediction so far accounted for +ring lasers shot–noise. +Introduction +– +Light +based +interferometers +have +reached an extremely high level of sensitivity, reliability, +and robustness. In most common interferometers, two +separate beams, possibly coming from the same source, +are injected in two separate paths and recombined to in- +terfere so that differences in path-lengths even smaller +than 10−14 times the wavelength can be resolved [1]. +While such measurement scheme is possible thanks to +the wave-nature of light, that shows-up as the interfer- +ence of coherent beams, the corpuscular nature of light +sets the intrinsic limit to the sensitivity attainable by in- +terference. This limit is known as (photon) shot–noise +and it is frequency independent. It intrinsically comes +from the stochastic fluctuations in the photon number +that, for coherent beams, are Poissonian distributed and +so are the obtained photo-electrons [2]. +Interferometer topology can be quite different. For ex- +ample, it is possible to have paths defined by four mirrors +located at the vertices of a square, thus defining a ring +cavity where the two light beams circulates in clockwise +and counter– clockwise directions. In this case, the two +paths are equals, frequency jitters are negligible, and the +interference of the two counter propagating beams carries +information on the non reciprocal effects connected to the +direction of circulation. If the frame supporting the four +mirrors rotates, the two counter propagating beams com- +plete the path at different times. In such a configuration, +the interference measures the time derivative of the dif- +ference in phase acquired by the two beams, rather than +the path spatial difference. This feature is the well known +Sagnac effect, named after the French physicist George +Sagnac [3, 4]. +Sagnac interferometers, in particular the active ver- +sions also known as Ring Laser Gyroscopes (RLGs), +are commonly used to measure inertial angular rotation. +When connected to the Earth crust, they can be used to +measure continuously the absolute angular rotation rate +of the Earth. Thanks to their large bandwidth and high +dynamic range, they can detect strong earthquakes and +seismological signals in the frequency window ∼ 0.01÷30 +Hz, as well as tiny geodetic signals in the very low fre- +quency domain (< 10−3 Hz), showing an adequate sen- +sitivity to probe General Relativity (GR) effects such as +the Lense-Thirring and de Sitter [5]. +Moreover, other non reciprocal effects related to propa- +gation of the two light beams and connected to the space +time structure or symmetries, can be investigated by +RLGs, leading to results relevant in fundamental physics +[6–8] when sensitivity of 5 · 10−14rad/s or better are +reached, corresponding to 1 part in 109 of the Earth ro- +tation rate for Earth based apparata. At the same time, +Sagnac interferometers are good candidates for investi- +gating the interplay between GR and quantum systems +and effects [9–13]. +As any interferometer, sensitivity of Sagnac ones is lim- +ited by the photon shot–noise. +Since the first model, +elaborated in 1982 by Cresser et al. [14], following the +concepts described in Ref. [15], it has been widely ac- +cepted that in Sagnac interferometers the two counter- +propagating beams are independent. +The correspond- +ing shot–noise can be evaluated accordingly (see, e.g., +[16, 17]): for example, in GINGERINO [18], a prototype +arXiv:2301.01386v1 [quant-ph] 3 Jan 2023 + +2 +of the RLG array GINGER located inside the Gran Sasso +National Laboratory of INFN, Italy, the model evaluates +a shot–noise of about 18 prad/sec Hz−1/2, taking into +account that its square optical cavity has side length of +3.6 m, total losses are 120 ppm, and the output power is +10 nW. However, in RLGs the two beams are generated +inside the rotating cavity, where the same volume of ac- +tive medium emits toward the two opposite directions. +Therefore, the laser equations for the two counterpropa- +gating beam amplitudes are coupled to each other [19]. +While classical amplitude equations are effective for cal- +culating the time dependence of mean values, inherent +fluctuations requires a quantum description of the field +modes, i.e. classical amplitudes have to be replaced by +quantum field operators. Once the equations are trans- +ferred into a quantum frame, coupling of the two different +modes implies the setting of some mutual correlation that +may affect the noise features of the device and possibly +its fundamental shot–noise limit. +Beating the quantum limit in gyroscopes has attracted +interest in recent years, owing to the appealing possibil- +ities of further improving their sensitivity. For passive +gyroscopes [20], in analogy with what has been proposed +[21] and then realised in Michelson interferometers (see +[22] and references therein), the use of externally injected +quantum states has been considered in different config- +urations [23–26] and experimentally realised [27–29] for +going beyond the standard quantum limit. Other authors +have considered the coupling of the ring modes to two– +level atoms for realising effective mode coupling and so +generating quantum correlation that may induce quan- +tum enhancement [30]. Very recently, an experimental +work reported a sensitivity below the standard shot-noise +for phase estimation in a gyroscope equipped with a liq- +uid crystal light valve (LCLV) for direct frequency mea- +surement [31]. +Recently, we have found that the ultimate sensitivity +of the GINGERINO prototype is not consistent with the +shot–noise calculated by the above mentioned indepen- +dent beams model [32, 33]. In that case the final sensitiv- +ity has been evaluated by subtracting from the data all +the known signals by linear regression methods and cor- +recting for the laser dynamics [34]. In this Letter we re- +port further measurements giving a conclusive proof that +the noise limit of the instrument is well below the con- +ventionally predicted shot–noise. Here, the noise floor is +estimated by subtracting data obtained from two equiva- +lent beating optical signals at the two outputs of a single +beam–splitter. +By principle, so doing we trace-out all +the possible rotational signals providing an upper limit +for the unavoidable quantum noise source. +RLG senses the projection of the angular velocity vec- +tor ⃗Ω on the area of the closed polygonal cavity. The ori- +entation of this area in space is determined by the area +versor ⃗n. The relationship between the Sagnac pulse fre- +quency ωs and the angular rotation rate Ω reads +ωs = 8π A +λLΩ cos θ , +(1) +where A is the area of the cavity, L the perimeter, λ the +wavelength of the light, and θ the angle between ⃗n and +⃗Ω. +So far, large RLGs have been dedicated to very low fre- +quency measurements [35, 37] below 30 Hz, where phys- +ical and geophysical investigations are relevant. In this +range, apart from those of scientific interest, there are +signals of different nature such as, human activity, mi- +croseismicity of the crust generated by the ocean, tides +and polar motion, temperature and pressure variations, +that may reduce the instrument sensitivity. Despite that, +available measurements show sensitivity ranging from the +nrad/s to tens of prad/s [17, 32, 36]. It is convenient to +express the sensitivity as angular rotation rate, and in +order to avoid confusion, the angular frequency will be +indicated as small cap ω and the corresponding angular +velocity as capital Ω, the two quantities are connected by +the geometrical scale factor of Eq. (1). +Data analysis and GINGERINO – GINGERINO has +shown evidence of a limiting noise smaller than expected +[33]. In order to gain useful insights into such unexpected +result, we have improved the setup with the aim to obtain +a direct estimation of the stochastic noise itself, hereafter +denoted ωT n. In principle, ΩT n ≥ Ωsn, where Ωsn indi- +cates the shot–noise, being ΩT n the sum of various noise +contributions. Contrarily to Ωsn, ΩT n is not a flat noise +and, for GINGERINO, it shows the limit of 2−3 prad/s in +1 s measurement time. The corresponding Modified Al- +lan Deviation reaches the value of 2.1±0.01 frad/s in 2.5 +days of integration time, that corresponds to 4.3 · 10−11 +the Earth rotation rate. +The two counter–propagating beams leaving the cav- +ity of our RLG prototypes are combined at a beam– +splitter placed at one of the cavity corners. +The two +resulting mixed beams, observed by two identical photo- +diodes, contain the measured beat note ωm. Since in this +general treatment an ideal behavior is assumed, neglect- +ing any laser systematics, we will consider ωm = ωs the +signal of interest. Without loss of generality, it is pos- +sible to state that photodiode signals can be expressed +as Si = Ag · (−1)i · (cos (ωs + ωn) · t) + φn) + Vni, with +i = 1, 2, where Ag is a gain factor, ωn indicates the +stochastic noise affecting the frequency itself, φn is the +stochastic term of the phase, and Vni is any noise added +outside the cavity [38]. The reconstructed frequency sig- +nal from each photodiode is defined by ωi = ωs + ωT ni, +where ωT ni takes into account all noise terms at once, +since it is not possible to discriminate among different +noise sources. Therefore, ωT ni has to be considered an +upper limit to ωn. In this configuration, the two mea- +surements are independent one another and each of them +contains the frequency signal ωs plus the sum of differ- + +3 +ent noise contributions: the noise of the two laser beams +in the cavity, mainly of stochastic nature, and the noise +picked up outside the cavity, containing disturbances in- +duced by the environment and stochastic terms. +In order to have a better estimate of the limiting +noise and of the signal, we consider the signal difference +(S = S1 − S2) and define ωd = ωs + ωT nd the corre- +sponding frequency signal. It is straightforward to note +that, considering the stochastic noise, ωd has a signal to +noise ratio +√ +2 larger than the single photodiode mea- +surements, because the Sagnac signal is doubled while +the stochastic noise is increased by a +√ +2 factor. More- +over, in ωT nd disturbances produced outside the cavity, +and common to both photodiodes, are cancelled out. +Let us consider ωn12, defined as the difference ωn12 = +ω1 − ω2, that contains the quadratic sum of all stochas- +tic terms of the two interference signals and the differ- +ence between the disturbances of environmental origin +recorded by the two detectors, similarly to ωT nd. We can +consider ωn12 as an upper limit to the stochastic noise +generated inside the cavity and simple manipulations +lead to ωn12 ∼ 2 · ωT nd. The factor 2 has been checked +with simulated data. In summary, ωd provides the best +angular rotation rate estimation, while ωT nd = ωn12/2 +measures its sensitivity noise limit. +At this point, it is necessary to take into account the +data analysis procedure. The procedure adopted for fre- +quency estimation is based on the Hilbert transform. We +first recover the phase from the analytic signal and then +evaluate the frequency ω by differentiation. In general, +interferograms, and monobeams intensities, are acquired +at 5 kSa/s. The subsequent analysis is performed with +no down sampling. When the analysis is focused below 1 +Hz, a digital band pass filter centered on the mean beat- +ing frequency and with a ±12 Hz width is applied before +the Hilbert transform [39]. The band–pass filter is not +used for high frequency investigation. +It is worth noticing that performances of the frequency +estimation procedure must be evaluated by simulation as +it is based on a non linear transformation of data. Three +main noise sources are identified: the white frequency +noise ωn, the white phase noise φn and a phase diffusion +noise φW modeled as a Wiener process. +Figure 1 shows the response of the reconstruction pro- +cedure to the injection of these three types of noise. In +particular, we report the Amplitude Spectral Distribu- +tion (ASD) of the injected noise ωn (green) and of the cor- +responding reconstructed signal (purple), as well as the +ASD of the reconstructed signal injecting φn, φn = ωn · ¯t +with ¯t = 0.02 s integration time (red), and φW (yellow). +The contribution of the white stochastic frequency +noise ωn is reconstructed by the analysis process as a fre- +quency white noise a factor 20 higher in the low frequency +range (10−2 ÷ 20 Hz), which grows linearly at higher +frequencies. At frequencies above 20 Hz, its behaviour +becomes indistinguishable from that reconstructed when +FIG. 1. +ASD of the injected noise Ωn (green) and of the +corresponding reconstructed signal (purple); ASD of the re- +constructed frequency obtained by injecting φn, with ¯t = 0.02 +s integration time (red), and ϕW (yellow). The injected noise +level is 20 prad/s Hz−1/2. +the white phase noise φn is injected, that produces a +power spectrum proportional to frequency over the full +frequency span. On the other hand, the phase diffusion +noise, simulated as a Wiener process, produces a con- +stant ASD, a factor of 2 higher than the level of the +injected noise. It’s worth noticing that all ASD of the +reconstructed signals show a discontinuity at the Sagnac +frequency. +Experimental spectra of RLG prototypes - +We have +analysed experimental data produced by four distinct +large frame RLG prototypes [40]. +In Fig. 2 we report +the ASD for G–Wettzell [17], GINGERINO, and GP2 +[41] while the ASD of ROMY [35], that shows very sim- +ilar behaviour, is not reported. Typically, the frequency +range below 0.1 Hz is affected by laser systematics and +contains signals of geophysical origin, for this reason it +is not suitable for any noise investigation. The minimum +of the ASD is in the frequency window 0.1 ÷ 1 Hz, where +microseismicity originated by the oceans is present. The +region above 5 Hz contains regular signals but also a char- +acteristic tail linearly growing with frequency. +Despite big differences, due to the different structure +and location, all three ASD show the characteristic high +frequency behavior linearly growing with frequency. This +feature, being compatible with a flat phase noise, indi- +cates that, at least in this range of frequencies, there is +a stochastic noise floor dominated by a frequency inde- +pendent phase noise. +Because of its noisy location, GP2 data show larger +noise (approximately a factor of ten above the other pro- +totypes). Moreover, different disturbances are affecting +the cavity at low frequency, as the microseismicity from +oceans, well visible in G. However, in the region around 1 + +106 +104 +102 +100 +10-2 +10°4 +10~3 +10-2 +10~1 +100 +101 +102 +103 +Frequency [Hz]4 +FIG. 2. +ASD of the data, expressed as angular rotation +rate, of G Wettzell, GINGERINO, and GP2. The high fre- +quency part of the spectrum shows a very characteristic tail +constantly rising with frequency. G, owing to its monolithic +structure, is very quiet and 1 hour has been used for the ASD, +while for GINGERINO 15 minutes of time have been selected. +GP2 is 1.6 m in side, and it is located in a rather noisy envi- +ronment, that explains the occurrence of a larger noise. Data +from G are acquired at 2 kSa/s, the cut-off occurring around +0.5 kHz is due to the analysis procedure. +Hz, where the curves exhibit their minimum, amplitude is +smaller than the one expected assuming Ωn = 18 prad/s +in 1 s of measurement time, which would correspond to +0.4 nrad/s after the reconstruction. +ΩT n was evaluated using 20 days of GINGERINO +data starting from October 29, 2022. +The whole set +of data have been acquired at 5 kSa/s. +The high fre- +quency part of the spectrum has been investigated us- +ing small portions of data, corresponding typically to 3 +hours, and avoiding any filtering around the beat note +(280.4 Hz). The low frequency part, from DC up to 5 +Hz, has been analysed following the standard procedure +of GINGERINO which evaluates the different terms: ωm, +ωs0, ωξ, and ωns, the latter relating to null shift [42]. +In the following we report the results using ωs0, which +is the best approximation of the Sagnac frequency, since +it takes into account the back-scatter noise, avoiding the +use of linear regression usually employed to subtract ef- +fects of electronic origin (ωξ), and null shift due to the +laser dynamics (ωns). However, it has been checked that +very similar results are obtained using the beat notes it- +self, ωm, or estimating the true Sagnac signal ωs by linear +regression [33]. +The ASD of reconstructed signals Ωd and Ωn12 are +shown in Fig. 3: noise floors at high frequency are in +good agreement with each other. +Fig. 4 compares Ωd and ΩT n, with the latter exhibiting +above 0.1 Hz the characteristic phase noise behavior, and +being almost flat at lower frequency, with a level around +FIG. 3. ASD of Ωn12/2 (red) compared with Ωd (blue) spec- +trum at high frequency. The noise floor agreement is good. +Some peaks due to electronics or environmental origin have +been removed. +FIG. 4. ASD of ΩT n (red) compared with Ωd (blue) spectrum +at low frequency. In this spectrum two hours of data around +the big Mw 5.9 event have been removed (see [42]); when +included, the low frequency bump increases. +2 prad/s Hz−1/2, a factor 10 below the expected shot– +noise, and 200 times below the one obtained by taking +into account the analysis procedure. +Fig. 5 reports the corresponding Overlapped and Mod- +ified Allan Deviations (obtained by STABLE32), demon- +strating levels of 4 and 2.63 frad/s in approximately 2.4 +days of integration time, respectively, corresponding to +1.23 and 1.87 in 1010 the Earth rotation rate, a level suf- +ficient for detecting fundamental physics effects with an +array of RLGs [7, 8]. +Summary - It is proved that, below 0.1 Hz, the large +RLG prototype GINGERINO shows a limiting noise floor + +10-6 +GINGERINO +G +GP2 +10-7 +10~10 +10-1 +100 +101 +102 +103 +Frequency[Hz]10-8 +10-9 +10-10 +102 +103 +Frequency [Hz]10-6 +10-7 +10-8 +10-9 +10 +10 +10~12 +10~13 +10-8 +10-6 +10°4 +10-2 +100 +Frequency [Hz]5 +FIG. 5. +Overlapped and Modified Allan Deviation of ΩT n +expressed in rad/s. +The plot have been obtained by using +STABLE32 freely available at: http://www.stable32.com/ +in the prad/s Hz−1/2 range, well below what expected +for the shot–noise in this type of apparatus taking for +granted the independent beam model [14]. This experi- +mental noise limit has been obtained by subtracting two +independent rotation signals. These signals come from +the two outputs of a single beam–splitter placed at one +of the cavity corners to let the counter propagating beams +interfere. So doing, the estimated noise level represents +an upper limit to the inherent quantum noise affecting +the apparatus. While this experimental finding suggests +that a complete quantum model of the system should +take into account the complex interdependent dynamics +of the counter–propagating beams, it gives a conclusive +proof of the feasibility of fundamental physics measure- +ments once an array of RLGs is available. In a forthcom- +ing study we will develop a model that, tracing back from +the detector scheme, accounts for all the possible inter- +actions between the counter–propagating beams and the +laser medium. 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Lett. 129, +113901 (2022). + +Date: 11/21/22 +Time: 16:37:53 +Data Points 1 thru 8699594 of 869959. +au=2.0000000e-01 +File:beat notes subtracted +FREQUENCY STABILITY +beat notes subtracted +11-0 +Tau +Sigma +Mod + sigma +2.00e-01 +9.30e-12 +9.10e-12 +4.00e-01 +5.58e- +3.79e- +12 +8.00e-01 +2.76e- +1.30e-12 +.60e+00 +1.26e +12 +4.80e +.20e+00 +6.80e- +13 +6.40e+00 +.82e- +1.3 +1.79e- +13 +28e+01 +.59e +1.3 +1.61e- +56e+01 +13 +1.30e- +13 +12e+01 +8e +1.3 +14 +1.02e+02 +5.65e- +5.61e14 +.73e-14 +A +10e+02 +8.19e+02 +.83e +17e- +14 +.64e+03 +2.43e- +14 +.28e+03 +14 +2.40e- +14 +.55e+03 +2.57e +14 +1.96e +14 +e+04 +1.90e +14 +1.43e +14 +.62e+04 +1.48e- +14 +1.05e: +14 +.24e+04 +8.76e +5.73e +15 +4.23e +05e+05 +t05 +.00e +15 +sr-01 +10-1 +100 +101 +102 +103 +104 +105 +106 +Averaging + Time, +T, +Seconds6 +[32] A.D.V. Di Virgilio, et al., Phys. Rev. Res. 2, 032069(R) +(2020). +[33] A.D.V. Di Virgilio, et al., Eur. Phys. J. C 81, 400 (2021). +[34] A.D.V. Di Virgilio, N. Beverini, G. 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C 82, 824 (2022). +[38] The last term includes noise equivalent power of the pho- +todiode, or amplitude fluctuation of the light at the de- +tector, etc. +[39] Different filters can be adopted: in the present analysis +we used a filter based on FFT. +[40] The considered prototypes, all but ROMY employing +square optical cavities, are: (i) G of the geodetic observa- +tory of Wettzell, Germany, located on the Earth surface +inside a pressure tight bunker and featuring a monolithic +structure in ZERODUR with 16 m perimeter and mir- +rors optically contacted to the central rigid structure; (ii) +GINGERINO, located inside the very quiet environment +of the Gran Sasso underground laboratory of INFN, Italy, +consisting of a heterolytic (HL) structure done attach- +ing together different rigid pieces in granite and mirrors +contained inside stainless steel boxes, forming a 14.4 m +perimeter cavity; (iii) ROMY, an array RLGs located in +the geophysical observatory close to Munich, composed +of 4 HL triangular RLGs with 36 m perimeter; (iv) GP2, +with a 6.4 m perimeter and HL structure, located inside +the basement of the INFN Pisa Section, Italy. +[41] E. Maccioni, N. Beverini, G. Carelli, G. Di Somma, A. Di +Virgilio, and P. Marsili, Appl. Opt. 61, 9256-9261 (2022). +[42] GINGERINO is free running, the temperature is stable +at the level of fractions of a degree, accordingly the cav- +ity perimeter changes causing a large number of mode +jumps. In the considered period, for two times the inter- +ferometer has operated in split mode for several hours, +approximately 50 minutes are missing, since it has been +necessary to restart the data acquisition system. In this +time window a large earthquake swarm about 200 − 300 +km apart is contained, for this reason 2 hours of data +around the big shock with Mw 5.6 of November 9, 2022, +have been removed. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Italy 5INFN - LNL ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Viale dell’Universit‘a 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 35020 Legnaro (PD),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Italy 6Department of Civil and Mechanical Engineering – DICEM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Universit´a di Cassino e Lazio Meridionale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 03043,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Cassino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Italy 7Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Universit`a di Camerino Via Madonna delle Carceri 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' I-62032 Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Italy and 8INFN Sez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' di Perugia, Via A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Pascoli, 06123 Perugia, Italy (Dated: January 5, 2023) Absolute angular rotation rate measurements with sensitivity better than prad/sec would be beneficial for fundamental science investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' On this regard, large frame Earth based ring laser gyroscopes are top instrumentation as far as bandwidth, long term operation and sensitivity are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Their classical sensitivity limit is given by the shot–noise of the two beams counter propagating inside the cavity usually considered as two independent propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Thus, it is given by the sum of the shot–noise associated to each beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Here we prove that the GINGERINO active ring laser prototype upper limiting noise allows an unprecedented sensitivity close to 10−15 rad/sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' This is more than a factor 10 better than the theoretical prediction so far accounted for ring lasers shot–noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Introduction – Light based interferometers have reached an extremely high level of sensitivity, reliability, and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In most common interferometers, two separate beams, possibly coming from the same source, are injected in two separate paths and recombined to in- terfere so that differences in path-lengths even smaller than 10−14 times the wavelength can be resolved [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' While such measurement scheme is possible thanks to the wave-nature of light, that shows-up as the interfer- ence of coherent beams, the corpuscular nature of light sets the intrinsic limit to the sensitivity attainable by in- terference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' This limit is known as (photon) shot–noise and it is frequency independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' It intrinsically comes from the stochastic fluctuations in the photon number that, for coherent beams, are Poissonian distributed and so are the obtained photo-electrons [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Interferometer topology can be quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' For ex- ample, it is possible to have paths defined by four mirrors located at the vertices of a square, thus defining a ring cavity where the two light beams circulates in clockwise and counter– clockwise directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this case, the two paths are equals, frequency jitters are negligible, and the interference of the two counter propagating beams carries information on the non reciprocal effects connected to the direction of circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' If the frame supporting the four mirrors rotates, the two counter propagating beams com- plete the path at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In such a configuration, the interference measures the time derivative of the dif- ference in phase acquired by the two beams, rather than the path spatial difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' This feature is the well known Sagnac effect, named after the French physicist George Sagnac [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Sagnac interferometers, in particular the active ver- sions also known as Ring Laser Gyroscopes (RLGs), are commonly used to measure inertial angular rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' When connected to the Earth crust, they can be used to measure continuously the absolute angular rotation rate of the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Thanks to their large bandwidth and high dynamic range, they can detect strong earthquakes and seismological signals in the frequency window ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='01÷30 Hz, as well as tiny geodetic signals in the very low fre- quency domain (< 10−3 Hz), showing an adequate sen- sitivity to probe General Relativity (GR) effects such as the Lense-Thirring and de Sitter [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Moreover, other non reciprocal effects related to propa- gation of the two light beams and connected to the space time structure or symmetries, can be investigated by RLGs, leading to results relevant in fundamental physics [6–8] when sensitivity of 5 · 10−14rad/s or better are reached, corresponding to 1 part in 109 of the Earth ro- tation rate for Earth based apparata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' At the same time, Sagnac interferometers are good candidates for investi- gating the interplay between GR and quantum systems and effects [9–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' As any interferometer, sensitivity of Sagnac ones is lim- ited by the photon shot–noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Since the first model, elaborated in 1982 by Cresser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [14], following the concepts described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [15], it has been widely ac- cepted that in Sagnac interferometers the two counter- propagating beams are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The correspond- ing shot–noise can be evaluated accordingly (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=', [16, 17]): for example, in GINGERINO [18], a prototype arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='01386v1 [quant-ph] 3 Jan 2023 2 of the RLG array GINGER located inside the Gran Sasso National Laboratory of INFN, Italy, the model evaluates a shot–noise of about 18 prad/sec Hz−1/2, taking into account that its square optical cavity has side length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='6 m, total losses are 120 ppm, and the output power is 10 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' However, in RLGs the two beams are generated inside the rotating cavity, where the same volume of ac- tive medium emits toward the two opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Therefore, the laser equations for the two counterpropa- gating beam amplitudes are coupled to each other [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' While classical amplitude equations are effective for cal- culating the time dependence of mean values, inherent fluctuations requires a quantum description of the field modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' classical amplitudes have to be replaced by quantum field operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Once the equations are trans- ferred into a quantum frame, coupling of the two different modes implies the setting of some mutual correlation that may affect the noise features of the device and possibly its fundamental shot–noise limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Beating the quantum limit in gyroscopes has attracted interest in recent years, owing to the appealing possibil- ities of further improving their sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' For passive gyroscopes [20], in analogy with what has been proposed [21] and then realised in Michelson interferometers (see [22] and references therein), the use of externally injected quantum states has been considered in different config- urations [23–26] and experimentally realised [27–29] for going beyond the standard quantum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Other authors have considered the coupling of the ring modes to two– level atoms for realising effective mode coupling and so generating quantum correlation that may induce quan- tum enhancement [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Very recently, an experimental work reported a sensitivity below the standard shot-noise for phase estimation in a gyroscope equipped with a liq- uid crystal light valve (LCLV) for direct frequency mea- surement [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Recently, we have found that the ultimate sensitivity of the GINGERINO prototype is not consistent with the shot–noise calculated by the above mentioned indepen- dent beams model [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In that case the final sensitiv- ity has been evaluated by subtracting from the data all the known signals by linear regression methods and cor- recting for the laser dynamics [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this Letter we re- port further measurements giving a conclusive proof that the noise limit of the instrument is well below the con- ventionally predicted shot–noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Here, the noise floor is estimated by subtracting data obtained from two equiva- lent beating optical signals at the two outputs of a single beam–splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' By principle, so doing we trace-out all the possible rotational signals providing an upper limit for the unavoidable quantum noise source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' RLG senses the projection of the angular velocity vec- tor ⃗Ω on the area of the closed polygonal cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The ori- entation of this area in space is determined by the area versor ⃗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The relationship between the Sagnac pulse fre- quency ωs and the angular rotation rate Ω reads ωs = 8π A λLΩ cos θ , (1) where A is the area of the cavity, L the perimeter, λ the wavelength of the light, and θ the angle between ⃗n and ⃗Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' So far, large RLGs have been dedicated to very low fre- quency measurements [35, 37] below 30 Hz, where phys- ical and geophysical investigations are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this range, apart from those of scientific interest, there are signals of different nature such as, human activity, mi- croseismicity of the crust generated by the ocean, tides and polar motion, temperature and pressure variations, that may reduce the instrument sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Despite that, available measurements show sensitivity ranging from the nrad/s to tens of prad/s [17, 32, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' It is convenient to express the sensitivity as angular rotation rate, and in order to avoid confusion, the angular frequency will be indicated as small cap ω and the corresponding angular velocity as capital Ω, the two quantities are connected by the geometrical scale factor of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Data analysis and GINGERINO – GINGERINO has shown evidence of a limiting noise smaller than expected [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In order to gain useful insights into such unexpected result, we have improved the setup with the aim to obtain a direct estimation of the stochastic noise itself, hereafter denoted ωT n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In principle, ΩT n ≥ Ωsn, where Ωsn indi- cates the shot–noise, being ΩT n the sum of various noise contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Contrarily to Ωsn, ΩT n is not a flat noise and, for GINGERINO, it shows the limit of 2−3 prad/s in 1 s measurement time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The corresponding Modified Al- lan Deviation reaches the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='01 frad/s in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='5 days of integration time, that corresponds to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='3 · 10−11 the Earth rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The two counter–propagating beams leaving the cav- ity of our RLG prototypes are combined at a beam– splitter placed at one of the cavity corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The two resulting mixed beams, observed by two identical photo- diodes, contain the measured beat note ωm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Since in this general treatment an ideal behavior is assumed, neglect- ing any laser systematics, we will consider ωm = ωs the signal of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Without loss of generality, it is pos- sible to state that photodiode signals can be expressed as Si = Ag · (−1)i · (cos (ωs + ωn) · t) + φn) + Vni, with i = 1, 2, where Ag is a gain factor, ωn indicates the stochastic noise affecting the frequency itself, φn is the stochastic term of the phase, and Vni is any noise added outside the cavity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The reconstructed frequency sig- nal from each photodiode is defined by ωi = ωs + ωT ni, where ωT ni takes into account all noise terms at once, since it is not possible to discriminate among different noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Therefore, ωT ni has to be considered an upper limit to ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this configuration, the two mea- surements are independent one another and each of them contains the frequency signal ωs plus the sum of differ- 3 ent noise contributions: the noise of the two laser beams in the cavity, mainly of stochastic nature, and the noise picked up outside the cavity, containing disturbances in- duced by the environment and stochastic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In order to have a better estimate of the limiting noise and of the signal, we consider the signal difference (S = S1 − S2) and define ωd = ωs + ωT nd the corre- sponding frequency signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' It is straightforward to note that, considering the stochastic noise, ωd has a signal to noise ratio √ 2 larger than the single photodiode mea- surements, because the Sagnac signal is doubled while the stochastic noise is increased by a √ 2 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' More- over, in ωT nd disturbances produced outside the cavity, and common to both photodiodes, are cancelled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Let us consider ωn12, defined as the difference ωn12 = ω1 − ω2, that contains the quadratic sum of all stochas- tic terms of the two interference signals and the differ- ence between the disturbances of environmental origin recorded by the two detectors, similarly to ωT nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' We can consider ωn12 as an upper limit to the stochastic noise generated inside the cavity and simple manipulations lead to ωn12 ∼ 2 · ωT nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The factor 2 has been checked with simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In summary, ωd provides the best angular rotation rate estimation, while ωT nd = ωn12/2 measures its sensitivity noise limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' At this point, it is necessary to take into account the data analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The procedure adopted for fre- quency estimation is based on the Hilbert transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' We first recover the phase from the analytic signal and then evaluate the frequency ω by differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In general, interferograms, and monobeams intensities, are acquired at 5 kSa/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The subsequent analysis is performed with no down sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' When the analysis is focused below 1 Hz, a digital band pass filter centered on the mean beat- ing frequency and with a ±12 Hz width is applied before the Hilbert transform [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The band–pass filter is not used for high frequency investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' It is worth noticing that performances of the frequency estimation procedure must be evaluated by simulation as it is based on a non linear transformation of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Three main noise sources are identified: the white frequency noise ωn, the white phase noise φn and a phase diffusion noise φW modeled as a Wiener process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Figure 1 shows the response of the reconstruction pro- cedure to the injection of these three types of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In particular, we report the Amplitude Spectral Distribu- tion (ASD) of the injected noise ωn (green) and of the cor- responding reconstructed signal (purple), as well as the ASD of the reconstructed signal injecting φn, φn = ωn · ¯t with ¯t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='02 s integration time (red), and φW (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The contribution of the white stochastic frequency noise ωn is reconstructed by the analysis process as a fre- quency white noise a factor 20 higher in the low frequency range (10−2 ÷ 20 Hz), which grows linearly at higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' At frequencies above 20 Hz, its behaviour becomes indistinguishable from that reconstructed when FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ASD of the injected noise Ωn (green) and of the corresponding reconstructed signal (purple);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ASD of the re- constructed frequency obtained by injecting φn, with ¯t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='02 s integration time (red), and ϕW (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The injected noise level is 20 prad/s Hz−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' the white phase noise φn is injected, that produces a power spectrum proportional to frequency over the full frequency span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' On the other hand, the phase diffusion noise, simulated as a Wiener process, produces a con- stant ASD, a factor of 2 higher than the level of the injected noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' It’s worth noticing that all ASD of the reconstructed signals show a discontinuity at the Sagnac frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Experimental spectra of RLG prototypes - We have analysed experimental data produced by four distinct large frame RLG prototypes [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 2 we report the ASD for G–Wettzell [17], GINGERINO, and GP2 [41] while the ASD of ROMY [35], that shows very sim- ilar behaviour, is not reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Typically, the frequency range below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='1 Hz is affected by laser systematics and contains signals of geophysical origin, for this reason it is not suitable for any noise investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The minimum of the ASD is in the frequency window 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='1 ÷ 1 Hz, where microseismicity originated by the oceans is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The region above 5 Hz contains regular signals but also a char- acteristic tail linearly growing with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Despite big differences, due to the different structure and location, all three ASD show the characteristic high frequency behavior linearly growing with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' This feature, being compatible with a flat phase noise, indi- cates that, at least in this range of frequencies, there is a stochastic noise floor dominated by a frequency inde- pendent phase noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Because of its noisy location, GP2 data show larger noise (approximately a factor of ten above the other pro- totypes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Moreover, different disturbances are affecting the cavity at low frequency, as the microseismicity from oceans, well visible in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' However, in the region around 1 106 104 102 100 10-2 10°4 10~3 10-2 10~1 100 101 102 103 Frequency [Hz]4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ASD of the data, expressed as angular rotation rate, of G Wettzell, GINGERINO, and GP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The high fre- quency part of the spectrum shows a very characteristic tail constantly rising with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' G, owing to its monolithic structure, is very quiet and 1 hour has been used for the ASD, while for GINGERINO 15 minutes of time have been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' GP2 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='6 m in side, and it is located in a rather noisy envi- ronment, that explains the occurrence of a larger noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Data from G are acquired at 2 kSa/s, the cut-off occurring around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='5 kHz is due to the analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Hz, where the curves exhibit their minimum, amplitude is smaller than the one expected assuming Ωn = 18 prad/s in 1 s of measurement time, which would correspond to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='4 nrad/s after the reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ΩT n was evaluated using 20 days of GINGERINO data starting from October 29, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The whole set of data have been acquired at 5 kSa/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The high fre- quency part of the spectrum has been investigated us- ing small portions of data, corresponding typically to 3 hours, and avoiding any filtering around the beat note (280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='4 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The low frequency part, from DC up to 5 Hz, has been analysed following the standard procedure of GINGERINO which evaluates the different terms: ωm, ωs0, ωξ, and ωns, the latter relating to null shift [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In the following we report the results using ωs0, which is the best approximation of the Sagnac frequency, since it takes into account the back-scatter noise, avoiding the use of linear regression usually employed to subtract ef- fects of electronic origin (ωξ), and null shift due to the laser dynamics (ωns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' However, it has been checked that very similar results are obtained using the beat notes it- self, ωm, or estimating the true Sagnac signal ωs by linear regression [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The ASD of reconstructed signals Ωd and Ωn12 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 3: noise floors at high frequency are in good agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 4 compares Ωd and ΩT n, with the latter exhibiting above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='1 Hz the characteristic phase noise behavior, and being almost flat at lower frequency, with a level around FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ASD of Ωn12/2 (red) compared with Ωd (blue) spec- trum at high frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The noise floor agreement is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Some peaks due to electronics or environmental origin have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ASD of ΩT n (red) compared with Ωd (blue) spectrum at low frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this spectrum two hours of data around the big Mw 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='9 event have been removed (see [42]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' when included, the low frequency bump increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 2 prad/s Hz−1/2, a factor 10 below the expected shot– noise, and 200 times below the one obtained by taking into account the analysis procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 5 reports the corresponding Overlapped and Mod- ified Allan Deviations (obtained by STABLE32), demon- strating levels of 4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='63 frad/s in approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='4 days of integration time, respectively, corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='23 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='87 in 1010 the Earth rotation rate, a level suf- ficient for detecting fundamental physics effects with an array of RLGs [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Summary - It is proved that, below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='1 Hz, the large RLG prototype GINGERINO shows a limiting noise floor 10-6 GINGERINO G GP2 10-7 10~10 10-1 100 101 102 103 Frequency[Hz]10-8 10-9 10-10 102 103 Frequency [Hz]10-6 10-7 10-8 10-9 10 10 10~12 10~13 10-8 10-6 10°4 10-2 100 Frequency [Hz]5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Overlapped and Modified Allan Deviation of ΩT n expressed in rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' The plot have been obtained by using STABLE32 freely available at: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='stable32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='com/ in the prad/s Hz−1/2 range, well below what expected for the shot–noise in this type of apparatus taking for granted the independent beam model [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' This experi- mental noise limit has been obtained by subtracting two independent rotation signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' These signals come from the two outputs of a single beam–splitter placed at one of the cavity corners to let the counter propagating beams interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' So doing, the estimated noise level represents an upper limit to the inherent quantum noise affecting the apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' While this experimental finding suggests that a complete quantum model of the system should take into account the complex interdependent dynamics of the counter–propagating beams, it gives a conclusive proof of the feasibility of fundamental physics measure- ments once an array of RLGs is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In a forthcom- ing study we will develop a model that, tracing back from the detector scheme, accounts for all the possible inter- actions between the counter–propagating beams and the laser medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' A full quantum picture is required so to have a theoretical shot–noise estimation to be compared with the presented experimental finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ACKNOWLEDGEMENT The authors thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Schreiber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Kodet, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Igel and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Brotzer for providing the data of G Wettzell and ROMY to investigate the high frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' ∗ Corresponding author alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='porzio@na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='it [1] Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=', Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Gagatsos, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Zhuang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Guha, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Applied 14, 034065 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Vahlbruch, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Schnabel, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 35, 1665 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Cai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Ma, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Sun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Gao, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 113, 261103 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Fink, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Steinlechner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Handsteiner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Dowling, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' A 105, 023716 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Howell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Kahn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Grynszpan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Cohen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Residori, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Bortolozzo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='30e-12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='10e-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='00e-01 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='58e- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='79e- 12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='00e-01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='76e- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='30e-12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='60e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='26e 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='80e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='20e+00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='80e- 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='40e+00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='82e- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='79e- 13 28e+01 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='59e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='61e- 56e+01 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='30e- 13 12e+01 8e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='3 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='02e+02 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='65e- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='61e14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='73e-14 A 10e+02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='19e+02 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='83e 17e- 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='64e+03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='43e- 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='28e+03 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='40e- 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='55e+03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='57e 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='96e 14 e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='90e 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='43e 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='62e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='48e- 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='05e: 14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='24e+04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='76e 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='73e 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='23e 05e+05 t05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='00e 15 sr-01 10-1 100 101 102 103 104 105 106 Averaging Time, T, Seconds6 [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Di Virgilio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 2, 032069(R) (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Di Virgilio, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=', Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' C 81, 400 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Di Virgilio, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Beverini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Carelli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Ciampini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Fuso, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' C 82, 824 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [38] The last term includes noise equivalent power of the pho- todiode, or amplitude fluctuation of the light at the de- tector, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [39] Different filters can be adopted: in the present analysis we used a filter based on FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [40] The considered prototypes, all but ROMY employing square optical cavities, are: (i) G of the geodetic observa- tory of Wettzell, Germany, located on the Earth surface inside a pressure tight bunker and featuring a monolithic structure in ZERODUR with 16 m perimeter and mir- rors optically contacted to the central rigid structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' (ii) GINGERINO, located inside the very quiet environment of the Gran Sasso underground laboratory of INFN, Italy, consisting of a heterolytic (HL) structure done attach- ing together different rigid pieces in granite and mirrors contained inside stainless steel boxes, forming a 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='4 m perimeter cavity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' (iii) ROMY, an array RLGs located in the geophysical observatory close to Munich, composed of 4 HL triangular RLGs with 36 m perimeter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' (iv) GP2, with a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='4 m perimeter and HL structure, located inside the basement of the INFN Pisa Section, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [41] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Maccioni, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Beverini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Carelli, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Di Somma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Di Virgilio, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Marsili, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' 61, 9256-9261 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' [42] GINGERINO is free running, the temperature is stable at the level of fractions of a degree, accordingly the cav- ity perimeter changes causing a large number of mode jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In the considered period, for two times the inter- ferometer has operated in split mode for several hours, approximately 50 minutes are missing, since it has been necessary to restart the data acquisition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In this time window a large earthquake swarm about 200 − 300 km apart is contained, for this reason 2 hours of data around the big shock with Mw 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content='6 of November 9, 2022, have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} +page_content=' In summary the full data set is con- tinuous with 4% of missing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdAzT4oBgHgl3EQfbfxY/content/2301.01386v1.pdf'} diff --git a/kb_51/content/tmp_files/kb_51.pdf.txt b/kb_51/content/tmp_files/kb_51.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e0e7da5d9348d2238b5c5bd7b56685a9fbc8904d --- /dev/null +++ b/kb_51/content/tmp_files/kb_51.pdf.txt @@ -0,0 +1,1051 @@ +Semantic Network Model for Sign +Language Comprehension + +Xinchen Kang (22f3d431-dcc0-42a5-8e2b-c26464e0654d) +Beijing Union University, China +Dengfeng Yao (ae88317c-d091-4f41-97f1-b1e6be00ca68) +Beijing Union University, China +Minghu Jiang (ea1cc43b-eee9-4185-8d97-edeac9186268) +Tsinghua University, China +Yunlong Huang (cc1ddbf2-64b9-4c51-b553-0ebe52a8d645) +Tsinghua University, China +Fanshu Li (64642396-6e95-456f-a4f2-47ff81a23d6e) +Beijing Union University, China +ABSTRACT +In this study, the authors propose a computational cognitive model for sign language (SL) perception and +comprehension with detailed algorithmic descriptions based on cognitive functionalities in human +language processing. The semantic network model (SNM) that represents semantic relations between +concepts is used as a form of knowledge representation. The proposed model is applied in the +comprehension of sign language for classifier predicates. The spreading activation search method is +initiated by labeling a set of source nodes (e.g. concepts in the semantic network) with weights or +"activation," and then iteratively propagating or "spreading" that activation out to other nodes linked to +the source nodes. The results demonstrate that the proposed search method improves the performance of +sign language comprehension in the SNM. +Keywords: Attention, Cognitive Processing, Comprehension, Decision-Tree, Game Theory, Linguistics, +Perception, Semantic Network, Sign Language +INTRODUCTION +Sign language (SL) comprehension is a fundamental task for computational linguists. Two types of +algorithms have been proposed: (1) rule-based methods (Supalla, 1982), and (2) statistical methods +(Bauer & Heinz, 2000; Huenerfauth, 2005). Rule-based methods lack the capability of planning the +elements in the entire scene (Liddell, 2003). The method of modeling infinite natural language input +through finite rules, especially minor rules, barely meets all requirements of SL processing (Yao et al., +2017). Therefore, statistical methods are the preferred type of algorithm for SL comprehension. Statistical +models can be applied to spoken languages. Given the abundant data resources of spoken languages in the +digitalized Internet age, statistical models can be applied readily. However, the raw and annotated corpora +of SLs are insufficient because collecting and annotating SL videos are tedious and difficult. Data sparsity +consequently remains as the most serious problem when applying statistical models onto SLs. For + +example, the real-time factor (RTF) of the SL video corpus is 100; that is, an hour corpus requires at least +100 hours of annotation (Dreuw et al., 2008b). + +Simulating SL comprehension using traditional statistical models and machine-learning methods +isdifficult. Thus, reliable methods for establishing a signer’s 3-D model (which is the process of +developing a mathematical representation of any three-dimensional surface of moving trajectories of +signers in the space for SL via specialized software) for SL corpus building and technologies for +annotating a large-scale SL video corpus automatically must be developed. Unlike the spoken language +that is “a set of values that change with the passage of time” (Huenerfauth, 2005), SL does not have a +writing system and thus cannot be saved in any form of written texts. + +The natural language-processing system relies on texts to process spoken languages. This system records +only the written text that corresponds to speech flows and relies only on the literacy of the user. On the +other hand, the SL system comprises information from multiple modalities. Examples of such information +are the hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye +gazing, body gestures, and facial expressions. The considerable information from multiple channels in SL +conveys linguistic meaning. This multi-modality nature of SL poses difficulties for the coding of SLs into +a linear single-channeled character string. In addition, SLs have writing systems, such as the Sign Writing +system (Sutton, 2010), ASL-phabet (Supalla et al., 2008), and HamNoSys (Prillwitz et al., 1989). +However, these systems have a limited number of users (Johnston, 2004). + +Many linguistic details are lost because of the multi-modality nature of SL during the translation of SL +into its corresponding writing system. SLs may be understood by directly matching the visual–spatial +characteristics of SL with the semantic units in the brain rather than applying written texts as an +interpreting medium. Here, semantic units are generally used for processing natural languages; these units +or nodes contain some information, which are used as knowledge representations form semantic units +(Geva et al., 2000). Such direct matching also represents the most natural way of comprehending SLs in +the brain. From this perspective, the authors present a computational cognitive model for SL +comprehension that is based on the cognitive functionalities of the human brain combined with a +knowledge representation theory of artificial intelligence (Shuklin, 2001). + +Visual–spatial mechanisms are exploited to express the grammatical structures and functions in SL. +Visual–spatial perception, memory, and mental transformations are prerequisites to grammatical +processing in SL (Emmorey & Corina, 1990) and are central to visual mental imagery (Farah, 1988). A +series of experiments have been conducted to investigate visual attention (Neville et al., 1998). Movement +recognition in peripheral vision is important in sign perception because the signers mainly look at the face +instead of tracking the hands when they communicate through SL (Siple, 1978). Therefore, identification +of lexical-level information depends on the peripheral vision system when signs are produced. The +recognition of movement directions is the selective function of peripheral vision (Bonnet, 1977). + +Whether deaf people only have a strong peripheral vision or efficiently allocate attention to peripheral +vision remains unclear. Stivalet et al. (1998) showed that visual attention processing can be changed by +auditory deprivation. They determined that deaf people do not shift their attention when processing the +information (i.e., alphabet set) presented in the central vision field, whereas hearing subjects must shift +their attention to search for the alphabet set continuously. Smith et al. (1998) also found that lack of +auditory input causes weak and selective (or highly distributed) visual attention among deaf children. +Stivalet et al. (1998) proposed that effective visual processing is caused by intermodal sensory +compensation; that is, the strong allocation of visual attention can be attributed to neuron reorganization +caused by auditory deprivation from birth. Recent magnetic resonance imaging evidence supports this +hypothesis (Bavelier et al., 2000). + + +These findings are selective attention cases, in which attention selectively processes certain stimuli but +ignores other stimuli. The cases refer to the selective orientation and concentration of the senses (i.e., +visual, auditory, taste, and tactile senses) and consciousness (i.e., awareness) of people on certain targets +(towards other factors). Studies on attention have failed to describe human attention at the biological level +in detail, as a person cannot focus continuously because the brain automatically suppresses activity when +attention reaches its limits. + +Emmorey and Reilly (2013) determined that when locations in a signing space (SL expressions streaks +the space) function topographically, spatial changes tend to be noticed easily. Thus, location information +indicating the spatial position of associated referents can be encoded and stored semantically in memory. +However, spatial locations with a primary distinguishing function of referents are encoded in a different +way and tend to be discarded from memory once the referential function is no longer required by context +(Emmorey & Reilly, 2013). Bavelier et al. (2001) claimed that only the posterior middle temporal gyrus +and the medial superior temporal cortex of deaf signers are highly active while perceiving movements in +peripheral vision. This phenomenon is unobservable in hearing signers who have skillfully grasped signs, +indicating that auditory deprivation results in a shift to stronger movement attention in the visual +periphery. Deaf people can easily reply to the attention and visual monitoring of their peri-personal space +(Bavelier et al., 2001). + +Neville et al. (1998) determined that the classic language area in the left hemisphere, particularly the left +perisylvian, of both deaf and hearing subjects is activated when reading English sentences. The right +hemisphere, including the right perisylvian, of deaf people is also activated. They argued that spatial +processing is of great importance to sign grammar. Thus, the SL comprehension process of deaf people +employs neurons at both high and low levels in the neural network, which are connected with each other +by edges, and generates high-level features via feature combination processes that are realized by +combining the weight on the edges. For example, low-level visual edge features are assembled, processed, +and sent to the high level to form the angle, shape, and other higher features (Bertasius et al., 2015). High- +level neurons form features that gradually approximate the semantics in turn, such as simple shapes, +simple targets, and real objects. The activation of high-level neurons during the reconstruction process +also reacts with the low-level neurons and adjusts and corrects deviations and losses (Bertasius et al., +2015); a temporal pattern appears in the horizontal structure connection. The neurons can make +predictions of the state at the next point of every time point through a horizontal connection based on the +information of their current status (Hawkins et al., 2009). +SEMANTIC NETWORK MODEL (SNM) +Model of semantic networks (SNs) are generally used for processing natural languages (Shuklin, 2001). +SNs, as knowledge representations, are extensible and have been used to model mental disturbances +(Geva et al., 2000). The semantic network (which is a network that represents semantic relations between +concepts, is used as a form of knowledge representation, here it is based on SL information processing of +human brain cortex. The edges connect different nodes in the network and represent the strength or +weakness of the correlation. After being set up, the semantic network is stored in long-term memory for +future retrieval and extraction to be encoded as semantic memory. Outside stimuli at a certain time can be +the demand of a person on specific knowledge and information to activate the demand on the extraction +of useful information of long-term memory (Sedikides & Skowronski, 1991). The activation process of +the stored network works in a form of spreading in the memory (Collins & Quillian, 1972). + +The semantic model, which is based on SL information processing of human brain cortex, is developed +accordingly. Different areas of the brain cortex are involved in the processing and are connected in a +hierarchical manner. Low-level information from sense organs is first processed in the primary +information-processing regions of the brain cortex and is then transferred to high-level regions for further + +processing, such as abstracting, integrating, and interpreting. The detailed description and illustration of +this hierarchical structure are summarized in Figure 1. +Figure 1. Hierarchical structure. Low-level areas in the hierarchy generate specific information that +increases speed and contain further details, whereas high-level areas form stable spatial invariance, +change slowly, and show high-level semantic object expression (Adapted from Yao et al., 2015) + + +In SL communication, both substantial and semantic information (substantial information includes hand +shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body +gestures, and facial expressions, and semantic information is represented into semantic concepts by these +substantial SL information) almost exclusively relies on signs. However, accurate SL information +analysis and prediction remain as challenging tasks in the field of natural SL processing. Three main tasks +are, namely, capturing, decoding, and extracting the physical characteristics and relationship of signs +(perception stage), matching the decoded cognitive representations with the stored semantic information +(memory stage), and completing the machine translation process of SL information (judgment stage). +This process of cognitive processing and understanding during SL communication is based on the PMJ +principle of “from the definition and extraction/annotation of cognitive representation (Stage P) to the +feature storage in line with the cognitive economy principles (Stage M), and then to the output of the +classification and judgment (Stage J).” + +The P→M→J (PMJ) principle exhibits a complete fine processing frame, the detailed illustration, and +description of SL comprehension frame based on the PMJ principle is summarized in Figure 2. +Figure 2. SL comprehension frame based on the PMJ principle. Perception refers to acquiring sign +information through selective attention. The information is limitedly processed by the brain if prominence +is given to useful and important information. Other information may be filtered out or suppressed when +sources for information processing are limited. Memory refers to the spreading activation process, in +which input information is coded, and one intends to store the information for a short period. Judgment + +targeted +wer +targeted sen antics +sign semantic concept +semantics +rea +representation +Area H +hign deve semantid +high-level semartic feature +feature detectors +Area A +low-evel senantic +low-level semantic feature +low-level s emanticf eature +feature detectors +cogritive +representation +Higher leve visual +handshape +location +oriention +moveme rt +No-manual feature +AreaV4 +abstractions +Edge +outine +Simple shape +Area V1 +detecton + edge feat ure +ed ge feature +Retina +oves +pixel +pixel +pixel +pixel +physical +characteristics +一refers to the process in which the perceived information or the information stored in memory is +compared, matched, or classified, and a decision or prediction is made. After the spreading activation, +the network records the attention features of users and activates their future preferences (Yao et al., +2015) + + +Concepts are in the form of network storage. The different concepts are stored in different functional +areas in both hemispheres of the human brain. The same or similar concepts are stored in same or +adjacent regions of brain. Specific information of entities in the outside world, such as humans, animals, +or tools, is represented by the concept network in the human brain. This concept network (A concept is an +abstract idea representing the fundamental characteristics of what it represents. Concept network consists +of these abstract concepts) is, in turn, connected with the lexical network from mental lexicon in the left +temporal lobe. Such specific information from mental lexicon will be employed to facilitate SL +production during which the SL users generate classifier hand shapes under the guidance of the +knowledge and rules of SL classifier predicates (Valli & Lucas, 2000). Here, classifier predicates are +made by combining small meaningful unites to create bigger units, the main units being the hand shape +and the movement. This condition implies that findings from brain research can provide knowledge and +guidance for the cognitive computational modeling of classifier predicate comprehension. In order to +obtain a deep understanding of sign lexical semantics, a cognitive processing model, which is based on +the cognitive mechanism of human brain, is established. The cognitive processing model would activate +the concept network of the associated classifier hand shapes in the brain. Here, classifier predicates differ +from traditional linguistic units. Traditional methods, such as the syntactic tree, cannot satisfy the +generation of the classifier predicates (Huenerfauth et al., 2006). +DECISION-TREE BASED ALGORITHMIC METHODS +The authors use SNs as the knowledge representation and organization mode of SL. The relationship in +semantic networks represents a type of information among nodes. Nodes with a complicated relationship +with other nodes contain additional information. Such nodes require further effort to be understood. +Consequently, the authors simulate selective attention (i.e., the processing of visual or auditory input +based on whether it is relevant or important). They selected particular representations to enter perceptual +awareness and therefore guide behavior. Through this process, less relevant information is suppressed by +humans using the proposed algorithmic methods to accentuate the nodes selectively and suppress the +unessential nodes (Chelazzi et al., 2013). + +The emergence of 3-D-based sensors, such as Kinect by Microsoft and Leap Motion (Yao et al., 2014), +has improved studies on sign recognition from video-based to 3-D-based sign recognition. However, this +transformation makes traditional video-based SL recognition methods inapplicable to 3-D-based SL + +Perception +Memon +Judgment +Attentional enha ncement and suppress ion +0 +Spread activation +Interactive activationrecognition technologies. Large training data are required for valid recognition in 3-D-based SL +recognition technologies because of the low operation efficiency of the rotatable joint-based sorter and the +matching techniques for sign signal recognition. Yao et al. (2014) proposed a decision tree-based +algorithm. The algorithm aims to achieve a high-precision and real-time performance of SL automatic +perception according to the features of Leap Motion. The authors adopted this method as the first step of +SL comprehension. +Attention Function +The authors propose the following attention function: + + ������������������������ = +∑ +������������������������������������ +������������������������������������ ������������ +∑ ������������������������������������ +������������������������ +(1) + +where ∑ +������������������������������������ +������������������������������������ ������������ + denotes the sum of the semantic relation weights around the semantic node x, ∑ ������������������������������������ +denotes the sum of all semantic relation weights, and ������������������������ represents the activation value on the semantic +node x after the spreading activation process. +Semantic Matching +Cognitive units in the memory network compete with one another based on certain rules to obtain more of +the limited attention resources and more energy for a more active state. SL comprehension supports +interactive activation models (Gutierrez et al., 2012). Therefore, judgment is the outcome of the attention +competition game in the spreading activation, which is a search algorithm. The search algorithm is +initiated by labeling a set of source nodes (e.g. concepts in a semantic network) with weights +or ”activation,” and then iteratively propagating or “spreading” that activation out to other nodes linked to +the source nodes processes of the human brain (Crestani, 1997; Preece, 1981). + +A semantic matching algorithm based on activation spreading modes is proposed to determine the most +appropriate semantic information. Activation starts to spread from the corresponding nodes of the signs +presented by the signer. The activation value of the stimulus node (i.e., signs to be perceived before the +start of spreading) must be calculated first. In particular, the increment in the interest value of object +concept must be calculated, and this concept node must be used as an initial node for the spread study. +Activation spreads to the neighboring nodes, which usually have a lower activation value than the source +value. Therefore, introducing an activation attenuation factor for decreasing activation over the path +length in the closed interval [0…1] is mandatory. That is, for every propagation through an edge a loss of +activation is considered (Neumann et al., 1993; Rocha et al., 2004). The activation spreading process can +be expressed as follows: + + ������������������������(������������ + 1) = ������������������������(������������)������������������������������������(1 − ������������) +(2) + +where ������������������������(������������ + 1) represents that the value is spread from node x to y at time, t+1, ������������������������(������������) represents the +activation value that was spread at node x at time, t, ������������������������������������ signifies the link between nodes x and y, and δ is +an attenuation factor used to describe the energy loss caused in the activation spreading process (Jiang & +Tan, 2006). + +Spreading activation theory states that the activation of human memory “chunks” (the content of any +buffer is limited to a single declarative unit of knowledge, called a chunk) is determined by two factors +(Anderson et. al., 2004; Anderson, 2013), namely, the use history of the memory chunk and the +correlation between the memory chunk and the current retrieval information. These two factors calculate +the weights and determine whether the chunk is activated and selected. This assumption has been verified + +by experimental cognitive psychology, and the calculation model has been established (Roelofs, 1992). +The authors must use moments to express the distance in each activation time with the current time. Time +units may be per hour as a unit and may also be the day. With the day as a unit, we can count the +historical value in the previous day as the activation value of the first day. The algorithm based on the +theory of memory activation can improve SL understanding, which is sometimes highly sensitive to time. + +At node y, the largest number of neighbor nodes is (n-1); thus, the maximum of ������������������������(������������ + 1) can be +expressed as: + + ������������������������(������������) = [������������1, ������������2, … , ������������������������]������������ + +i.e., the initial value of the semantic network. Where I1, I2, …, In are these activation value of neighbor +nodes. + +If activation spreads from a node in many directions, then its adjacent nodes obtain a low activation value. +The adjacent nodes give a feedback value of their resonance energy (i.e., contributing structure with the +lowest potential energy) to the co-adjacent nodes after they absorb the activation value. The following +equation is therefore used: + +Iz(t + 1) = Oz(t) + ∑ +Ox(t)Λxz(1-δ) +all actived x + (3) + +where Oz(t) denotes the activation value of node z at time t. + +Given that the quantity of activated information is limited, the nodes that obtain less resonance +information are equivalently inhibited and are less likely to be activated. The activation value distribution +in the resonance process conforms to the human attention model. +Attention Game Process +Cognitive units in memory network compete with one another by certain rules to increase the possibility +of obtaining more human attention resources and more energy that will improve activity. This +phenomenon is called a game process. The authors use game theory (Myerson, 1997), which is the study +of mathematical models of conflict and cooperation between intelligent rational decision-makers and +attempts to achieve the largest cognitive gains with the least energy possible, as a reference to simulate +the attention enhancement and suppression processes that are selective attention processes. In other +words, when visually searching for a non-spatial feature or a perceptual feature, selectively enhancing the +sensitivity to that specific feature plays a role in directing attention. When people are told to look for +motion, then motion will capture their attention, but attention is not captured by motion if they are told to +look for color (Reynolds & Chelazzi, 2004). Activated results consistent with cognitive features can then +be obtained. The authors assume that the game contains n nodes. ������������������������ +′ and ������������������������ +" are the two selectable +strategies for node i, and they represent the acceptance and non-acceptance of the change in the attention +function (i.e., ������������������������ +′, ������������������������ +" ∈ ������������������������ ). The corresponding gain can be represented by ������������������������ +′ and ������������������������ +" , and ������������������������ +′, ������������������������ +" ∈ ������������������������. N +nodes are assumed to reach an agreement before participating in the game to introduce the Nash +equilibrium (i.e., each node only selects a specific strategy). The authors let ������������∗ = (������������1 +∗, … , ������������������������∗) represent the +agreement, where ������������������������ +∗ is the strategy of the node i specified in the agreement. Nodes comply with this +agreement only when the benefit from complying with the agreement is larger than that from not +complying. This agreement constitutes Nash equilibrium if any node abides by this agreement. Thus, the +Nash equilibrium is written as follows: + + ������������������������(������������������������ +∗, ������������−������������ +∗ ) ≥ ������������������������(������������������������, ������������−������������ +∗ ), ∀si ∈ Si +(4) + + +where the combination of strategy ������������∗ = (������������1 +∗, … , ������������������������∗) is a Nash equilibrium. Given that other nodes select +������������−������������ +∗ = (������������1 +∗, … , ������������������������−1 +∗ +, ������������������������+1 +∗ +, … , ������������������������∗), ������������������������ +∗ is the optimal strategy of each node i (Myerson, 1997). + +The attention game process determines whether the nodes need adjustment or need to be changed on the +basis of the attention function. The activation energy distribution will reach a state consistent with the +human attentive distribution after adjusting the activated value distribution. Nodes of the spread SNs have +their own activation energy threshold values. The source node in the attention game process that +represents a presented sign has the maximum activation value O in the present SNs. All equidistant nodes +will participate in the game based on the attention function. The nodes with low activation energy +(defined as the minimum energy required to start a chemical reaction) of a reaction is denoted by Ea and +given in units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol)), threshold must be +removed through a screening process to prevent them from participating in the enhancement and +suppression processes of activating the most likely node. In the proposed screening, the authors ignore the +nodes with a significantly low activation value to be activated in the enhancement process instead of +lowering the possibility for other nodes to be activated. + +The difference between the attentive readjustment in the present attention game process and the previous +attentive allocation causes the instability in the overall cognitive structure of users to decrease knowledge +credibility. Thus, a new cognitive structure must be determined at a cost as follows: + +Cost(t, i, si, ui, SN) = �n−1 ∑ +�Ii(t + 1) − Oi(t)� +2 +n +i=1 + (5) + +where ������������������������(������������ + 1) denotes the activation value that is conveyed from one node at time t+1 to node i, and 0i +(t) denotes the activation value of node i at time t. Therefore, the total cost is attributed to the change in +the activation energy of all nodes in the SN. The goal of judgment is to achieve the overall optimal gain +with a minimal computing cost. The gain function in the attention game process must then be determined. +As the optimal strategy for node i, ������������������������ +∗ must minimize the distribution change that refers to the distribution +change in the activation values of the overall network changed by the decision. The amount of spreading +activation energy is fixed in the total process of activation spread in the SN; thus, the semantic node +energy enhancement must be accompanied by reduced node energy. The attention parameters are affected +by the overall distribution change in activation energy. The activation energy enhancement increases the +impossibility of activating this node. Such activation is the ultimate purpose of each node (i.e., the node +obtains the gain). Accordingly, the gain function is presented as follows: + +Gain(t, i, si, ui, SN) = +�∑ +Ix∈{neighbor node}(t+1) +num(all x) +j=1 +−∑ +Ox∈{neighbor node}(t) +num(all x) +j=1 +�(1−δ) +num(all x) + +(6) + +where SN represents the current semantic network, num(all x) represents the number of neighbor nodes x +of node i, ∑ +Ox∈{neighbor node}(t) +num(all x) +j=1 + denotes the sum of the activation value that was spread of all +node i neighboring nodes at time t, the gain function is expressed as the attention gain of neighbor nodes +x of node i after the enhancement and suppression processes, it represents the benefit a node gets by +unilaterally changing their strategy. + +The utility function of the attention game process can be determined as follows: + +Max�������������������������(������������, ������������������������ +∗, ������������−������������ +∗ )� = Gain(������������, ������������, ������������������������, ������������������������, ������������������������������������) − Cost(������������, ������������, ������������������������, ������������������������, ������������������������������������) +(7) + + +where ������������������������ +∗ is the optimal strategy of each node i, ������������−������������ +∗ is the strategies set of other nodes except node i. only +when ������������������������(������������, ������������������������ +∗, ������������−������������ +∗ ) reaches the maximum, ������������������������ +∗ is a Nash equilibrium of node i. The utility of the other +nodes will be affected by the decision of all other nodes because of the fact that the total quantity of +activation energy is fixed (i.e., attention is limited) in the attention game process. When each node selects +a decision for itself, it also considers the possible decision of other nodes and selects the “Nash +equilibrium point” with maximum utility. This scenario is consistent with classical game theory. The +authors select a Nash equilibrium decision for each node through the utility function of the attention game +that is defined by Equation (7). +METHODS +Data Sets and Experimental Settings +All data from the authors’ experiments are obtained from the Tsinghua University–Chinese SL Corpus +(TH–SLC). The data mainly comprise SL expressions of idiom stories and life fragments of deaf students. +No automatic annotation software based on videos is currently available because the annotation process +for SL videos is time consuming and requires expert knowledge in dual language (i.e., Chinese language +and Chinese SL). Video annotation is also time consuming. Specifically, it takes about 30 hours for the +annotation RTF (real-time factor) of a parliamentary speech (i.e., One hour of speech requires 30 hours of +annotation). However, the annotation RTF (real-time factor) for a full annotation of all manual and non- +manual components of an SL video can reach up to 100 hours (Dreuw & Ney, 2008a). Therefore, such a +corpus is significantly small. For example, the Aachen Boston database contains American SL and has +annotated 201 English sentences (Dreuw & Ney, 2008a). The authors spent a year collecting more than +2000 sentences, but only 416 sentences containing 2496 signs were marked. + +The authors asked 20 deaf students to select 300 sign pairs from 2469 annotated signs in TH–SLC and to +judge the relevance of the sign pairs. The correlation values range from 0.0 to 1.0. For convenience, a +five-point scale is used to assess the correlation. The sign pairs were obtained using a marked correlation. +The authors establish an SN based on the word similarity computing method of HowNet (Liu & Li, 2002) +to determine the connection weight of the network to validate the effects of the proposed model. The +authors introduce the continuous bag-of-words (CBOW that predicts the current word from a window of +surrounding context words. The order of context words does not influence the prediction (CBOW +assumption) model (Mikolov et al., 2013), and the HowNet (Liu & Li, 2002) method as the baseline +methods using the same recommended parameters. The efficiency of the utility function of the attention +game process is evaluated in terms of word correlation computation, and the model complexity is +analyzed. +Word Relatedness Computation +Each model in this task needs to compute the semantic correlation of the given sign pair. The correlation +between the experimental results of the model and human judgment reflects upon the model’s +performance. The authors selected 290 signs for the closed set and 10 signs for the open set. + +Spearman’s correlation between model correlation score and human judgment correlation score was +calculated for comparison. Spearman correlation coefficient is defined as the Pearson correlation +coefficient among the ranked variables (Myers & Well, 2003). For a sample of size N, original data ������������������������, ������������������������ +are converted into grade data������������������������, ������������������������, the correlation coefficient ρ is defined as follows: + +ρ = 1 − +6 ∑ di +2 +n(n2−1) +(8) + + +where the difference between the observations of the two variable levels is set as ������������������������ = ������������������������ − ������������������������. If there is +no duplicate value in the data, and two variables are completely monotonic correlation, the Spearman +correlation coefficient is +1 or -1. +RESULTS +For CBOW, the correlation scores of the two words are calculated using the cosine similarity of word +embedding (Mikolov et al., 2013). The evaluative results of the baseline methods and the proposed SNM +method in the closed test and in all test sets are shown in Table 1. +Table 1. Evaluative results +Data Set +Closed Test +All Test Sets +(Including Open Test) +Spearman’s Rank +Correlation Coefficient +Method +290 pairs +300 pairs + +CBOW (baseline method) +0.4843 +0.4869 +0.4136 +Word similarity computing +based on HowNet +0.6157 +0.6174 +0.6052 +Proposed SNM method +0.6951 +0.7063 +0.6437 + +The evaluation results show that the proposed SNM method is better than the baseline method in 290 and +300 word pairs. This finding indicates that the cognitive mechanism of sign comprehension is essential to +understanding the meaning of signs. The internal structure, such as location, orientation, hand shape, and +movement, contains rich semantic information. However, deep learning methods, such as CBOW, +consider the external context, but ignore the internal structure. + +Using the computing method of word similarity based on HowNet results in only a rough semantic +computation. For example, adding 10 new sign pairs negligibly changes the performance of these +methods. In other words, these methods can still handle new signs with improved performance. The +semantic correlation of these new sign pairs calculated by the proposed method is close to human +judgment. Figure 3 shows the quantitative analysis of the attention game process for two signs. Each hand +shape of the two signs has at least 20 related semantic lexicons. The stimulus information and +permutation of each node are shown in the first and second columns from high to low according to the +activated value after the activation spreading process. Only 10 semantic lexicons that are maximally +activated are shown. The permutation of each node is shown in columns three to seven from high to low +according to the activation value after the end of the first to fifth attention games. The top 10 lexicons are +also shown. The semantic lexicons in the blue background rank high after the games, those in the green +background rank low after the games, and those in the white background are unchanged. +Figure 3. Examples of attention games. The semantic lexicons in the blue background rank high after the +games, those in the green background rank low after the games, and those in the white background are +unchanged. This trend shows that the ranking of other semantic lexicons below slightly changes after the +semantic lexicon that ranks highest becomes unchanged. This condition is due to the source that +corresponds to the attention model being determined after several game processes. + + + +Figure 3 also shows that significant changes occur during the ranking of the semantic lexicons in the first +and second instances after the first several games, whereas only a few changes occur in the following +stimulus games. This trend shows that the ranking of lower semantic lexicons slightly change after the +semantic lexicon that ranks highest becomes unchanged. This condition is due to the source that +corresponds to the attention model being determined after several game processes. Attention is also +assigned to other nodes in accordance with the attention game process. Humans reach a steady state after +thinking about problems constantly, and the result negligibly changes if they rethink. Nearly no change is +observed in the result after several rounds. Several semantic lexicons related to the signs are contained in +the text set; thus, a few possible changes occur. The result of the attention game model conforms to +human cognitive rules to a certain degree. + +Attention is also assigned to other nodes in accordance with the attention game process (here, efforts have +been made in modeling according to the mechanism of human attention). The result of the SNM conforms +to human cognitive rules to a certain degree (Gutierrez et al., 2012). For example, the authors assume that +deaf people understand the signs shown in Figure 3. Deaf people usually search for many familiar and +specific nouns or signs in a spreading activation mode to comprehend classifier predicates. After all +activated values are calculated; the activated nodes are graded and sorted. A high-activated value of the +node indicates the importance of the interested object or concept represented by the node. This shows that +deaf people are familiar with the concept node. Similar to the attention game process shown in Figure 3, + +activation +activation +activation +activation +activation +activation +value +value +value +value +value +value +sorting +sorting after +sorting after +sorting after +sorting after +sorting after +after +the first +the second +the 3rd +the 4th +the 5th +spread +attention +attention +attention +attention +attention +input +activation +game +game +game +game +game +A +handshape +good +General +General +General +General +poog +Reliable +Beheaded +Beheaded +Beheaded +Beheaded +Defend +Advanced +Support +Support +Support +Support +Maintain +Strange +Keep +Keep +Keep +Keep +Protect +General +General +General +General +General +Beheaded +Teacher +Teacher +Teacher +Teacher +Teacher +Support +Marshal +Reliable +Reliable +Reliable +Reliable +General +Madam +Advanced +Advanced +Advanced +Advanced +Teacher +Ancestors +Protect +Protect +Protect +Protect +Reliable +Defend +Maintain +Maintain +Maintain +Maintain +Y +handshape +human +Animal +cat +cat +cat +stand +animal +Burial +dog +Sop +dog +run +burial +Frustration +horse +horse +horse +lie +Frustration +Future +human +human +human +Resistance +future +Voltage +Ambassador +Ambassador +Ambassador +Protest +Voltage +Ambassador +coach +coach +coach +Control +Ambassador +Coach +stand +stand +stand +Incite +coach +Guide +run +run +run +Exploitation +Guide +Blind +lie +lie +lie +Recalcitrant +Blind +Opponent +Resistance +Resistance +Resistancethe high-ranked semantic lexicon is a cat or dog after several rounds. This result shows that the most +common subjects for deaf people are typical subjects that represent classifier predicates. +DISCUSSION +Compared with that of existing models, the complexity of the proposed model is reflected mainly on the +computational cost of the memory stage and the judgment stage (i.e., the computational cost of spreading +activation and the attention game at time (t + 1)). The cost is a dynamic value and related to two factors, +namely, the activation state of the current sign and the current cycle as the first activation of the sign. +Therefore, the value changes regardless of the choice of the user. This outcome is consistent with the +strong dynamics of sign information, which can reflect the influence of information in different periods. +In the memory stage, the time complexity of computing ������������������������(������������) is unity; thus, the time complexity is +related to the total amount N of activation energy and cycle times. The time complexity of each activation +in each cycle is n × 1 = n. Space complexity is the storage space of each node and the semantic relation +weight according to semantic similarity (semantic similarity can be estimated by defining a topological +similarity, by using ontologies to define the distance between terms/concepts). Therefore, unlike the +general model such as cobweb theorem model and vector space model, where the SNM increases the +overhead in time complexity and space complexity. The model also increases the matching time of query +nodes and weights in the current activation. However, the overhead at this time can provide more +effective results than an invalid spreading and can be accepted by users. + +In the judgment stage, when the node selects the game strategy to change its activated energy value, the +convergence speed of adjusting the cognitive benefits to its own utility maximum “Nash equilibrium” is +an important measure of evaluating the SNM (i.e., the cycle times of an attention game process). For the +attention game, the Nash decision of different semantic nodes must minimize the change cost of the +activation energy distribution of the entire network. The Nash equilibrium point decision for each node is +selected using the utility function defined in the SNM. This process is repeated until the overall network +activation energy distribution change is less than the specified threshold. The node needs to solve n-order +nonlinear equations in every cycle. Therefore, the performance of the convergence speed of the SNM is +indicated by the number of game cycles that the network requires to reach the Nash equilibrium point +(i.e., the computing times of calculating the corresponding equation by each node in a game process). The +square root of the sum of the variance of activation value ������������������������(������������ + 1) of each adjusted node is directly +reflected by the rate of convergence in the game process. + +To verify its effectiveness, the attention game model is compared with the traditional model in terms of +load balancing. In the traditional method, the activation value of each node is certain (i.e., the value is not +enhanced or inhibited). The experimental results are shown in Figure 4. The results show that the load +balance performance of the attention game model is better than that of the traditional model because the +attention game model adjusts the activation strategy after the activation of each node. When the change +cost of the energy distribution of the entire network activation is larger than the specified threshold, the +human brain adjusts the strategy to inhibit the activation energy value in the next cycle. In doing so, the +free competition and distribution of attention for each node according to the attention game model can be +assured. The result is obtained through the overall competition. The load of attention of the network is +balanced. The traditional model assumes that the activation energy value of each node is certain because +the brain activation energy resource amount is constant in a period of time. The brain selects the node +with a low activation energy value and performs the allocation of attention. This allocation causes the +attention load of several nodes to be excessively large or unutilized. +Figure 4. Comparison of load balance. The load balance performance of the SNM is better than that of +the traditional model because the SNM adjusts the activation strategy after the activation of each node + + + +The proposed SNM model used Nash equilibrium to simulate the energy activation process. In order to +quantitatively analyze the effects of Nash equilibrium, the authors compared the SNM with the cobweb +theorem model (Pashigian, 2008) in terms of different activation energy amounts. The cobweb theorem is +expressed as follows: + +������������(������������ + 1) = ������������(������������) + ������������ ��������������������������(������������)� − �������������������������′(������������)�� +(9) + +where r is the adjustment parameter of the activation value, �������������������������(������������)� is the activation function of a node, +������������(������������) is the activation value at time t, �������������������������′(������������)� is the attention allocation function, ������������′(������������) is the +expectation activation value at time t, and �������������������������(������������)� − �������������������������′(������������)�is the excessive demand function that +represents the actual gaps between the activation value and activated allocated value. A large gap +indicates a high activation value of the Nash Equilibrium of node. The parameter (r) indicates the actual +speed and strength of adjusting the activation value according to the attention distribution condition in the +last moment. When r > 0 it indicates that the adjustment direction of the activation value is consistent +with the direction of the demand function. + +The amount of activation energy Ea is assumed to be 100 kJ/mol. Figure 5 shows the result of comparing +the attention utilization between the game model and the cobweb model. The attention amount (attention +is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information, +while ignoring other perceivable information. Attention amount refers to as the allocation size of limited +processing resources), is less than 100 kJ/mol. If the attention amount is insufficient, then attention +resources can only meet part of the node demand, and the resource utilization rate of the SNM will +become higher than that of the cobweb model. When attention supply exceeds the demand of a node, the +cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle. The SNM +meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the +cobweb model. + +Comparisonofloadbalance +9 +8 +nodes +hhhl +6 +Numberof +L +4 +3 +1 +0 +No.1 +No.2 +No.3 +No.4 +No.5 +No.6No.7 +No.8 +No.9 +No.10 +Numberofactivationenergy amount +Iattentiongamemodel +cobwebtheoremmodeFigure 5. Comparison of activation energy values. After the change in the initial value of the activation +energy, the number of iterations increases depending on the difference between the initial activation +energy value in the cobweb model and the balanced energy value. The iteration of the attention game +model can be adjusted according to the difference in the activation energy between supply and demand. A +sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the +supply and demand + +Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash +equilibrium. As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in +the SNM, and the activation energy is 120 kJ/mol in total. If the initial value of the activation energy is +changed, then the initial activation energy value of the cobweb model is higher than the energy +equilibrium value and requires abundant cycle time. The SNM in each cycle can adjust the activation +energy according to the variance of the activation energy. The variance and adjustment range are large, +and the SNM eventually reaches the Nash equilibrium point. +Figure 6. Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium. +When the supply falls short of demand, attention resources can only meet the demands of several nodes, +and the resource utilization rate of the SNM becomes higher than that of the cobweb model. If the supply +exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet +only part of the demands of nodes. However, the SNM can meet the demands of all nodes, and its +resource utilization rate is higher than that of the cobweb model +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +40 +60 +80 +100 +120 +Attention Resource Utilization(percentage) +Amount of activation energy(KJ/mol) +Comparison of activation energy values +attention game model +cobweb theorem model + + +CONCLUSION +The authors presented a new model for SL comprehension based on spatial information. This process uses +game theory to simulate the human attention suppression and enhancement process. This process also +joins the forgetting function of human memory traces to compute the initial state of the node. Memory is +encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and +temporal plane. Although the semantic network provides a functional view of how knowledge may be +organized in the brain, it does not provide a clear model of how semantic memory might be presented in +the brain (see Cacha et al., 2017). Spreading activation reveals that information can be stored in SNs for a +long time, in which a network node is a linguistic concept and the nodes are connected through the +correlation. An algorithmic method is proposed according to selective functions, and its effectiveness was +verified using an example. The results show that the proposed method improves the performance of SL +comprehension. +ACKNOWLEDGMENT +The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for +helping in stimulus preparation and data collection. This paper forms an expanded and revised version of +a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive +Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015. The authors are grateful to Dr. +Raymond Chiong, and two anonymous referees for their helpful comments. +Conflict of Interest +The authors of this publication declare there is no conflict of interest. +Funding Agency +This research was supported by the Beijing Municipal Natural Science Foundation [4202028]; National +Social Science Foundation of China [21BYY106]; National Natural Science Foundation of China +[62036001, 61866035, 61966033]; Premium Funding Project for Academic Human Resources + +Comparison of cycle times +40 +Number of Iterations +35 +30 +25 +20 +15 +10 +5 +0 +1 +5 +10 +15 +20 +25 +30 +Activationvalue +attentiongamemodel +cobwebtheoremmodelDevelopment in Beijing Union University [BPHR2019CZ05]; Jiangsu Province Key R&D Program +(Industry Prospects and Key Core Technologies) [BE2020047]; and the characteristic-disciplines oriented +research project in Beijing Union University [KYDE40201702]. +REFERENCES +Anderson, J. R., Bothell, D., Byrne, M. 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Gallaudet +University Press. + + + diff --git a/kb_51/content/tmp_files/load_file.txt b/kb_51/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d652510553bbcd750f3d1f22f5eec4d668f4b774 --- /dev/null +++ b/kb_51/content/tmp_files/load_file.txt @@ -0,0 +1,836 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf,len=835 +page_content='Semantic Network Model for Sign Language Comprehension Xinchen Kang (22f3d431-dcc0-42a5-8e2b-c26464e0654d) Beijing Union University, China Dengfeng Yao (ae88317c-d091-4f41-97f1-b1e6be00ca68) Beijing Union University, China Minghu Jiang (ea1cc43b-eee9-4185-8d97-edeac9186268) Tsinghua University, China Yunlong Huang (cc1ddbf2-64b9-4c51-b553-0ebe52a8d645) Tsinghua University, China Fanshu Li (64642396-6e95-456f-a4f2-47ff81a23d6e) Beijing Union University, China ABSTRACT In this study, the authors propose a computational cognitive model for sign language (SL) perception and comprehension with detailed algorithmic descriptions based on cognitive functionalities in human language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic network model (SNM) that represents semantic relations between concepts is used as a form of knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The proposed model is applied in the comprehension of sign language for classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The spreading activation search method is initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' concepts in the semantic network) with weights or "activation,"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and then iteratively propagating or "spreading" that activation out to other nodes linked to the source nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The results demonstrate that the proposed search method improves the performance of sign language comprehension in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Keywords: Attention, Cognitive Processing, Comprehension, Decision-Tree, Game Theory, Linguistics, Perception, Semantic Network, Sign Language INTRODUCTION Sign language (SL) comprehension is a fundamental task for computational linguists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Two types of algorithms have been proposed: (1) rule-based methods (Supalla, 1982), and (2) statistical methods (Bauer & Heinz, 2000; Huenerfauth, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Rule-based methods lack the capability of planning the elements in the entire scene (Liddell, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The method of modeling infinite natural language input through finite rules, especially minor rules, barely meets all requirements of SL processing (Yao et al., 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, statistical methods are the preferred type of algorithm for SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Statistical models can be applied to spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Given the abundant data resources of spoken languages in the digitalized Internet age, statistical models can be applied readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, the raw and annotated corpora of SLs are insufficient because collecting and annotating SL videos are tedious and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Data sparsity consequently remains as the most serious problem when applying statistical models onto SLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For example, the real-time factor (RTF) of the SL video corpus is 100; that is, an hour corpus requires at least 100 hours of annotation (Dreuw et al., 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Simulating SL comprehension using traditional statistical models and machine-learning methods isdifficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Thus, reliable methods for establishing a signer’s 3-D model (which is the process of developing a mathematical representation of any three-dimensional surface of moving trajectories of signers in the space for SL via specialized software) for SL corpus building and technologies for annotating a large-scale SL video corpus automatically must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Unlike the spoken language that is “a set of values that change with the passage of time” (Huenerfauth, 2005), SL does not have a writing system and thus cannot be saved in any form of written texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The natural language-processing system relies on texts to process spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This system records only the written text that corresponds to speech flows and relies only on the literacy of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' On the other hand, the SL system comprises information from multiple modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Examples of such information are the hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body gestures, and facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The considerable information from multiple channels in SL conveys linguistic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This multi-modality nature of SL poses difficulties for the coding of SLs into a linear single-channeled character string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In addition, SLs have writing systems, such as the Sign Writing system (Sutton, 2010), ASL-phabet (Supalla et al., 2008), and HamNoSys (Prillwitz et al., 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, these systems have a limited number of users (Johnston, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Many linguistic details are lost because of the multi-modality nature of SL during the translation of SL into its corresponding writing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' SLs may be understood by directly matching the visual–spatial characteristics of SL with the semantic units in the brain rather than applying written texts as an interpreting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Here, semantic units are generally used for processing natural languages; these units or nodes contain some information, which are used as knowledge representations form semantic units (Geva et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Such direct matching also represents the most natural way of comprehending SLs in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' From this perspective, the authors present a computational cognitive model for SL comprehension that is based on the cognitive functionalities of the human brain combined with a knowledge representation theory of artificial intelligence (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Visual–spatial mechanisms are exploited to express the grammatical structures and functions in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Visual–spatial perception, memory, and mental transformations are prerequisites to grammatical processing in SL (Emmorey & Corina, 1990) and are central to visual mental imagery (Farah, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' A series of experiments have been conducted to investigate visual attention (Neville et al., 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Movement recognition in peripheral vision is important in sign perception because the signers mainly look at the face instead of tracking the hands when they communicate through SL (Siple, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, identification of lexical-level information depends on the peripheral vision system when signs are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The recognition of movement directions is the selective function of peripheral vision (Bonnet, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Whether deaf people only have a strong peripheral vision or efficiently allocate attention to peripheral vision remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (1998) showed that visual attention processing can be changed by auditory deprivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' They determined that deaf people do not shift their attention when processing the information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., alphabet set) presented in the central vision field, whereas hearing subjects must shift their attention to search for the alphabet set continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (1998) also found that lack of auditory input causes weak and selective (or highly distributed) visual attention among deaf children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (1998) proposed that effective visual processing is caused by intermodal sensory compensation; that is, the strong allocation of visual attention can be attributed to neuron reorganization caused by auditory deprivation from birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Recent magnetic resonance imaging evidence supports this hypothesis (Bavelier et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' These findings are selective attention cases, in which attention selectively processes certain stimuli but ignores other stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The cases refer to the selective orientation and concentration of the senses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., visual, auditory, taste, and tactile senses) and consciousness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., awareness) of people on certain targets (towards other factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Studies on attention have failed to describe human attention at the biological level in detail, as a person cannot focus continuously because the brain automatically suppresses activity when attention reaches its limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Emmorey and Reilly (2013) determined that when locations in a signing space (SL expressions streaks the space) function topographically, spatial changes tend to be noticed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Thus, location information indicating the spatial position of associated referents can be encoded and stored semantically in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, spatial locations with a primary distinguishing function of referents are encoded in a different way and tend to be discarded from memory once the referential function is no longer required by context (Emmorey & Reilly, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Bavelier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2001) claimed that only the posterior middle temporal gyrus and the medial superior temporal cortex of deaf signers are highly active while perceiving movements in peripheral vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This phenomenon is unobservable in hearing signers who have skillfully grasped signs, indicating that auditory deprivation results in a shift to stronger movement attention in the visual periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Deaf people can easily reply to the attention and visual monitoring of their peri-personal space (Bavelier et al., 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Neville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (1998) determined that the classic language area in the left hemisphere, particularly the left perisylvian, of both deaf and hearing subjects is activated when reading English sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The right hemisphere, including the right perisylvian, of deaf people is also activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' They argued that spatial processing is of great importance to sign grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Thus, the SL comprehension process of deaf people employs neurons at both high and low levels in the neural network, which are connected with each other by edges, and generates high-level features via feature combination processes that are realized by combining the weight on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For example, low-level visual edge features are assembled, processed, and sent to the high level to form the angle, shape, and other higher features (Bertasius et al., 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' High- level neurons form features that gradually approximate the semantics in turn, such as simple shapes, simple targets, and real objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation of high-level neurons during the reconstruction process also reacts with the low-level neurons and adjusts and corrects deviations and losses (Bertasius et al., 2015); a temporal pattern appears in the horizontal structure connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The neurons can make predictions of the state at the next point of every time point through a horizontal connection based on the information of their current status (Hawkins et al., 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' SEMANTIC NETWORK MODEL (SNM) Model of semantic networks (SNs) are generally used for processing natural languages (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' SNs, as knowledge representations, are extensible and have been used to model mental disturbances (Geva et al., 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic network (which is a network that represents semantic relations between concepts, is used as a form of knowledge representation, here it is based on SL information processing of human brain cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The edges connect different nodes in the network and represent the strength or weakness of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' After being set up, the semantic network is stored in long-term memory for future retrieval and extraction to be encoded as semantic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Outside stimuli at a certain time can be the demand of a person on specific knowledge and information to activate the demand on the extraction of useful information of long-term memory (Sedikides & Skowronski, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation process of the stored network works in a form of spreading in the memory (Collins & Quillian, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic model, which is based on SL information processing of human brain cortex, is developed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Different areas of the brain cortex are involved in the processing and are connected in a hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Low-level information from sense organs is first processed in the primary information-processing regions of the brain cortex and is then transferred to high-level regions for further processing, such as abstracting, integrating, and interpreting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The detailed description and illustration of this hierarchical structure are summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Low-level areas in the hierarchy generate specific information that increases speed and contain further details, whereas high-level areas form stable spatial invariance, change slowly, and show high-level semantic object expression (Adapted from Yao et al., 2015) In SL communication, both substantial and semantic information (substantial information includes hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body gestures, and facial expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and semantic information is represented into semantic concepts by these substantial SL information) almost exclusively relies on signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, accurate SL information analysis and prediction remain as challenging tasks in the field of natural SL processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Three main tasks are, namely, capturing, decoding, and extracting the physical characteristics and relationship of signs (perception stage), matching the decoded cognitive representations with the stored semantic information (memory stage), and completing the machine translation process of SL information (judgment stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This process of cognitive processing and understanding during SL communication is based on the PMJ principle of “from the definition and extraction/annotation of cognitive representation (Stage P) to the feature storage in line with the cognitive economy principles (Stage M), and then to the output of the classification and judgment (Stage J).”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The P→M→J (PMJ) principle exhibits a complete fine processing frame, the detailed illustration, and description of SL comprehension frame based on the PMJ principle is summarized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' SL comprehension frame based on the PMJ principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Perception refers to acquiring sign information through selective attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The information is limitedly processed by the brain if prominence is given to useful and important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Other information may be filtered out or suppressed when sources for information processing are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Memory refers to the spreading activation process, in which input information is coded, and one intends to store the information for a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Judgment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='targeted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='wer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='targeted sen antics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='sign semantic concept ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='semantics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='rea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Area H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='hign deve semantid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='high-level semartic feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='feature detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Area A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='low-evel senantic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='low-level semantic feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='low-level s emanticf eature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='feature detectors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='cogritive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='representation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Higher leve visual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='handshape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='location ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='oriention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='moveme rt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='No-manual feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='AreaV4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='abstractions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='outine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Simple shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Area V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='detecton ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='edge feat ure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='ed ge feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Retina ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='oves ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='pixel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='physical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='characteristics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='一refers to the process in which the perceived information or the information stored in memory is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='compared, matched, or classified, and a decision or prediction is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' After the spreading activation, the network records the attention features of users and activates their future preferences (Yao et al., 2015) Concepts are in the form of network storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The different concepts are stored in different functional areas in both hemispheres of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The same or similar concepts are stored in same or adjacent regions of brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Specific information of entities in the outside world, such as humans, animals, or tools, is represented by the concept network in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This concept network (A concept is an abstract idea representing the fundamental characteristics of what it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Concept network consists of these abstract concepts) is, in turn, connected with the lexical network from mental lexicon in the left temporal lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Such specific information from mental lexicon will be employed to facilitate SL production during which the SL users generate classifier hand shapes under the guidance of the knowledge and rules of SL classifier predicates (Valli & Lucas, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Here, classifier predicates are made by combining small meaningful unites to create bigger units, the main units being the hand shape and the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This condition implies that findings from brain research can provide knowledge and guidance for the cognitive computational modeling of classifier predicate comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In order to obtain a deep understanding of sign lexical semantics, a cognitive processing model, which is based on the cognitive mechanism of human brain, is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The cognitive processing model would activate the concept network of the associated classifier hand shapes in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Here, classifier predicates differ from traditional linguistic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Traditional methods, such as the syntactic tree, cannot satisfy the generation of the classifier predicates (Huenerfauth et al., 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' DECISION-TREE BASED ALGORITHMIC METHODS The authors use SNs as the knowledge representation and organization mode of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The relationship in semantic networks represents a type of information among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Nodes with a complicated relationship with other nodes contain additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Such nodes require further effort to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Consequently, the authors simulate selective attention (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the processing of visual or auditory input based on whether it is relevant or important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' They selected particular representations to enter perceptual awareness and therefore guide behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Through this process, less relevant information is suppressed by humans using the proposed algorithmic methods to accentuate the nodes selectively and suppress the unessential nodes (Chelazzi et al., 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The emergence of 3-D-based sensors, such as Kinect by Microsoft and Leap Motion (Yao et al., 2014), has improved studies on sign recognition from video-based to 3-D-based sign recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, this transformation makes traditional video-based SL recognition methods inapplicable to 3-D-based SL Perception Memon Judgment Attentional enha ncement and suppress ion 0 Spread activation Interactive activationrecognition technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Large training data are required for valid recognition in 3-D-based SL recognition technologies because of the low operation efficiency of the rotatable joint-based sorter and the matching techniques for sign signal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2014) proposed a decision tree-based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The algorithm aims to achieve a high-precision and real-time performance of SL automatic perception according to the features of Leap Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors adopted this method as the first step of SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Attention Function The authors propose the following attention function: ������������������������ = ∑ ������������������������������������ ������������������������������������ ������������ ∑ ������������������������������������ ������������������������ (1) where ∑ ������������������������������������ ������������������������������������ ������������ denotes the sum of the semantic relation weights around the semantic node x, ∑ ������������������������������������ denotes the sum of all semantic relation weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and ������������������������ represents the activation value on the semantic node x after the spreading activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Semantic Matching Cognitive units in the memory network compete with one another based on certain rules to obtain more of the limited attention resources and more energy for a more active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' SL comprehension supports interactive activation models (Gutierrez et al., 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, judgment is the outcome of the attention competition game in the spreading activation, which is a search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The search algorithm is initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' concepts in a semantic network) with weights or ”activation,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and then iteratively propagating or “spreading” that activation out to other nodes linked to the source nodes processes of the human brain (Crestani, 1997; Preece, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' A semantic matching algorithm based on activation spreading modes is proposed to determine the most appropriate semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Activation starts to spread from the corresponding nodes of the signs presented by the signer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation value of the stimulus node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., signs to be perceived before the start of spreading) must be calculated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In particular, the increment in the interest value of object concept must be calculated, and this concept node must be used as an initial node for the spread study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Activation spreads to the neighboring nodes, which usually have a lower activation value than the source value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, introducing an activation attenuation factor for decreasing activation over the path length in the closed interval [0…1] is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' That is, for every propagation through an edge a loss of activation is considered (Neumann et al., 1993; Rocha et al., 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation spreading process can be expressed as follows: ������������������������(������������ + 1) = ������������������������(������������)������������������������������������(1 − ������������) (2) where ������������������������(������������ + 1) represents that the value is spread from node x to y at time, t+1, ������������������������(������������) represents the activation value that was spread at node x at time, t, ������������������������������������ signifies the link between nodes x and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and δ is an attenuation factor used to describe the energy loss caused in the activation spreading process (Jiang & Tan, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Spreading activation theory states that the activation of human memory “chunks” (the content of any buffer is limited to a single declarative unit of knowledge, called a chunk) is determined by two factors (Anderson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' al., 2004; Anderson, 2013), namely, the use history of the memory chunk and the correlation between the memory chunk and the current retrieval information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' These two factors calculate the weights and determine whether the chunk is activated and selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This assumption has been verified by experimental cognitive psychology, and the calculation model has been established (Roelofs, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors must use moments to express the distance in each activation time with the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Time units may be per hour as a unit and may also be the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' With the day as a unit, we can count the historical value in the previous day as the activation value of the first day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The algorithm based on the theory of memory activation can improve SL understanding, which is sometimes highly sensitive to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' At node y, the largest number of neighbor nodes is (n-1); thus, the maximum of ������������������������(������������ + 1) can be expressed as: ������������������������(������������) = [������������1, ������������2, … , ������������������������]������������ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the initial value of the semantic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Where I1, I2, …, In are these activation value of neighbor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' If activation spreads from a node in many directions, then its adjacent nodes obtain a low activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The adjacent nodes give a feedback value of their resonance energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., contributing structure with the lowest potential energy) to the co-adjacent nodes after they absorb the activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The following equation is therefore used: Iz(t + 1) = Oz(t) + ∑ Ox(t)Λxz(1-δ) all actived x (3) where Oz(t) denotes the activation value of node z at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Given that the quantity of activated information is limited, the nodes that obtain less resonance information are equivalently inhibited and are less likely to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation value distribution in the resonance process conforms to the human attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Attention Game Process Cognitive units in memory network compete with one another by certain rules to increase the possibility of obtaining more human attention resources and more energy that will improve activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This phenomenon is called a game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors use game theory (Myerson, 1997), which is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers and attempts to achieve the largest cognitive gains with the least energy possible, as a reference to simulate the attention enhancement and suppression processes that are selective attention processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In other words, when visually searching for a non-spatial feature or a perceptual feature, selectively enhancing the sensitivity to that specific feature plays a role in directing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When people are told to look for motion, then motion will capture their attention, but attention is not captured by motion if they are told to look for color (Reynolds & Chelazzi, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Activated results consistent with cognitive features can then be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors assume that the game contains n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' ������������������������ ′ and ������������������������ " are the two selectable strategies for node i, and they represent the acceptance and non-acceptance of the change in the attention function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., ������������������������ ′, ������������������������ " ∈ ������������������������ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The corresponding gain can be represented by ������������������������ ′ and ������������������������ " , and ������������������������ ′, ������������������������ " ∈ ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' N nodes are assumed to reach an agreement before participating in the game to introduce the Nash equilibrium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., each node only selects a specific strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors let ������������∗ = (������������1 ∗, … , ������������������������∗) represent the agreement, where ������������������������ ∗ is the strategy of the node i specified in the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Nodes comply with this agreement only when the benefit from complying with the agreement is larger than that from not complying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This agreement constitutes Nash equilibrium if any node abides by this agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Thus, the Nash equilibrium is written as follows: ������������������������(������������������������ ∗, ������������−������������ ∗ ) ≥ ������������������������(������������������������, ������������−������������ ∗ ), ∀si ∈ Si (4) where the combination of strategy ������������∗ = (������������1 ∗, … , ������������������������∗) is a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Given that other nodes select ������������−������������ ∗ = (������������1 ∗, … , ������������������������−1 ∗ , ������������������������+1 ∗ , … , ������������������������∗), ������������������������ ∗ is the optimal strategy of each node i (Myerson, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The attention game process determines whether the nodes need adjustment or need to be changed on the basis of the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation energy distribution will reach a state consistent with the human attentive distribution after adjusting the activated value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Nodes of the spread SNs have their own activation energy threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The source node in the attention game process that represents a presented sign has the maximum activation value O in the present SNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' All equidistant nodes will participate in the game based on the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The nodes with low activation energy (defined as the minimum energy required to start a chemical reaction) of a reaction is denoted by Ea and given in units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol)), threshold must be removed through a screening process to prevent them from participating in the enhancement and suppression processes of activating the most likely node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In the proposed screening, the authors ignore the nodes with a significantly low activation value to be activated in the enhancement process instead of lowering the possibility for other nodes to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The difference between the attentive readjustment in the present attention game process and the previous attentive allocation causes the instability in the overall cognitive structure of users to decrease knowledge credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Thus, a new cognitive structure must be determined at a cost as follows: Cost(t, i, si, ui, SN) = �n−1 ∑ �Ii(t + 1) − Oi(t)� 2 n i=1 (5) where ������������������������(������������ + 1) denotes the activation value that is conveyed from one node at time t+1 to node i, and 0i (t) denotes the activation value of node i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, the total cost is attributed to the change in the activation energy of all nodes in the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The goal of judgment is to achieve the overall optimal gain with a minimal computing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The gain function in the attention game process must then be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' As the optimal strategy for node i, ������������������������ ∗ must minimize the distribution change that refers to the distribution change in the activation values of the overall network changed by the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The amount of spreading activation energy is fixed in the total process of activation spread in the SN; thus, the semantic node energy enhancement must be accompanied by reduced node energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The attention parameters are affected by the overall distribution change in activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The activation energy enhancement increases the impossibility of activating this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Such activation is the ultimate purpose of each node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the node obtains the gain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Accordingly, the gain function is presented as follows: Gain(t, i, si, ui, SN) = �∑ Ix∈{neighbor node}(t+1) num(all x) j=1 −∑ Ox∈{neighbor node}(t) num(all x) j=1 �(1−δ) num(all x) (6) where SN represents the current semantic network, num(all x) represents the number of neighbor nodes x of node i, ∑ Ox∈{neighbor node}(t) num(all x) j=1 denotes the sum of the activation value that was spread of all node i neighboring nodes at time t, the gain function is expressed as the attention gain of neighbor nodes x of node i after the enhancement and suppression processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' it represents the benefit a node gets by unilaterally changing their strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The utility function of the attention game process can be determined as follows: Max�������������������������(������������, ������������������������ ∗, ������������−������������ ∗ )� = Gain(������������, ������������, ������������������������, ������������������������, ������������������������������������) − Cost(������������, ������������, ������������������������, ������������������������, ������������������������������������) (7) where ������������������������ ∗ is the optimal strategy of each node i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' ������������−������������ ∗ is the strategies set of other nodes except node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' only when ������������������������(������������, ������������������������ ∗, ������������−������������ ∗ ) reaches the maximum, ������������������������ ∗ is a Nash equilibrium of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The utility of the other nodes will be affected by the decision of all other nodes because of the fact that the total quantity of activation energy is fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., attention is limited) in the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When each node selects a decision for itself, it also considers the possible decision of other nodes and selects the “Nash equilibrium point” with maximum utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This scenario is consistent with classical game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors select a Nash equilibrium decision for each node through the utility function of the attention game that is defined by Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' METHODS Data Sets and Experimental Settings All data from the authors’ experiments are obtained from the Tsinghua University–Chinese SL Corpus (TH–SLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The data mainly comprise SL expressions of idiom stories and life fragments of deaf students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' No automatic annotation software based on videos is currently available because the annotation process for SL videos is time consuming and requires expert knowledge in dual language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., Chinese language and Chinese SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Video annotation is also time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Specifically, it takes about 30 hours for the annotation RTF (real-time factor) of a parliamentary speech (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., One hour of speech requires 30 hours of annotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, the annotation RTF (real-time factor) for a full annotation of all manual and non- manual components of an SL video can reach up to 100 hours (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, such a corpus is significantly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For example, the Aachen Boston database contains American SL and has annotated 201 English sentences (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors spent a year collecting more than 2000 sentences, but only 416 sentences containing 2496 signs were marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors asked 20 deaf students to select 300 sign pairs from 2469 annotated signs in TH–SLC and to judge the relevance of the sign pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The correlation values range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For convenience, a five-point scale is used to assess the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The sign pairs were obtained using a marked correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors establish an SN based on the word similarity computing method of HowNet (Liu & Li, 2002) to determine the connection weight of the network to validate the effects of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors introduce the continuous bag-of-words (CBOW that predicts the current word from a window of surrounding context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The order of context words does not influence the prediction (CBOW assumption) model (Mikolov et al., 2013), and the HowNet (Liu & Li, 2002) method as the baseline methods using the same recommended parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The efficiency of the utility function of the attention game process is evaluated in terms of word correlation computation, and the model complexity is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Word Relatedness Computation Each model in this task needs to compute the semantic correlation of the given sign pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The correlation between the experimental results of the model and human judgment reflects upon the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors selected 290 signs for the closed set and 10 signs for the open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Spearman’s correlation between model correlation score and human judgment correlation score was calculated for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Spearman correlation coefficient is defined as the Pearson correlation coefficient among the ranked variables (Myers & Well, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For a sample of size N, original data ������������������������, ������������������������ are converted into grade data������������������������, ������������������������, the correlation coefficient ρ is defined as follows: ρ = 1 − 6 ∑ di 2 n(n2−1) (8) where the difference between the observations of the two variable levels is set as ������������������������ = ������������������������ − ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' If there is no duplicate value in the data, and two variables are completely monotonic correlation, the Spearman correlation coefficient is +1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' RESULTS For CBOW, the correlation scores of the two words are calculated using the cosine similarity of word embedding (Mikolov et al., 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The evaluative results of the baseline methods and the proposed SNM method in the closed test and in all test sets are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Evaluative results Data Set Closed Test All Test Sets (Including Open Test) Spearman’s Rank Correlation Coefficient Method 290 pairs 300 pairs CBOW (baseline method) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='4843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='4869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='4136 Word similarity computing based on HowNet 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6052 Proposed SNM method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='7063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6437 The evaluation results show that the proposed SNM method is better than the baseline method in 290 and 300 word pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This finding indicates that the cognitive mechanism of sign comprehension is essential to understanding the meaning of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The internal structure, such as location, orientation, hand shape, and movement, contains rich semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, deep learning methods, such as CBOW, consider the external context, but ignore the internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Using the computing method of word similarity based on HowNet results in only a rough semantic computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For example, adding 10 new sign pairs negligibly changes the performance of these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In other words, these methods can still handle new signs with improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic correlation of these new sign pairs calculated by the proposed method is close to human judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 3 shows the quantitative analysis of the attention game process for two signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Each hand shape of the two signs has at least 20 related semantic lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The stimulus information and permutation of each node are shown in the first and second columns from high to low according to the activated value after the activation spreading process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Only 10 semantic lexicons that are maximally activated are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The permutation of each node is shown in columns three to seven from high to low according to the activation value after the end of the first to fifth attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The top 10 lexicons are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic lexicons in the blue background rank high after the games, those in the green background rank low after the games, and those in the white background are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Examples of attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The semantic lexicons in the blue background rank high after the games, those in the green background rank low after the games, and those in the white background are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This trend shows that the ranking of other semantic lexicons below slightly changes after the semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This condition is due to the source that corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 3 also shows that significant changes occur during the ranking of the semantic lexicons in the first and second instances after the first several games, whereas only a few changes occur in the following stimulus games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This trend shows that the ranking of lower semantic lexicons slightly change after the semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This condition is due to the source that corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Attention is also assigned to other nodes in accordance with the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Humans reach a steady state after thinking about problems constantly, and the result negligibly changes if they rethink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Nearly no change is observed in the result after several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Several semantic lexicons related to the signs are contained in the text set; thus, a few possible changes occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The result of the attention game model conforms to human cognitive rules to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Attention is also assigned to other nodes in accordance with the attention game process (here, efforts have been made in modeling according to the mechanism of human attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The result of the SNM conforms to human cognitive rules to a certain degree (Gutierrez et al., 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For example, the authors assume that deaf people understand the signs shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Deaf people usually search for many familiar and specific nouns or signs in a spreading activation mode to comprehend classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' After all activated values are calculated; the activated nodes are graded and sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' A high-activated value of the node indicates the importance of the interested object or concept represented by the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This shows that deaf people are familiar with the concept node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Similar to the attention game process shown in Figure 3, ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Exploitation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Guide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Blind ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='lie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='lie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='lie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Recalcitrant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Blind ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Opponent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Resistance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Resistance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='Resistancethe high-ranked semantic lexicon is a cat or dog after several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This result shows that the most common subjects for deaf people are typical subjects that represent classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' DISCUSSION Compared with that of existing models, the complexity of the proposed model is reflected mainly on the computational cost of the memory stage and the judgment stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the computational cost of spreading activation and the attention game at time (t + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The cost is a dynamic value and related to two factors, namely, the activation state of the current sign and the current cycle as the first activation of the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, the value changes regardless of the choice of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This outcome is consistent with the strong dynamics of sign information, which can reflect the influence of information in different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In the memory stage, the time complexity of computing ������������������������(������������) is unity; thus, the time complexity is related to the total amount N of activation energy and cycle times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The time complexity of each activation in each cycle is n × 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Space complexity is the storage space of each node and the semantic relation weight according to semantic similarity (semantic similarity can be estimated by defining a topological similarity, by using ontologies to define the distance between terms/concepts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, unlike the general model such as cobweb theorem model and vector space model, where the SNM increases the overhead in time complexity and space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The model also increases the matching time of query nodes and weights in the current activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, the overhead at this time can provide more effective results than an invalid spreading and can be accepted by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In the judgment stage, when the node selects the game strategy to change its activated energy value, the convergence speed of adjusting the cognitive benefits to its own utility maximum “Nash equilibrium” is an important measure of evaluating the SNM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the cycle times of an attention game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' For the attention game, the Nash decision of different semantic nodes must minimize the change cost of the activation energy distribution of the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The Nash equilibrium point decision for each node is selected using the utility function defined in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This process is repeated until the overall network activation energy distribution change is less than the specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The node needs to solve n-order nonlinear equations in every cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Therefore, the performance of the convergence speed of the SNM is indicated by the number of game cycles that the network requires to reach the Nash equilibrium point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the computing times of calculating the corresponding equation by each node in a game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The square root of the sum of the variance of activation value ������������������������(������������ + 1) of each adjusted node is directly reflected by the rate of convergence in the game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' To verify its effectiveness, the attention game model is compared with the traditional model in terms of load balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In the traditional method, the activation value of each node is certain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='e., the value is not enhanced or inhibited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The experimental results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The results show that the load balance performance of the attention game model is better than that of the traditional model because the attention game model adjusts the activation strategy after the activation of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When the change cost of the energy distribution of the entire network activation is larger than the specified threshold, the human brain adjusts the strategy to inhibit the activation energy value in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In doing so, the free competition and distribution of attention for each node according to the attention game model can be assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The result is obtained through the overall competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The load of attention of the network is balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The traditional model assumes that the activation energy value of each node is certain because the brain activation energy resource amount is constant in a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The brain selects the node with a low activation energy value and performs the allocation of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This allocation causes the attention load of several nodes to be excessively large or unutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Comparison of load balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The load balance performance of the SNM is better than that of the traditional model because the SNM adjusts the activation strategy after the activation of each node The proposed SNM model used Nash equilibrium to simulate the energy activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In order to quantitatively analyze the effects of Nash equilibrium, the authors compared the SNM with the cobweb theorem model (Pashigian, 2008) in terms of different activation energy amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The cobweb theorem is expressed as follows: ������������(������������ + 1) = ������������(������������) + ������������ ��������������������������(������������)� − �������������������������′(������������)�� (9) where r is the adjustment parameter of the activation value, �������������������������(������������)� is the activation function of a node, ������������(������������) is the activation value at time t, �������������������������′(������������)� is the attention allocation function, ������������′(������������) is the expectation activation value at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' and �������������������������(������������)� − �������������������������′(������������)�is the excessive demand function that represents the actual gaps between the activation value and activated allocated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' A large gap indicates a high activation value of the Nash Equilibrium of node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The parameter (r) indicates the actual speed and strength of adjusting the activation value according to the attention distribution condition in the last moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When r > 0 it indicates that the adjustment direction of the activation value is consistent with the direction of the demand function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The amount of activation energy Ea is assumed to be 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 5 shows the result of comparing the attention utilization between the game model and the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The attention amount (attention is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information, while ignoring other perceivable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Attention amount refers to as the allocation size of limited processing resources), is less than 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' If the attention amount is insufficient, then attention resources can only meet part of the node demand, and the resource utilization rate of the SNM will become higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When attention supply exceeds the demand of a node, the cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The SNM meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Comparisonofloadbalance 9 8 nodes hhhl 6 Numberof L 4 3 1 0 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='1 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='3 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='4 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='5 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='7 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='8 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='9 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='10 Numberofactivationenergy amount Iattentiongamemodel cobwebtheoremmodeFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Comparison of activation energy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' After the change in the initial value of the activation energy, the number of iterations increases depending on the difference between the initial activation energy value in the cobweb model and the balanced energy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The iteration of the attention game model can be adjusted according to the difference in the activation energy between supply and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' A sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the supply and demand Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in the SNM, and the activation energy is 120 kJ/mol in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' If the initial value of the activation energy is changed, then the initial activation energy value of the cobweb model is higher than the energy equilibrium value and requires abundant cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The SNM in each cycle can adjust the activation energy according to the variance of the activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The variance and adjustment range are large, and the SNM eventually reaches the Nash equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' When the supply falls short of demand, attention resources can only meet the demands of several nodes, and the resource utilization rate of the SNM becomes higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' If the supply exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet only part of the demands of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' However, the SNM can meet the demands of all nodes, and its resource utilization rate is higher than that of the cobweb model 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='2 40 60 80 100 120 Attention Resource Utilization(percentage) Amount of activation energy(KJ/mol) Comparison of activation energy values attention game model cobweb theorem model CONCLUSION The authors presented a new model for SL comprehension based on spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This process uses game theory to simulate the human attention suppression and enhancement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This process also joins the forgetting function of human memory traces to compute the initial state of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Memory is encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and temporal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Although the semantic network provides a functional view of how knowledge may be organized in the brain, it does not provide a clear model of how semantic memory might be presented in the brain (see Cacha et al., 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Spreading activation reveals that information can be stored in SNs for a long time, in which a network node is a linguistic concept and the nodes are connected through the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' An algorithmic method is proposed according to selective functions, and its effectiveness was verified using an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The results show that the proposed method improves the performance of SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' ACKNOWLEDGMENT The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for helping in stimulus preparation and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' This paper forms an expanded and revised version of a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' The authors are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Raymond Chiong, and two anonymous referees for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Conflict of Interest The authors of this publication declare there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='signwriting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='org/archive/docs7/sw0636_SignWriting_Alphabet_Manual_2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content='pdf Yao, D., Jiang, M., Abulizi, A., & You, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2014, October) Decision-tree-based algorithm for 3D sign classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In 2014 12th International Conference on Signal Processing (ICSP) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' 1200- 1204).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Yao, D., Jiang, M., Abulizi, A., & Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2015, July).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Cognitive computing on Chinese Sign Language perception and comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' In 2015 IEEE 14th International Conference on Cognitive Informatics & Cognitive Computing (ICCI* CC) (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' 90-97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Yao, D., Jiang, M., Huang, Y., Abulizi, A., & Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Study of sign segmentation in the text of Chinese sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Universal Access in the Information Society, 16(3), 725-737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Valli, C., & Lucas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' (2000) Linguistics of American Sign Language: An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} +page_content=' Gallaudet University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kb_51/content/kb_51.pdf'} diff --git a/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/2301.03455v1.pdf.txt b/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/2301.03455v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fceb7e17c78defef48968cc294cae7e3f5dc424 --- /dev/null +++ b/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/2301.03455v1.pdf.txt @@ -0,0 +1,546 @@ + + + + + + + + +Datenkompetenz im Physikstudium – ein Erfahrungsbericht +Michael Krieger, Heiko B. Weber +Lehrstuhl für Angewandte Physik, Department Physik, Friedrich-Alexander-Universität Erlangen- +Nürnberg +Christopher van Eldik +Erlangen Centre for Astroparticle Physics, Department Physik, Friedrich-Alexander-Universität +Erlangen-Nürnberg + +Daten werden als entscheidende Ressource des 21. Jahrhunderts betrachtet1. In der Physik gibt es +eine große Bandbreite von wohlstrukturierten, jahrelangen digitalen Datenströmen aus +Großexperimenten (z. B. Astronomie, Teilchenphysik) bis hin zu manueller und hochgradig individuell +geprägter Datenaufnahme in Laborexperimenten (z. B. Festkörperphysik, Optik). Wären alle +Forschungsdaten der Vergangenheit systematisch erfasst und offen zugänglich, könnten unter +Umständen völlig neue Erkenntnisse aus deren Analyse resultieren. Dabei sind nicht nur die +Rohdaten selbst, sondern insbesondere auch die Metadaten von größter Bedeutung. Letztere +beschreiben Messaufbau, experimentelle Parameter, Einheiten, Kontext, Versionen, etc. und +erzeugen damit ein möglichst vollständiges und nachvollziehbares Bild. Je reichhaltiger und +systematischer Metadaten erfasst werden, umso besser wird man aus der Kombination +verschiedener Datensätze etwas lernen können. +Als Leitgedanke dieses sogenannten Forschungsdatenmanagements dienen die FAIR-Prinzipien: +Daten sollen auffindbar (findable), offen zugänglich (accessible), standardisiert und dialogfähig +(interoperable) sowie wiederverwertbar (reusable) gespeichert werden. Man kann sich die weitere +Verwendung vielleicht so vorstellen, dass eine WissenschaftlerIn in 10 Jahren einen Computer durch +all unsere Daten surfen lässt und dabei Zusammenhänge entdeckt, die den einzelnen +Forschungsgruppen verborgen geblieben sind. Bund und Länder haben mit der groß angelegten +Initiative für eine Nationale Forschungsdateninfrastuktur (NFDI)2 erhebliche Fördermittel +bereitgestellt, um die wissenschaftlichen Disziplinen diesbezügliche Konzepte ausarbeiten zu lassen. +Vermitteln unsere Physikcurricula in dieser sich verändernden Welt die passenden +Datenverarbeitungskompetenzen? +In der Physik besteht breiter Konsens, dass die Studierenden fundierte Kenntnisse in Analysis, +linearer Algebra, etc. haben müssen, um im Verlauf des Studiums physikalische Konzepte zu +verstehen. Man hat sich hier weltweit auf einen gewissen Kanon geeinigt. Unser Eindruck ist, dass ein +solcher Konsens bezüglich Datenkompetenzen nicht existiert. Beim Erstkontakt mit Daten, also +insbesondere in den Praktika der Experimentalphysik, werden zwar elementare Datenevaluation und +systematische Dokumentation eingeübt, diese genügen aber nicht den steigenden Ansprüchen der +Forschung und der Berufspraxis, sich zunehmend mit der Analyse großer Datenmengen zu befassen. +Es scheitert oft schon an einem Konsens über eine geeignete Einstiegsprogrammiersprache. +Am Department Physik der Friedrich-Alexander-Universität Erlangen-Nürnberg haben wir in den +letzten Jahren kleinere Anpassungen im Physikcurriculum vorgenommen, die wir in diesem Artikel +vorstellen. Datenkompetenz wurde früh im Bachelorstudium platziert, woraus sich erhebliche +Vorteile für den weiteren Studienverlauf ergeben haben. Die Autoren können sich des Eindrucks + +1 https://ec.europa.eu/info/sites/default/files/communication-european-strategy-data-19feb2020_en.pdf +2 https://www.nfdi.de/ + + + + + + + + + +nicht erwehren, dass die Studierenden in puncto Datenkompetenz schnell auf die Überholspur +gehen; wir sehen bereits jetzt in unseren Arbeitsgruppen, dass sie sich als treibende Kräfte hin zu +einem modernen Forschungsdatenmanagement erweisen. +Obligatorisches Datenverarbeitungspraktikum +Wie führt man Physikstudierende möglichst frühzeitig an computergestützte Datenauswertung +heran? Wie begegnet man der Herausforderung, dass manche Studierende bereits aus der Schule +oder durch eigene Aneignung Programmiererfahrung vorweisen können, für andere das Thema aber +wie ein Buch mit sieben Siegeln erscheint? +Unsere persönliche Erfahrung ist, dass sich das Erlernen einer Programmiersprache am einfachsten +durch begleitetes Learning-by-doing anhand physikalischer Fragestellungen gestalten lässt. Die +intensive eigene Beschäftigung in Form kleinerer Programmieraufgaben fördert ein tiefes +Verständnis und legt die Grundlage für die anstehenden Herausforderungen in der +Datenauswertung. Das Bearbeiten von Aufgaben in Teams stärkt den gerade zu Beginn des Studiums +so wichtigen Austausch mit KommilitonInnen und fördert früh den Aufbau einer wissenschaftlichen, +lösungsorientierten Diskussionskultur. Dabei dürfen vor lauter Programmieren die Inhalte des +Fachstudiums nicht in den Hintergrund treten. Nur wenn der Anwendungsbezug klar ersichtlich ist, +werden sich ein nachhaltiger Kompetenzgewinn und die Motivation einstellen, das Gelernte im +weiteren Studienverlauf zu nutzen. +Wir haben in diesem Sinne – erstmals zum Wintersemester 2018/19 – ein verpflichtendes +Computerpraktikum „Datenverarbeitung in der Physik“ (DV-Praktikum) in das Curriculum unseres +Bachelor-Studiengangs aufgenommen. Im DV-Praktikum werden unter Anleitung von Tutoren +Programmieraufgaben im Team bearbeitet und anschließend elektronisch in Form eines fertigen +Programms eingereicht und bewertet. Begleitet wird das wöchentliche zweistündige DV-Praktikum +von einem interaktiven Online-Kurs, durch den die Studierenden genau die Grundlagenkenntnisse +erwerben, die für die Bearbeitung der nächsten Praktikumseinheiten benötigt werden. Dabei +beschränken sich die Kursinhalte nicht auf das Erlernen einer Programmiersprache. Das +Hauptaugenmerk liegt vielmehr in der Vermittlung grundlegender Datenverarbeitungs- und +Datenanalyse-Kompetenzen. So enthält der Kurs beispielsweise einführende Module in +Messunsicherheiten und Fehlerfortpflanzung, Interpolation, Fitten von Daten, Statistik-Funktionen +und Monte-Carlo-Simulationen, Fourier-Transformationen, numerische Integration, Lösung von +Differentialgleichungen sowie grafische Aufbereitung von Ergebnissen. Auch ein einführendes Kapitel +zu „Guter wissenschaftlicher Praxis“ ist Teil des Kurses. Alle diese Aspekte sind selbst für +computeraffine Studierende neu und fordern damit alle Teilnehmenden gleichermaßen. +Als Programmiersprache kommt Python zum Einsatz. Die Grundlagen von Python sind auch für +Anfänger schnell zu erlernen, und die Erfahrung zeigt, dass gute Python-Kenntnisse das Erlernen +anderer gebräuchlicher Sprachen (z.B. C++, Java) erheblich vereinfachen. Wichtiger noch für unsere +Zwecke: Neben der umfangreichen Python-Standardbibliothek stehen mit „numpy“3, „scipy“4 und +„matplotlib“5 frei zugängliche und gut kuratierte Basis-Bibliotheken zur numerischen +Datenverarbeitung, Statistik und grafischen Aufbereitung zur Verfügung, die in der Physik- +Community und weit darüber hinaus zur Anwendung kommen. Im DV-Praktikum werden konsequent +Jupyter-Notebooks6 verwendet; sie ermöglichen, Python-Code innerhalb eines Webbrowser-Fensters + +3 https://numpy.org +4 https://scipy.org +5 https://matplotlib.org +6 https://jupyter.org + + + + + + + + + +zu erstellen und laufen zu lassen. Dabei werden Python-Code und Ergebnisse (Text und Grafiken) +gemeinsam in einem Dokument dargestellt und gespeichert und können auf diese Weise sehr einfach +mit anderen Interessierten geteilt werden. Jupyter-Notebooks verfügen zudem über dieselben +umfangreichen (Text-)Formatierungs-Werkzeuge, die in Wikis zur Anwendung kommen, inklusive +Formelsatz, Tabellen und Einbindung von Bildern (siehe Abb. 1). +Nach einer sehr erfolgreichen Pilotphase haben wir das DV-Praktikum auf besonderen Wunsch der +Studierenden bereits im ersten Fachsemester verankert. Um dafür Platz im dicht gedrängten +Stundenplan zu schaffen, beginnt das Grundpraktikum deshalb bei uns erst im zweiten Semester. +Dies bietet für unsere Studierenden eine perfekte Gelegenheit, die erworbenen Datenverarbeitungs- +Kompetenzen gewinnbringend nicht nur bei der Auswertung der Praktikumsversuche einzusetzen, +sondern auch im gesamten Studium. +Spielerischer Zugang zur Physik in Kursvorlesungen +Mit der Entscheidung für die Programmiersprache Python und der Einbindung ins Erlanger +Physikcurriculum kann man in Vorlesungen, Übungen und Praktika auf diese Kompetenzen +zurückgreifen. Mit jeder Anwendung wird dieser Kenntnisstand weiter vertieft, das gilt +gleichermaßen für Studierende und Lehrende. Das ermöglicht völlig neue Aufgabentypen, bei denen +man komplexere, manchmal auch realitätsnähere oder aktuellere physikalische Probleme behandeln +kann, als wenn man nur auf analytische Kompetenzen zurückgreifen könnte. Jeder Dozent kann für +sich selbst entscheiden, wo analytische Verfahren und wo numerischen Verfahren didaktisch sinnvoll +sind. +Erhebliche Vorteile hat die Numerik bei der Visualisierung. Dies ermöglicht eine zweite +Herangehensweise an physikalische Probleme: die spielerische. Hierbei lassen sich Parameter schnell +variieren und die Ergebnisse in verschiedenen grafischen Auftragungen darstellen – in dieser Qualität +und Geschwindigkeit ist dies mit traditionellen Methoden innerhalb einer Übung nicht zu schaffen. +Bei manchen wird dabei Neugierde geweckt. Beispiel Fourier-Analyse: Was passiert, wenn man im +Signal nur gerade Terme zulässt, oder nur ungerade? Man würde sich wünschen, dass Studierende +befähigt sind, solche Fragestellungen sowohl analytisch als auch numerisch zu lösen. Besonders +profitieren die Studierenden, die den Vergleich anstellen. +Ein anderes Beispiel betrifft das Misstrauen aufmerksamer Studierender gegenüber Näherungen, die +für analytische Lösungswege häufig unvermeidbar sind. Numerisch quantifiziert und visualisiert lässt +sich oft viel leichter nachvollziehen, ob und unter welchen Umständen diese Näherungen gut +begründbar sind. Wir sind der Überzeugung, dass Lehrveranstaltungen der experimentellen und der +theoretischen Physik gleichermaßen von dieser Art spielerischer Datenkompetenz profitieren. +Elektronisches Laborbuch im Praktikum +Für ein zukunftsweisendes Forschungsdatenmanagement ist das traditionell handgeschriebene +Laborbuch nicht geeignet. Die nachhaltige Verfügbarkeit von Daten und Metadaten lässt sich nur +durch sorgfältige elektronische Protokollierung z. B. in Form eines elektronischen Laborbuchs +erreichen; so wird von Beginn an eine strukturierte Datenerfassung ermöglicht. +Unsere Universität hat sich im Rahmen eines Pilotprojektes für ein open-source elektronisches +Laborbuch (ELN) entschieden (openBIS7; siehe Infobox). Dieses wird aktuell in verschiedenen +Forschungsgruppen ausprobiert und evaluiert. + +7 https://openbis.ch/ + + + + + + + + + +Wir haben uns entschlossen, als Vertiefung des DV-Praktikums dieses ELN-System im Physikstudium +einzuführen. Geeignet ist dafür das bestehende Elektronikpraktikum mit seinen 11 Versuchstagen im +4. Fachsemester. Dieses bearbeitet ein Programm von einfachen elektrischen Messungen über +komplexer werdende analoge und digitale Schaltungen bis hin zu Microcontrollerprogrammierung +und Analog/Digital-Wandlung. Bislang gaben wir eine Struktur vor, die Messdaten auf vorgegebenen +Pfaden eines Netzwerklaufwerks zu speichern, wohingegen die Experimentvorbereitung und die +Metadaten sorgfältig, aber ohne Formvorgaben in einem gebundenen Papier-Laborbuch +dokumentiert wurden. Im Frühjahr 2022 vollzogen wir den Übergang zum ELN, der technisch +gesehen relativ einfach war, da die Arbeitsplätze sowieso mit einheitlichen Computern ausgestattet +waren. Die Lernziele und Inhalte des Elektronikpraktikums blieben unberührt. +Wir haben nur einige wenige Vorgaben implementiert: Beim ersten Login sehen die +PraktikumsteilnehmerInnen ihren Laborbuch-space, auf den die 2er-Teams gemeinsamen Zugriff +haben. Innerhalb dessen sind bereits alle Versuchstage als projects und darunter die einzelnen +Experimente und Aufgaben als experiments und experimental steps angelegt. Die kursiv gesetzten +Begrifflichkeiten sind von openBIS vorgesehene Standardelemente. Die gesamte Struktur wird von +uns per Python-Script angelegt. +Die elektronischen Laborbuchseiten (experimental steps) enthalten drei vorgegebene Freitextfelder +für (i) die Ziele des Experiments, (ii) die Experimentbeschreibung und (iii) die Dokumentation der +Resultate. Messdaten können bei kleineren Messreihen entweder per Hand in Tabellen erfasst +werden oder als Rohdatendatei in das ELN geladen werden. Auf diese Weise werden beschreibende +Metadaten und Rohdaten von den Studierenden während des Experimentierens an einem Ort +zusammengeführt (siehe Abb. 2). Der besondere Vorteil des ELN wird bei Nutzung der +Pythonschnittstelle deutlich. Aufbauend auf ihre mittlerweile dreisemestrige Pythonkompetenz +können unsere Studierenden hiervon unmittelbar profitieren. Mit einer einzigen Programmzeile +lassen sich die Daten direkt aus dem ELN zur weiteren Auswertung und Visualisierung laden (vgl. Abb. +3). Das geht im Praktikum genauso wie zu Hause. Hierzu wurde ein Pythonmodul8 zur Verfügung +gestellt, das Befehle der PyBIS-Bibliothek kombiniert. +Die Studierenden fanden schnell heraus, wie man Fotos von ihrem Smartphone in das ELN einbinden +kann. So wurden handgezeichnete Skizzen oder dokumentierende Aufnahmen der +Versuchsaufbauten integriert. Das gleiche gilt natürlich auch für Screenshots relevanter Passagen aus +externen Dokumenten, z. B. Benutzerhandbüchern von Messgeräten. Die Studierenden haben das +elektronische Laborbuch sofort akzeptiert und ausschließlich genutzt, obwohl die Verwendung des +ELNs in der Pilotphase freigestellt war. +Natürlich bietet dieses neue Medium auch Gelegenheit, die Arbeitsabläufe sinnvoller zu gestalten. +Ein Beispiel: Die Praktikumsvorbereitung erfolgte bislang basierend auf einem Fragenkatalog im +Laborbuch. Im Verlauf des Praktikums wurde diese zusammenhängende Darstellung aufgrund der +chronologischen Notation in einem Papierbuch oft nicht mehr beachtet. Im ELN lassen sich +Vorbereitung und Experimentdurchführung viel besser verzahnen, in dem die Antworten auf die +Vorbereitungsfragen und zusätzliche Notizen in die entsprechenden Freitextfelder (Ziele und +Experimentbeschreibung) der entsprechenden Experimente eingetragen werden. Vorbereitung, +Fragenkatalog und Antworten der Studierenden erscheinen am Ende in einer wohlgeordneten +Darstellung, die für die Studierenden sehr viel besser lesbar ist. Es sei auch erwähnt, dass die +Betreuer sich durch ein Pythonskript eine exzerpierte Version der Vorbereitung ansehen können, um + +8 https://github.com/FAU-PHYSIK-EP/load_openBIS_data_into_python + + + + + + + + + +sich einen Überblick über den Vorbereitungsstand der Praktikumsgruppen zu verschaffen, ohne sich +immer wieder durch die einzelnen Experimente im ELN klicken zu müssen. +Gegenüber traditionellen Laborbüchern erlaubt ein ELN, Nachbesserungen, Plagiate oder +Fälschungen leichter zu verhindern. Die Studierenden wissen natürlich bereits von ihrer +Internetnutzung, dass jede Handlung in einem elektronischen System prinzipiell genau +nachvollziehbar sein könnte. Somit sollte gar nicht erst der Gedanke aufkommen, Daten zu +“frisieren”. +Die Betreuer haben wiederum über den Rohdatenzugriff eine Möglichkeit, die Versuchsaufbauten im +Verlauf der Zeit bei etwaigen Ergebnisabweichungen zu überprüfen und so technischen Defekten auf +die Schliche zu kommen. +Nach Abschluss einer Lehrveranstaltung sollten Studierende eine vollständige Dokumentation für +sich behalten können. Neben der Weiternutzung des ELN im Fortgang des Studiums gibt es auch die +Möglichkeit, die ELN-Inhalte zu exportieren. +Wirkung auf die Forschungsebene +Nachdem die Studierenden den Umgang mit dem ELN im 4. Semester erlernt haben, bringen sie ihre +Kompetenzen während der Forschungsphase zur Bachelor- oder Masterarbeit unmittelbar in die +Arbeitsgruppen ein. Dort muss die Erfassung der Forschungsdaten allerdings wesentlich +strukturierter und unmittelbar an die Arbeitsabläufe in der jeweiligen Forschungsgruppe angepasst +erfolgen. Diese Aufgabe steht der großen Mehrheit noch bevor und ist Gegenstand der NFDI- +Initiative. Auch wir sind gerade dabei, unsere Arbeitsabläufe in der Forschung systematisch zu +erfassen und abzubilden. Dabei helfen Konzepte aus der Informatik, z. B. Entity-Relationship- +Diagramme, wie sie für Datenbanken erstellt werden. Hier hilft der ELN-Einsatz im Praktikum: Wir +werden Konzepte mit stärker strukturierten Laborbüchern zunächst im Praktikum erproben und +evaluieren. Aufgrund der abgeschlossenen Praktikumseinheiten sind dabei kurze Lern- und +Innovationszyklen möglich, deren Erfahrungen wir dann in die längerfristig angelegte und +komplexere Dokumentation der Forschung in unserer Arbeitsgruppe einbringen können. +Wir sehen es als ein Erfolgsrezept, frühzeitig im Physikcurriculum zeitgemäße Datenkompetenz zu +vermitteln. Die durchgängige Verwendung einer einheitlichen und in den Naturwissenschaften +mächtigen Programmiersprache wie Python verschafft neue Möglichkeiten und verändert die Lehre +und die Forschung nachhaltig. Wir sehen einen unmittelbaren Einfluss auf den Übungsbetrieb und +den Praktikumsbetrieb im Bachelor- und Masterstudium sowie zusätzliche Kompetenzen der +Studierenden, die sie in der Forschungsphase zur Bachelor- und Masterarbeit in die Arbeitsgruppen +einbringen können. Darüber hinaus sehen wir einen sehr positiven Einfluss auf die sich verändernde +Forschungslandschaft, die viel stärker als früher von einem aktiven Forschungsdatenmanagement +geprägt sein wird. + +Heiko B. Weber und Michael Krieger forschen am Erlanger Department Physik an +festkörperphysikalischen Fragestellungen, leiten das Elektronikpraktikum und arbeiten aktiv im NFDI- +Konsortium FAIRmat mit. +Christopher van Eldik forscht am selben Department an astroteilchenphysikalischen Fragestellungen. +Als Studiendekan hat er die Umstellung des Curriculums mitgeprägt und leitet die Lehrveranstaltung +“Datenverarbeitung in der Physik”. + + + + + + + + + + + +Abbildungen: + +Abb. 1: Beispielanalyse zur Bestimmung der Fallbeschleunigung aus dem +Datenverarbeitungspraktikum im 1. Fachsemester. In solchen interaktiven Jupyter-Notebooks +werden Python-Code, Dokumentation und Abbildungen auf einer Ebene dargestellt. + +AnalysederMessdaten +Wir laden die Daten, extrahieren die Zeiten von den beiden Zahlern in separate Arrays und korrigieren direkt den Zahler-Offset. +data=np.loadtxt('g_measurements.dat') +# readinmeasured times,correct forcalibrationoffset +t1 = data[:,0] - t1_calib_offset +t2 = data[:, 1] - t2_calib_offset +N = data.shape[0] +t1_mean = np.mean(t1) +t2_mean =np.mean(t2) +delta_t1_mean = np.std(t1)/ np.sqrt(N) +delta_t2_mean=np.std(t2)/np.sqrt(N) +print('mean time1:((:.5f)+/-(:.5f))s'.format(t1_mean,delta_t1_mean)) +print('mean time2:({:.5f)+/-(:.5f})s.format(t2_mean,delta_t2_mean)) +meantime1:(0.16221+/-0.00056)s +mean time 2:(0.32598+/-0.00073)s +PlottenderMesswerteverteilungen +fig,ax=plt.subplots(figsize=(9,6)) +#Histograms of corrected times +ax.hist(t1,fc='lightblue',label='counter 1') +ax.hist(t2,fc='orange',label='counter 2') +#Histograms of uncorrected times +ax.hist(t1 +t1_calib_offset,histtype='step',color='blue',label='counter 1 (uncorrected)') +ax.hist(t2+t1_calib_offset,histtype='step',color='brown',label='counter2(uncorrected)') +ax.set_xlabel('time[s]') +ax.set_ylabel('number of entries') +ax.legend() +plt.show() +30 +25 + 20 +of entri +10 +5 +counter 1 +counter 2 +counter 1 (uncorrected) +counter 2 (uncorrected) +0.15 +0.20 +0.25 +0.30 +0.35 +time [s] + + + + + + + + +Abb. 2: Ansicht des ELN während des ersten Versuchstages im Elektronikpraktikum: in der +Baumansicht haben wir den Praktikumsablauf vorgezeichnet. Die Studierenden haben während der +Vorbereitung auf das Praktikum bereits Ziele und Durchführung der einzelnen +Praktikumsexperimente (Teilaufgaben) inklusive Skizzen in die entsprechenden Eingabefelder notiert. +Ergebnisse und Beobachtungen werden während des Praktikums im gleichen Formular erfasst. +Hierfür stehen ein Freitextfeld, eine Tabelle für numerische Messdaten sowie der Dateiupload für +Messdatendateien zur Verfügung. + +X +口 +C +8 +https:/physics.openbis.data.fau.de/openbis/webapp/eln-lims/ ☆ +D += +Q Glot +Experimental Step: Aufgabe 1.1 +12 +t ++ New- +@ Upload +More . +7 +Lab Notebook + Others +Experimentalgoals: +A +1415000000EscSose22 +Bestimmung des Innenwiderstands des Digitalmultimeters im Modus Gleichstrommessung +Versuchstag1 +Aufgabe1 ++Aufgabe 1.1 + Aufgabe 1.2 +Experimental description: +Aufgabe 2 +VerbindungdesLabornetzgeratausgangs(Spannungsausgang) +mit dem Digitalmultimetereingang (Stromeingang); die angelegte +Itinete +Aufgabe 3 ++ +Aufgabe 4 +Spannung fallt direkt uber dem Innenwiderstand des +Aufgabe 5 +Digitalmultimeters ab ++ +Aufgabe6 +MessgroBen: Strom (Digitalmultimeter), Spannung ++ +Aufgabe 7 +(Labornetzgerat) +Labornetz gerat +Aufgabe 8(optiona +Messablauf: Spannung aufsteigend von 50 mV bis 1 V, dann +Versuchstag 2 +Spannungabsteigendvon500mVbis5mV +Versuchstag 3 +Stromlimit: 2.5 A !!! +Versuchstag 4 +Versuchstag 5 +Experimentalresults: +Versuchstag 6 +Beim Andern derangelegten Spannung klickt das Digitalmultimeter manchmal +Versuchstag 7 +(Messbereichsumschaltung?); dabei andern sich die gemessenen Strome deutlich. +Versuchstag 8 +Versuchstag 9 +Versuchstag11 +Spreadsheet: +Others (disabled) +U_up (mV) +I_up (mA) +c +U_down (mV) I_down (mA) +G +H +晶Inventory +50 +0.259 +500 +1057 +Stock +2 +100 +0.481 +250 +521 ++ +Utilities +3 +150 +0.737 +100 +203 +About +4 +200 +0.989 +60 +117 +< +5 +220 +1.085 +40 +14.5 +6 +240 +1.185 +35 +12.45 +7 +260 +102.1 +25 +8.46 +8 +280 +110.1 +20 +6.5 +9 +300 +118.1 +15 +4.4 + + + + + + + + +Abb. 3: Grafische Visualisierung von Messdaten aus dem Elektronikpraktikum (vgl. Abb. 2); mit +wenigen Programmzeilen laden die Studierenden ihre im Praktikum erfassten Messdaten direkt aus +dem ELN in ein Jupyter-Notebook, wo sie analysiert werden. + + + +jupyter +EP1 Aufgabe1 (autosaved) +Logout +File +1P3 +View +Insert +Cell +Kernel +Widgets +Help +Trusted +Python 3 O +In [13]: +import numpy as np +import matplotlib.pyplot as plt +import openbis # auf PyBIs basierendes Modul +# Login auf openBIs-Server +openbis.login('https://physics.openbis.data.fau.de') +Login to openBIS: https://physics.openbis.data.fau.de +Login successful +In [14]: +# TabelLendaten aus Experimental Step Laden (PermId=-20220429111309709-4228) +data=0penbis.get5preadsheetData(20220429111309709-4228') +In[15]: +plt.scatter(data[o],data[1]) +out[15]: + +2000 +1500 +1000 +500 +0 +200 +400 +600 +800 +1000 +In [16]:# Disconnect +openbis.logout() + + + + + + + +Infobox +Electronic Lab Notebooks (ELN) +Unter den verfügbaren open-source Softwarelösungen hat sich unsere Universität in der Pilotphase +für openBIS entschieden. Ein anderes in der Physik gängiges open-source System wäre elabFTW. +openBIS bringt viele Funktionalitäten mit sich und erlaubt, Datenstrukturen ähnlich einer Datenbank +zu definieren und zu verknüpfen. Dabei gilt es für den Nutzer, die Balance zwischen Effizienz +(einfache Formulare; Freitexteingabe) und Perfektion (experimentspezifische Formulare und +resultierende strukturierte Metadatenablage), Schnittstellen zu Forschungspartnern, etc zu finden. +Dies ist zu einem großen Teil eine aktuelle und andauernde Forschungsfrage in den verschiedenen +NFDI-Konsortien (z. B. FAIRmat). +Eine wichtige Funktionalität eines ELN-Systems ist die Exportschnittstelle. openBIS bietet hier sowohl +die Möglichkeit, gut lesbare, formatierte Dokumente zu erstellen als auch strukturierte Daten (JSON- +Format) zu exportieren – diese Interoperabilität ist eine Grundvoraussetzung für die Umsetzung der +FAIR-Prinzipien. + + + + + + + + + + + +Dieser Artikel ist erschienen im Physik Journal, Ausgabe 12/2022, Wiley-VCH, ISSN 1617-9439: +M. Krieger, H.B. Weber, C. van Eldik, Früh zur Datenkompetenz, Physik Journal 21 (2022) Nr. 12, S. 42 + diff --git a/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/load_file.txt b/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe8cea5f8e44c4bbf941bb82f97ac41cb2af969d --- /dev/null +++ b/kdE1T4oBgHgl3EQf0gUD/content/tmp_files/load_file.txt @@ -0,0 +1,257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf,len=256 +page_content='Datenkompetenz im Physikstudium – ein Erfahrungsbericht Michael Krieger, Heiko B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Weber Lehrstuhl für Angewandte Physik, Department Physik, Friedrich-Alexander-Universität Erlangen- Nürnberg Christopher van Eldik Erlangen Centre for Astroparticle Physics, Department Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg Daten werden als entscheidende Ressource des 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Jahrhunderts betrachtet1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' In der Physik gibt es eine große Bandbreite von wohlstrukturierten, jahrelangen digitalen Datenströmen aus Großexperimenten (z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Astronomie, Teilchenphysik) bis hin zu manueller und hochgradig individuell geprägter Datenaufnahme in Laborexperimenten (z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Festkörperphysik, Optik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wären alle Forschungsdaten der Vergangenheit systematisch erfasst und offen zugänglich, könnten unter Umständen völlig neue Erkenntnisse aus deren Analyse resultieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dabei sind nicht nur die Rohdaten selbst, sondern insbesondere auch die Metadaten von größter Bedeutung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Letztere beschreiben Messaufbau, experimentelle Parameter, Einheiten, Kontext, Versionen, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' und erzeugen damit ein möglichst vollständiges und nachvollziehbares Bild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Je reichhaltiger und systematischer Metadaten erfasst werden, umso besser wird man aus der Kombination verschiedener Datensätze etwas lernen können.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Als Leitgedanke dieses sogenannten Forschungsdatenmanagements dienen die FAIR-Prinzipien: Daten sollen auffindbar (findable), offen zugänglich (accessible), standardisiert und dialogfähig (interoperable) sowie wiederverwertbar (reusable) gespeichert werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Man kann sich die weitere Verwendung vielleicht so vorstellen, dass eine WissenschaftlerIn in 10 Jahren einen Computer durch all unsere Daten surfen lässt und dabei Zusammenhänge entdeckt, die den einzelnen Forschungsgruppen verborgen geblieben sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Bund und Länder haben mit der groß angelegten Initiative für eine Nationale Forschungsdateninfrastuktur (NFDI)2 erhebliche Fördermittel bereitgestellt, um die wissenschaftlichen Disziplinen diesbezügliche Konzepte ausarbeiten zu lassen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Vermitteln unsere Physikcurricula in dieser sich verändernden Welt die passenden Datenverarbeitungskompetenzen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' In der Physik besteht breiter Konsens, dass die Studierenden fundierte Kenntnisse in Analysis, linearer Algebra, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' haben müssen, um im Verlauf des Studiums physikalische Konzepte zu verstehen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Man hat sich hier weltweit auf einen gewissen Kanon geeinigt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Unser Eindruck ist, dass ein solcher Konsens bezüglich Datenkompetenzen nicht existiert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Beim Erstkontakt mit Daten, also insbesondere in den Praktika der Experimentalphysik, werden zwar elementare Datenevaluation und systematische Dokumentation eingeübt, diese genügen aber nicht den steigenden Ansprüchen der Forschung und der Berufspraxis, sich zunehmend mit der Analyse großer Datenmengen zu befassen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Es scheitert oft schon an einem Konsens über eine geeignete Einstiegsprogrammiersprache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Am Department Physik der Friedrich-Alexander-Universität Erlangen-Nürnberg haben wir in den letzten Jahren kleinere Anpassungen im Physikcurriculum vorgenommen, die wir in diesem Artikel vorstellen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Datenkompetenz wurde früh im Bachelorstudium platziert, woraus sich erhebliche Vorteile für den weiteren Studienverlauf ergeben haben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Autoren können sich des Eindrucks 1 https://ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='eu/info/sites/default/files/communication european strategy data 19feb2020_en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='pdf 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='nfdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='de/ nicht erwehren, dass die Studierenden in puncto Datenkompetenz schnell auf die Überholspur gehen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' wir sehen bereits jetzt in unseren Arbeitsgruppen, dass sie sich als treibende Kräfte hin zu einem modernen Forschungsdatenmanagement erweisen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Obligatorisches Datenverarbeitungspraktikum Wie führt man Physikstudierende möglichst frühzeitig an computergestützte Datenauswertung heran?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wie begegnet man der Herausforderung, dass manche Studierende bereits aus der Schule oder durch eigene Aneignung Programmiererfahrung vorweisen können, für andere das Thema aber wie ein Buch mit sieben Siegeln erscheint?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Unsere persönliche Erfahrung ist, dass sich das Erlernen einer Programmiersprache am einfachsten durch begleitetes Learning-by-doing anhand physikalischer Fragestellungen gestalten lässt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die intensive eigene Beschäftigung in Form kleinerer Programmieraufgaben fördert ein tiefes Verständnis und legt die Grundlage für die anstehenden Herausforderungen in der Datenauswertung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Das Bearbeiten von Aufgaben in Teams stärkt den gerade zu Beginn des Studiums so wichtigen Austausch mit KommilitonInnen und fördert früh den Aufbau einer wissenschaftlichen, lösungsorientierten Diskussionskultur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dabei dürfen vor lauter Programmieren die Inhalte des Fachstudiums nicht in den Hintergrund treten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Nur wenn der Anwendungsbezug klar ersichtlich ist, werden sich ein nachhaltiger Kompetenzgewinn und die Motivation einstellen, das Gelernte im weiteren Studienverlauf zu nutzen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wir haben in diesem Sinne – erstmals zum Wintersemester 2018/19 – ein verpflichtendes Computerpraktikum „Datenverarbeitung in der Physik“ (DV-Praktikum) in das Curriculum unseres Bachelor-Studiengangs aufgenommen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Im DV-Praktikum werden unter Anleitung von Tutoren Programmieraufgaben im Team bearbeitet und anschließend elektronisch in Form eines fertigen Programms eingereicht und bewertet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Begleitet wird das wöchentliche zweistündige DV-Praktikum von einem interaktiven Online-Kurs, durch den die Studierenden genau die Grundlagenkenntnisse erwerben, die für die Bearbeitung der nächsten Praktikumseinheiten benötigt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dabei beschränken sich die Kursinhalte nicht auf das Erlernen einer Programmiersprache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Das Hauptaugenmerk liegt vielmehr in der Vermittlung grundlegender Datenverarbeitungs- und Datenanalyse-Kompetenzen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' So enthält der Kurs beispielsweise einführende Module in Messunsicherheiten und Fehlerfortpflanzung, Interpolation, Fitten von Daten, Statistik-Funktionen und Monte-Carlo-Simulationen, Fourier-Transformationen, numerische Integration, Lösung von Differentialgleichungen sowie grafische Aufbereitung von Ergebnissen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Auch ein einführendes Kapitel zu „Guter wissenschaftlicher Praxis“ ist Teil des Kurses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Alle diese Aspekte sind selbst für computeraffine Studierende neu und fordern damit alle Teilnehmenden gleichermaßen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Als Programmiersprache kommt Python zum Einsatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Grundlagen von Python sind auch für Anfänger schnell zu erlernen, und die Erfahrung zeigt, dass gute Python-Kenntnisse das Erlernen anderer gebräuchlicher Sprachen (z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' C++, Java) erheblich vereinfachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wichtiger noch für unsere Zwecke: Neben der umfangreichen Python-Standardbibliothek stehen mit „numpy“3, „scipy“4 und „matplotlib“5 frei zugängliche und gut kuratierte Basis-Bibliotheken zur numerischen Datenverarbeitung, Statistik und grafischen Aufbereitung zur Verfügung, die in der Physik- Community und weit darüber hinaus zur Anwendung kommen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Im DV-Praktikum werden konsequent Jupyter-Notebooks6 verwendet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' sie ermöglichen, Python-Code innerhalb eines Webbrowser-Fensters 3 https://numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='org 4 https://scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='org 5 https://matplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='org 6 https://jupyter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='org zu erstellen und laufen zu lassen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dabei werden Python-Code und Ergebnisse (Text und Grafiken) gemeinsam in einem Dokument dargestellt und gespeichert und können auf diese Weise sehr einfach mit anderen Interessierten geteilt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Jupyter-Notebooks verfügen zudem über dieselben umfangreichen (Text-)Formatierungs-Werkzeuge, die in Wikis zur Anwendung kommen, inklusive Formelsatz, Tabellen und Einbindung von Bildern (siehe Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Nach einer sehr erfolgreichen Pilotphase haben wir das DV-Praktikum auf besonderen Wunsch der Studierenden bereits im ersten Fachsemester verankert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Um dafür Platz im dicht gedrängten Stundenplan zu schaffen, beginnt das Grundpraktikum deshalb bei uns erst im zweiten Semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dies bietet für unsere Studierenden eine perfekte Gelegenheit, die erworbenen Datenverarbeitungs- Kompetenzen gewinnbringend nicht nur bei der Auswertung der Praktikumsversuche einzusetzen, sondern auch im gesamten Studium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Spielerischer Zugang zur Physik in Kursvorlesungen Mit der Entscheidung für die Programmiersprache Python und der Einbindung ins Erlanger Physikcurriculum kann man in Vorlesungen, Übungen und Praktika auf diese Kompetenzen zurückgreifen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Mit jeder Anwendung wird dieser Kenntnisstand weiter vertieft, das gilt gleichermaßen für Studierende und Lehrende.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Das ermöglicht völlig neue Aufgabentypen, bei denen man komplexere, manchmal auch realitätsnähere oder aktuellere physikalische Probleme behandeln kann, als wenn man nur auf analytische Kompetenzen zurückgreifen könnte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Jeder Dozent kann für sich selbst entscheiden, wo analytische Verfahren und wo numerischen Verfahren didaktisch sinnvoll sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Erhebliche Vorteile hat die Numerik bei der Visualisierung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dies ermöglicht eine zweite Herangehensweise an physikalische Probleme: die spielerische.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Hierbei lassen sich Parameter schnell variieren und die Ergebnisse in verschiedenen grafischen Auftragungen darstellen – in dieser Qualität und Geschwindigkeit ist dies mit traditionellen Methoden innerhalb einer Übung nicht zu schaffen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Bei manchen wird dabei Neugierde geweckt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Beispiel Fourier-Analyse: Was passiert, wenn man im Signal nur gerade Terme zulässt, oder nur ungerade?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Man würde sich wünschen, dass Studierende befähigt sind, solche Fragestellungen sowohl analytisch als auch numerisch zu lösen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Besonders profitieren die Studierenden, die den Vergleich anstellen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Ein anderes Beispiel betrifft das Misstrauen aufmerksamer Studierender gegenüber Näherungen, die für analytische Lösungswege häufig unvermeidbar sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Numerisch quantifiziert und visualisiert lässt sich oft viel leichter nachvollziehen, ob und unter welchen Umständen diese Näherungen gut begründbar sind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wir sind der Überzeugung, dass Lehrveranstaltungen der experimentellen und der theoretischen Physik gleichermaßen von dieser Art spielerischer Datenkompetenz profitieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Elektronisches Laborbuch im Praktikum Für ein zukunftsweisendes Forschungsdatenmanagement ist das traditionell handgeschriebene Laborbuch nicht geeignet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die nachhaltige Verfügbarkeit von Daten und Metadaten lässt sich nur durch sorgfältige elektronische Protokollierung z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' in Form eines elektronischen Laborbuchs erreichen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' so wird von Beginn an eine strukturierte Datenerfassung ermöglicht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Unsere Universität hat sich im Rahmen eines Pilotprojektes für ein open-source elektronisches Laborbuch (ELN) entschieden (openBIS7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' siehe Infobox).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dieses wird aktuell in verschiedenen Forschungsgruppen ausprobiert und evaluiert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 7 https://openbis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='ch/ Wir haben uns entschlossen, als Vertiefung des DV-Praktikums dieses ELN-System im Physikstudium einzuführen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Geeignet ist dafür das bestehende Elektronikpraktikum mit seinen 11 Versuchstagen im 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Fachsemester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dieses bearbeitet ein Programm von einfachen elektrischen Messungen über komplexer werdende analoge und digitale Schaltungen bis hin zu Microcontrollerprogrammierung und Analog/Digital-Wandlung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Bislang gaben wir eine Struktur vor, die Messdaten auf vorgegebenen Pfaden eines Netzwerklaufwerks zu speichern, wohingegen die Experimentvorbereitung und die Metadaten sorgfältig, aber ohne Formvorgaben in einem gebundenen Papier-Laborbuch dokumentiert wurden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Im Frühjahr 2022 vollzogen wir den Übergang zum ELN, der technisch gesehen relativ einfach war, da die Arbeitsplätze sowieso mit einheitlichen Computern ausgestattet waren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Lernziele und Inhalte des Elektronikpraktikums blieben unberührt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wir haben nur einige wenige Vorgaben implementiert: Beim ersten Login sehen die PraktikumsteilnehmerInnen ihren Laborbuch-space, auf den die 2er-Teams gemeinsamen Zugriff haben.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Innerhalb dessen sind bereits alle Versuchstage als projects und darunter die einzelnen Experimente und Aufgaben als experiments und experimental steps angelegt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die kursiv gesetzten Begrifflichkeiten sind von openBIS vorgesehene Standardelemente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die gesamte Struktur wird von uns per Python-Script angelegt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die elektronischen Laborbuchseiten (experimental steps) enthalten drei vorgegebene Freitextfelder für (i) die Ziele des Experiments, (ii) die Experimentbeschreibung und (iii) die Dokumentation der Resultate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Messdaten können bei kleineren Messreihen entweder per Hand in Tabellen erfasst werden oder als Rohdatendatei in das ELN geladen werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Auf diese Weise werden beschreibende Metadaten und Rohdaten von den Studierenden während des Experimentierens an einem Ort zusammengeführt (siehe Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Der besondere Vorteil des ELN wird bei Nutzung der Pythonschnittstelle deutlich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Aufbauend auf ihre mittlerweile dreisemestrige Pythonkompetenz können unsere Studierenden hiervon unmittelbar profitieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Mit einer einzigen Programmzeile lassen sich die Daten direkt aus dem ELN zur weiteren Auswertung und Visualisierung laden (vgl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Das geht im Praktikum genauso wie zu Hause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Hierzu wurde ein Pythonmodul8 zur Verfügung gestellt, das Befehle der PyBIS-Bibliothek kombiniert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Studierenden fanden schnell heraus, wie man Fotos von ihrem Smartphone in das ELN einbinden kann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' So wurden handgezeichnete Skizzen oder dokumentierende Aufnahmen der Versuchsaufbauten integriert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Das gleiche gilt natürlich auch für Screenshots relevanter Passagen aus externen Dokumenten, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Benutzerhandbüchern von Messgeräten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Studierenden haben das elektronische Laborbuch sofort akzeptiert und ausschließlich genutzt, obwohl die Verwendung des ELNs in der Pilotphase freigestellt war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Natürlich bietet dieses neue Medium auch Gelegenheit, die Arbeitsabläufe sinnvoller zu gestalten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Ein Beispiel: Die Praktikumsvorbereitung erfolgte bislang basierend auf einem Fragenkatalog im Laborbuch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Im Verlauf des Praktikums wurde diese zusammenhängende Darstellung aufgrund der chronologischen Notation in einem Papierbuch oft nicht mehr beachtet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Im ELN lassen sich Vorbereitung und Experimentdurchführung viel besser verzahnen, in dem die Antworten auf die Vorbereitungsfragen und zusätzliche Notizen in die entsprechenden Freitextfelder (Ziele und Experimentbeschreibung) der entsprechenden Experimente eingetragen werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Vorbereitung, Fragenkatalog und Antworten der Studierenden erscheinen am Ende in einer wohlgeordneten Darstellung, die für die Studierenden sehr viel besser lesbar ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Es sei auch erwähnt, dass die Betreuer sich durch ein Pythonskript eine exzerpierte Version der Vorbereitung ansehen können, um 8 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='com/FAU PHYSIK EP/load_openBIS_data_into_python sich einen Überblick über den Vorbereitungsstand der Praktikumsgruppen zu verschaffen, ohne sich immer wieder durch die einzelnen Experimente im ELN klicken zu müssen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Gegenüber traditionellen Laborbüchern erlaubt ein ELN, Nachbesserungen, Plagiate oder Fälschungen leichter zu verhindern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Studierenden wissen natürlich bereits von ihrer Internetnutzung, dass jede Handlung in einem elektronischen System prinzipiell genau nachvollziehbar sein könnte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Somit sollte gar nicht erst der Gedanke aufkommen, Daten zu “frisieren”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Betreuer haben wiederum über den Rohdatenzugriff eine Möglichkeit, die Versuchsaufbauten im Verlauf der Zeit bei etwaigen Ergebnisabweichungen zu überprüfen und so technischen Defekten auf die Schliche zu kommen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Nach Abschluss einer Lehrveranstaltung sollten Studierende eine vollständige Dokumentation für sich behalten können.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Neben der Weiternutzung des ELN im Fortgang des Studiums gibt es auch die Möglichkeit, die ELN-Inhalte zu exportieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wirkung auf die Forschungsebene Nachdem die Studierenden den Umgang mit dem ELN im 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Semester erlernt haben, bringen sie ihre Kompetenzen während der Forschungsphase zur Bachelor- oder Masterarbeit unmittelbar in die Arbeitsgruppen ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dort muss die Erfassung der Forschungsdaten allerdings wesentlich strukturierter und unmittelbar an die Arbeitsabläufe in der jeweiligen Forschungsgruppe angepasst erfolgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Diese Aufgabe steht der großen Mehrheit noch bevor und ist Gegenstand der NFDI- Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Auch wir sind gerade dabei, unsere Arbeitsabläufe in der Forschung systematisch zu erfassen und abzubilden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Dabei helfen Konzepte aus der Informatik, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Entity-Relationship- Diagramme, wie sie für Datenbanken erstellt werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Hier hilft der ELN-Einsatz im Praktikum: Wir werden Konzepte mit stärker strukturierten Laborbüchern zunächst im Praktikum erproben und evaluieren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Aufgrund der abgeschlossenen Praktikumseinheiten sind dabei kurze Lern- und Innovationszyklen möglich, deren Erfahrungen wir dann in die längerfristig angelegte und komplexere Dokumentation der Forschung in unserer Arbeitsgruppe einbringen können.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wir sehen es als ein Erfolgsrezept, frühzeitig im Physikcurriculum zeitgemäße Datenkompetenz zu vermitteln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die durchgängige Verwendung einer einheitlichen und in den Naturwissenschaften mächtigen Programmiersprache wie Python verschafft neue Möglichkeiten und verändert die Lehre und die Forschung nachhaltig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Wir sehen einen unmittelbaren Einfluss auf den Übungsbetrieb und den Praktikumsbetrieb im Bachelor- und Masterstudium sowie zusätzliche Kompetenzen der Studierenden, die sie in der Forschungsphase zur Bachelor- und Masterarbeit in die Arbeitsgruppen einbringen können.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Darüber hinaus sehen wir einen sehr positiven Einfluss auf die sich verändernde Forschungslandschaft, die viel stärker als früher von einem aktiven Forschungsdatenmanagement geprägt sein wird.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Heiko B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Weber und Michael Krieger forschen am Erlanger Department Physik an festkörperphysikalischen Fragestellungen, leiten das Elektronikpraktikum und arbeiten aktiv im NFDI- Konsortium FAIRmat mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Christopher van Eldik forscht am selben Department an astroteilchenphysikalischen Fragestellungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Als Studiendekan hat er die Umstellung des Curriculums mitgeprägt und leitet die Lehrveranstaltung “Datenverarbeitung in der Physik”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Abbildungen: Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 1: Beispielanalyse zur Bestimmung der Fallbeschleunigung aus dem Datenverarbeitungspraktikum im 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Fachsemester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' In solchen interaktiven Jupyter-Notebooks werden Python-Code, Dokumentation und Abbildungen auf einer Ebene dargestellt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' AnalysederMessdaten Wir laden die Daten, extrahieren die Zeiten von den beiden Zahlern in separate Arrays und korrigieren direkt den Zahler-Offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' data=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="loadtxt('g_measurements." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="dat') # readinmeasured times,correct forcalibrationoffset t1 = data[:,0] - t1_calib_offset t2 = data[:, 1] - t2_calib_offset N = data." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='shape[0] t1_mean = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='mean(t1) t2_mean =np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='mean(t2) delta_t1_mean = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='std(t1)/ np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='sqrt(N) delta_t2_mean=np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='std(t2)/np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="sqrt(N) print('mean time1:((:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='5f)+/-(:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="5f))s'." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="format(t1_mean,delta_t1_mean)) print('mean time2:({:." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='5f)+/-(:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='5f})s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='format(t2_mean,delta_t2_mean)) meantime1:(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='16221+/-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='00056)s mean time 2:(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='32598+/-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='00073)s PlottenderMesswerteverteilungen fig,ax=plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='subplots(figsize=(9,6)) #Histograms of corrected times ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="hist(t1,fc='lightblue',label='counter 1') ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="hist(t2,fc='orange',label='counter 2') #Histograms of uncorrected times ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="hist(t1 +t1_calib_offset,histtype='step',color='blue',label='counter 1 (uncorrected)') ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="hist(t2+t1_calib_offset,histtype='step',color='brown',label='counter2(uncorrected)') ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="set_xlabel('time[s]') ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="set_ylabel('number of entries') ax." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='legend() plt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='show() 30 25 20 of entri 10 5 counter 1 counter 2 counter 1 (uncorrected) counter 2 (uncorrected) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='35 time [s] Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 2: Ansicht des ELN während des ersten Versuchstages im Elektronikpraktikum: in der Baumansicht haben wir den Praktikumsablauf vorgezeichnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Die Studierenden haben während der Vorbereitung auf das Praktikum bereits Ziele und Durchführung der einzelnen Praktikumsexperimente (Teilaufgaben) inklusive Skizzen in die entsprechenden Eingabefelder notiert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Ergebnisse und Beobachtungen werden während des Praktikums im gleichen Formular erfasst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Hierfür stehen ein Freitextfeld, eine Tabelle für numerische Messdaten sowie der Dateiupload für Messdatendateien zur Verfügung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' X 口 C 8 https:/physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='openbis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='de/openbis/webapp/eln-lims/ ☆ D = Q Glot Experimental Step: Aufgabe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='1 12 t + New- @ Upload More .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 7 Lab Notebook Others Experimentalgoals: A 1415000000EscSose22 Bestimmung des Innenwiderstands des Digitalmultimeters im Modus Gleichstrommessung Versuchstag1 Aufgabe1 +Aufgabe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='1 Aufgabe 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='2 Experimental description: Aufgabe 2 VerbindungdesLabornetzgeratausgangs(Spannungsausgang) mit dem Digitalmultimetereingang (Stromeingang);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' die angelegte Itinete Aufgabe 3 + Aufgabe 4 Spannung fallt direkt uber dem Innenwiderstand des Aufgabe 5 Digitalmultimeters ab + Aufgabe6 MessgroBen: Strom (Digitalmultimeter), Spannung + Aufgabe 7 (Labornetzgerat) Labornetz gerat Aufgabe 8(optiona Messablauf: Spannung aufsteigend von 50 mV bis 1 V, dann Versuchstag 2 Spannungabsteigendvon500mVbis5mV Versuchstag 3 Stromlimit: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='5 A !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Versuchstag 4 Versuchstag 5 Experimentalresults: Versuchstag 6 Beim Andern derangelegten Spannung klickt das Digitalmultimeter manchmal Versuchstag 7 (Messbereichsumschaltung?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' dabei andern sich die gemessenen Strome deutlich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Versuchstag 8 Versuchstag 9 Versuchstag11 Spreadsheet: Others (disabled) U_up (mV) I_up (mA) c U_down (mV) I_down (mA) G H 晶Inventory 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='259 500 1057 Stock 2 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='481 250 521 + Utilities 3 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='737 100 203 About 4 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='989 60 117 < 5 220 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='085 40 14.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='1 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='4 Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 3: Grafische Visualisierung von Messdaten aus dem Elektronikpraktikum (vgl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' Abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' mit wenigen Programmzeilen laden die Studierenden ihre im Praktikum erfassten Messdaten direkt aus dem ELN in ein Jupyter-Notebook, wo sie analysiert werden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content=' jupyter EP1 Aufgabe1 (autosaved) Logout File 1P3 View Insert Cell Kernel Widgets Help Trusted Python 3 O In [13]: import numpy as np import matplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='pyplot as plt import openbis # auf PyBIs basierendes Modul # Login auf openBIs-Server openbis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="login('https://physics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='openbis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} 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data=0penbis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content="get5preadsheetData(20220429111309709-4228') In[15]: plt." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdE1T4oBgHgl3EQf0gUD/content/2301.03455v1.pdf'} +page_content='scatter(data[o],data[1]) out[15]: 0, ∀1 ≤ i ≤ n}. +A ⪰ 0 means that A is positive semi-definite. For a vector x ∈ Rn, its 2-norm is denoted by ∥x∥. Given a +matrix A ∈ Rm×n, ∥A∥ is defined as its the largest singular value and A+ is its generalized inverse. +2 +Problem Setup and Preliminaries +Consider an L-layer feed-forward neural network y = f(x) described by the following recursive equations: +z0 = x, +zk+1 = σ(Wkzk + bk), +k = 0, . . . , L − 1 +y = WLzL + bL, +(1) +2 + +where x ∈ Rn0, zk ∈ Rnk, y ∈ RnL+1 are the network input, hidden unit of the kth layer and network output, +respectively. Here Wk ∈ Rnk+1×nk and bk ∈ Rnk+1 are the weight matrix and bias vector for the kth layer. We +make the following assumption on σ, which holds for commonly-used activation functions [29]. +Assumption 2.1. The nonlinear activation σ : R → R is piecewise differentiable and sloped restricted in [0, 1]. +Note also that if different channels have different activation functions, then we simply require that they all +satisfy the above assumption. +Definition 2.2. A feed-forward neural network f of the form (1) is said to be globally Lipschitz bounded by +γ > 0 (or simply γ-Lipschitz) if +∥f(x1) − f(x2)∥ ≤ γ∥x1 − x2∥, +∀x1, x2 ∈ Rn0. +(2) +Moreover, f is nonexpansive if it is 1-Lipschitz in ℓ2 norm. +The main goal of this work is to learn feed-forward networks (1) with certificated Lipschitz bound of γ, i.e., +min +θ +L(fθ) +s.t. +fθ is γ-Lipschitz +(3) +where L(·) is a loss function. Since it is NP-hard to compute the Lipschitz constant (i.e. the smallest Lipschitz +bound) of fθ. We need an accurate Lipschitz bound estimation so that the constraint in (3) does not lead to a +significant restriction on the model expressivity. +In [14], integral quadratic constraint (IQC) theory was applied to capture both monotonicity and 1- +Lipschitzness properties of σ, leading to a state-of-art tight Lipschitz bound estimation based on the following +linear matrix inequality (LMI), see details in Appendix A: +H := +� +� +γI +−U ⊤Λ +0 +−ΛU +2Λ − ΛW − W ⊤Λ +−Y ⊤ +0 +−Y +γI +� +� ⪰ 0 +(4) +where Λ ∈ Dn +++ with n = �L +k=1 nk, and +W = +� +����� +0 +W1 +... +... +... +0 +0 +· · · +WL−1 +0 +� +����� +, U = +� +���� +W0 +0 +... +0 +� +���� , +Y = +�0 +· · · +0 +WL +� +. +Although (4) can be converted into a convex constraint for Lipschitz bound estimation of a network with fixed +W, U, Y , the learning problem in (3) is highly nonconvex due to changeable W, U, Y . For even relatively small- +scale networks (e.g. ∼ 1000 neurons), the associate barrier terms or projections become a major computational +bottleneck. +Remark 2.3. The published paper [14] claimed that even tighter Lipschitz bounds could be achieved with a less +restrictive class of multipliers Λ than diagonal. However, this claim was false: a counterexample was presented +in [25], and an explanation of the error was presented in [23]. +3 +Model parameterization +In this section we will present a model parameterization (see Figure 1) such that the learning problem (3) with +complicated matrix inequality constraint (4) can be transformed into an unconstrained optimization problem. +3 + +σ +Ψ−1 +k +Wk+1 +σ +Wk +Ψk−1 +2BkA⊤ +k−1 +Ψk +2Bk+1A⊤ +k +Ψ−1 +k+1 +· · · +x +f(x) +RN ∋ θ = {dk, Xk, Yk} −→ Lip(fθ) ≤ γ +σ +dk ∈ Rk+1 +edk +e−dk +� +Xk +Yk +� +∈ R(nk+1+nk)×nk+1 +Cayley(·) +· · · +Figure 1: Direct parameterization for Lipschitz-bounded deep networks. +Definition 3.1. A mapping MΘ : θ ∈ Θ ⊆ RN �→ fθ is called a parameterization of DNNs with Lipschitz +bound of γ if fθ is γ-Lipschitz for any θ ∈ Θ. Furthermore, such mapping is called a direct parameterization if +Θ = RN. +The free parameter θ in the proposed direct parameterization consists of +dj ∈ Rnj+1, +j = 0, . . . , L − 1, +Xk ∈ Rnk+1×nk+1, Yk ∈ Rnk×nk+1, +k = 0, . . . , L. +Note that bias terms are dropped for simplicity. Based these parameters, we first construct +Ψj = diag +� +edj� +, +�Ak +Bk +�⊤ += Cayley +��Xk +Yk +�� +(5) +where the Cayley transform is defined as +Cayley +��X +Y +�� +:= +�(I + Z)−1(I − Z) +−2Y (I + Z)−1 +� +(6) +with Z = X − X⊤ + Y ⊤Y . Then, the weight matrices of (1) are given by +Wk = 2Ψ−1 +k BkA⊤ +k−1Ψk−1, +k = 0, . . . , L +(7) +where A−1 = I, Ψ−1 = +� +γ/2I and ΨL = +� +2/γI with γ as the prescribed Lipschitz bound. Notice that weight +k depends on parameters of index k and k − 1, i.e. there is an “interlacing” coupling between parameters and +weights. +The proposed approach is mainly based on the observation that the structure of H in (4) is a chordal graph. +Thus, any semi-definite matrix with such structure can be parameterized by H = PP ⊤ where +P = +� +���� +D−1 +−V0 +D0 +... +... +−VL +DL +� +���� . +Substituting the above parameterization into (4) yields (6), see detailed derivation in Appendix B. +The main theoretical results is that our parameterization is complete (necessary and sufficient) for the set of +DNNs satisfying the LMI constraint (4) of [14]. +4 + +Theorem 3.2. The forward network (1) satisfies the LMI condition (4) iff its weight matrices Wk can be +parameterized via (7). +The proof of this and all other theorems can be found in the appendix. +3.1 +Nonexpansive sandwich layer +The proposed parameterization can also be interpreted as a new layer type. By introducing new hidden units +hk = +√ +2A⊤ +k Ψkzk for k = 0, . . . L, we can rewrite the proposed γ-LBDN as +h0 =√γx +hk+1 = +√ +2A⊤ +k Ψkσ( +√ +2Ψ−1 +k Bkhk + bk) +y =√γBLhL + bL. +(8) +The core component of the above model is a sandwich-structured layer of the form: +hout = +√ +2A⊤Ψσ +�√ +2Ψ−1Bhin + b +� +(9) +where hin ∈ Rp, hout ∈ Rq are the layer input and output, respectively. Unlike the parameterization in (7), +consecutive layers in (8) does not have coupled free parameters, which allows for modular implementation. +Another advantage is that such representation can reveal some fundamental insights on the roles of Ψ, A and B. +Theorem 3.3. The layer (9) with Ψ, A, B constructed by (5) is nonexpansive. +To understand the role of Ψ, we look at a simple nonlinear activation layer which is obtained simply by +placing Ψ ∈ Dq +++ and its inverse after and before σ, i.e., +u = Ψσ(Ψ−1v + b). +(10) +Here Ψ can change the shape and shift the position of individual activation channel while keeping their slopes +within [0, 1], allowing the optimizer to search over a rich set of activations. +For the roles of A and B, we need to look at another special case of (9) where σ is the identity operator. +Then, (9) becomes a linear layer +hout = 2A⊤Bhin + ˆb. +(11) +As a direct corollary of Theorem 3.3, the above linear layer is nonexpansive, i.e., ∥2A⊤B∥ ≤ 1. We show that +such parameterization is complete for nonexpansive linear layers. +Proposition 3.4. A linear layer is nonexpansive iff its weight W satisfies W = 2A⊤B with A, B given by (5). +3.2 +Nonexpansive convolutional layer +Our proposed layer parameterization can also incorporate more structured linear operators such as convolution. +Let hin ∈ Rp×s×s be a p-channel image tensor with s × s spatial domain and hout ∈ Rq×s×s be q-channel output +tensor. We also let A ∈ Rq×q×s×s denote a multi-channel convolution operator and similarly for B ∈ Rq×p×s×s. +For the sake of simplicity, we assume that the convolutional operators A, B are circular and unstrided. Such +assumption can be easily related to plain and/or 2-strided convolutions, see [30]. Similar to (9), the proposed +convolutional layer can be rewritten as +Vec(hout) = +√ +2C⊤ +AΨsσ +�√ +2Ψ−1 +s CB Vec(hin) + b +� +(12) +where CA ∈ Rqs2×qs2, CB ∈ Rqs2×ps2 are the doubly-circular matrix representations of A and B, respectively. +For instance, Vec(B ∗ hin) = CB Vec(hin) where ∗ is the convolution operator. We choose Ψs = Ψ ⊗ Is with +Ψ = diag(ed) so that individual channel has a constant scaling factor. To ensure that (12) is nonexpansive, +5 + +Algorithm 1 Nonexpansive convolutional layer +Require: hin ∈ Rp×s×s, P ∈ R(p+q)×q×s×s, d ∈ Rq +1: +˜hin ← FFT(hin) +2: +Ψ ← diag(ed), +� ˜A +˜B +�∗ ← Cayley(FFT(P)) +3: +˜h[:, i, j] ← +√ +2Ψ−1 ˜B[:, :, i, j]˜hin[:, i, j] +4: +˜h ← FFT +� +σ(FFT−1(˜h) + b) +� +5: +˜hout[:, i, j] ← +√ +2A[:, :, i, j]∗Ψ˜h[:, i, j] +6: +hout ← FFT−1(˜hout) +we need to construct CA, CB using the Cayley transformation (6), which involves inverting a highly-structured +large matrix I + CZ ∈ Rqs2×qs2. +Thanks to the doubly-circular structure, we can perform efficient computation on the Fourier domain. +Taking a 2D case for example, circular convolution of two matrices is simply the elementwise product of their +representations in the Fourier domain [31]. In [30], the 2D convolution theorem was extended to multi-channel +circular convolutions of tensors, which are reduced to a batch of complex matrix-vector products in the Fourier +domain rather than elementwise products. For example, the Fourier-domain output related to the (i, j)th pixel +is a matrix-vector product: +FFT(B ∗ hin)[:, i, j] = ˜B[:, :, i, j]˜hin[:, i, j]. +where ˜B[:, :, i, j] ∈ Cq×p and ˜hin[:, i, j] ∈ Cp. Here ˜x = FFT(x) is the fast Fourier transformation (FFT) of a +multi-channel tensor x ∈ Rc1×···×cr×s×s: +FFT(x)[i1, . . . , ir, :, :] = Fsx[i1, . . . , ir, :, :]F∗ +s +where Fs[i, j] = 1 +se−2π(i−1)(j−1)ι/s with ι = √−1. Moreover, transposing or inverting a convolution is equivalent +to applying the complex version of the same operation to its Fourier domain representation – a batch of small +complex matrices: +FFT(A⊤)[:, :, i, j] = ˜A[:, :, i, j]∗, +FFT((I + Z)−1)[:, :, i, j] = (I + ˜Z[:, :, i, j])−1. +Since the FFT of a real tensor is Hermitian-symmetric, the batch size can be reduced to s × (⌊s/2⌋ + 1). +We now give both model parameterization and forward computation of a nonexpansive convolutional layer +in Algorithm 1. In line 1 and 6, we use the (inverse) FFT on the input/output tensor, which can be either/both +removed for multiple consecutive convolutional layers. In line 2, we perform the Cayley transformation of +convolutions in the Fourier domain, which involves s × (⌊s/2⌋ + 1) parallel complex matrix inverse of size q × q. +In line 3-5, all operations related to the (i, j)th term can be done in parallel. +3.3 +Comparison to Semi-Orthogonal Layers +In this section we will compare the proposed approach to a closely related method developed in [30], which also +applies the Cayley transform to construct non-expansive layers. We first compare from the layer point of view +as both approaches provide parameterization for nonexpansive layers. We show that our layer parameterization +is more general. Second, we compare the networks constructed from those two layers, in terms of layerwise +spectral bound �L +k=0 ∥Wk∥, which is a loose Lipschitz upper bound of the network Jacobian operator +Jcfθ = WL +L +� +k=1 +JL−kWL−k ∈ RnL+1×n0 +where Jk = Jcσ(Wkzk + bk) ∈ Jnk+1 ++ +with Jc as the generalized Clake Jacobian. It often leads to conservative +results when training a 1-Lipschitz network subject to the naive bound �L +k=0 ∥Wk∥ ≤ 1. We present some +6 + +theoretical analysis to show that our parameterization allows for both the individual layer and network spectral +bound to be larger than 1, while the network Lipschitz constant is still bounded by a weighted layerwise spectral +bound of 1. In the next section, we will illustrate this result using a toy example. +Layer-level comparison. +The core component in [30] is the parameterization of (semi)-orthogonal layers +via Cayley transformation. It takes a free variable P ∈ Rq×p and then produce (semi)-orthogonal weight matrix +via W = Cayley(P). +However, for the case p = q, such parameterization is incomplete as it is limited to the orthogonal matrices +without −1 eigenvalues. As shown in Proposition 3.4 our parameterization is complete, i.e., it includes any +weight matrix whose eigenvalues within the unit disk, at the the cost of extra q×q parameters. Taking p = q = 2 +for example, Cayley(P) only contains rotation matrices while 2A⊤B can also include reflection matrices, e.g., +we obtain W = diag(−1, 1) via (6) with +X = 1 +3 +� 0 +0 +2 +√ +2 +0 +� +, +Y = 1 +3 +� 1 +√ +2 +√ +2 +−1 +� +. +Network-level comparison. +The γ-LBDN constructed by semi-orthogonal layers has weight matrices of +Wk = +�√γ Cayley(Pk), +k = 0, L +Cayley(Pk), +k = 1, . . . , L − 1, +(13) +where Pk ∈ Rnk+1×nk is a free variable. It is easy to verify +∥W0∥ = √γ, ∥Wk∥ = 1, 0 ≤ k < L, ∥WL∥ = √γ, +which further provides the Lipschitz certification by ∥Jcf∥ ≤ �L +k=0 ∥Wk∥ = γ. However, such model set is quite +restrictive as the bound estimation is quite loose. Our parameterization uses more sophisticated weighting (i.e. +full matrices) for the spectral bounds, which leads to a more expressive model set. +Proposition 3.5. The forward network with (7) satisfy the following weighted spectral bounds: +� +� +� +� +� +� +� +∥B+ +0 Ψ0W0∥ ≤ √2γ, +���B⊤ +k ΨkWkΨ−1 +k−1 +� +A⊤ +k−1 +�+��� ≤ 2, +1 ≤ k < L, +∥WLΨ−1 +L−1 +� +A⊤ +L−1 +�+∥ ≤ √2γ, +which further implies that ∥Jcf∥ ≤ γ. +4 +Experiments +Our experiments have two goals: First, to show that our model parameterization can provide a tight Lipschitz +bounds, which we illustrate with simple curve-fitting tasks. Second, to examine the performance and scalability +of the proposed new layer parameterization on adversarially robustness image classification tasks, illustrated +with the MNIST and CIFAR-10 data sets. Throughout this section we use sandwich and orthogon to denote +the LBDNs constructed from the proposed sandwich layer and orthogonal layer from [30], respectively. Further +results and training details can be found in the appendix. +Tightness of Lipschitz Bounds. +We illustrate the tight Lipschitz bound of our model parameterization +some toy examples of curve fitting: +square wave: f(x) = +� +0, +x ∈ [−1, 0) ∪ [1, 2] +1, +x ∈ [−2, −1) ∪ [0, 1) +multi-sine: f(x) = sin(x) + 0.2 sin(5x), +x ∈ [0, 2π]. +(14) +7 + +−2 +−1 +0 +1 +2 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +x +y +True curve +Sandwich γ = 2 +Sandwich γ = 5 +MLP +Orthogon γ = 2 +Orthogon γ = 5 +0 +0.5π +π +1.5π +2π +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +x +y +Train data +True curve +Sandwich +Orthogon +MLP +(a) square wave +(b) multi-sine wave +Figure 2: Simple 1D curve-fitting tasks with Lipschitz bound constraints. +The results are illustrated in Fig. 2. Table 1 includes details of the model structures and numerical results of +the fitting. Note that we used wider layers for the orthogonal and MLP layers in order to approximately match +the total number of parameters with the sandwich layer. +For the square wave, the true function has no global Lipschitz bound due to the points of discontinuity. Thus +a function approximator will naturally try to find models with large local Lipschitz constant near those singular +points, and if a global γ-Lipschitz constraint is imposed this is a useful test of its accuracy.We evaluate the +tightness of Lipschitz bound using γ/γ where γ is an empirical lower Lipschitz bound obtained by a PGD-like +method and γ is the imposed upper bound, which was 1, 2, and 5 in the cases we tested. In Table 1 it can be +seen that our approach achieves a much tighter Lipschitz bounds for all three cases, roughly 99% versus 75% +for orthogonal layer. +In Figure 3 we break down the Lipschitz bounds and spectral norms over layers. It can be seen that both +the orthogonal layer zk → zk+1 and sandwich layer hk → hk+1 have quite tight Lipschitz bounds on a per-layer +basis of around 98%. However, for the complete network the sandwich layer achieves a much tighter bound +of 99.6% vs 75%.This illustrates the benefits of taking into account coupling between neighborhood layers, +thus allowing individual layers to have spectral norm greater than 1. We note that, for the sandwich model, +the layer-wise product of spectral norms reaches 122.2, illustrating how poor this commonly-used bound is +compared to our bound. +For the multi-sine, we have a sum of a low-frequency sine wave and high-frequency sine-waves, each of which +individually have a maximum slope (Lipschitz bound) of 1, and the true Lipschitz constant is 2 (achieved at +x = π). We fit Lipschitz-bounded MLP models using the proposed sandwich layers and orthogonal layers. It +can be seen in Table 1 that the tightness of our Lipschitz bound is 98.4%, whereas for the orthogonal layers +it is only 61.3%. The MLP model has a maximum slope of 4.051, despite the actual curve being fit having a +maximum slope of 2. +Adversarial Robust Training of Image Classifiers. +We next examine an application in image classifica- +tion, using the Lipschitz bounds to enhance adversarial robustness. +The first case we considered was the MNIST data set with fully-connected layers. In Figure 4 we observe +that the sandwich layer had lower test error than the orthogonal layer in all cases, illustrating the improved +flexibility. Both layers achieved have similarly tight Lipschitz bounds, however the were not nearly as tight as +in the curve fitting case. We note that with γ = 0.1 both models offer significant robustness advantages to +adversarial perturbations, while Sandwich offers better nominal test error that MLP, but Orthogonal degrades +test error. In fact, the sandwich layer with γ = 0.1 achieves similar test error to the orthogonal layer γ = 0.5 +and 1, while offering much better robustness. +For CIFAR10 we fit multi-layer convolutional models, and see similar trends. We note that this is, to our +knowledge, the first experimental result of multi-layer 2D convolutional models satisfying the Lipschitz bounds +8 + +Table 1: Hyperparameters and Lipschitz bounds for the 1D curve fitting tasks. +Square wave +Sine wave +Sandwich +Orthogon +MLP +Sandwich +Orthogon +MLP +Fc: depth×width +8×90 +8×128 +8×128 +8×90 +8×128 +8×128 +Num. params +139,512 +132,491 +132,481 +139,512 +132,491 +132,481 +Max. lr +0.01 +0.01 +0.01 +0.005 +0.05 +0.01 +Epochs +200 +200 +200 +400 +400 +400 +Emp. γ / Cert. γ +0.998/1.0 +0.738/1.0 +106.7/– +0.984/1.0 +0.613/1.0 +4.051/– +Emp. γ / Cert. γ +1.998/2.0 +1.472/2.0 +– +– +– +– +Emp. γ / Cert. γ +4.950/5.0 +3.355/5.0 +– +– +– +– +75 +80 +90 +100 +Input layer +Hidden layer 1 +Hidden layer 2 +Hidden layer 3 +Hidden layer 4 +Hidden layer 5 +Hidden layer 6 +Hidden layer 7 +Hidden layer 8 +Output layer +Full network +γ/γ (%) +Sandwich +Orthogon +1 +2 +3 +∥W ∥ +Figure 3: Left: empirical Lipschitz bound for curve fitting of a square wave. The lower bound γ is obtained +using PGD-like method. We observed tight layer Lipschitz bound for both orthogonal and sandwich layers +(≥ 98.1%). However, the propose sandwich layer has a much tighter Lipschitz bound for the entire network +(99.6% versus 75%). Right: the spectral norm of weight matrices. Our approach admits weight matrices with +spectral norm larger than 1. The layerwise product �L +k=0 ∥Wk∥ is about 122.2, which is much larger than that +of orthogonal layers. +9 + +10−1 +100 +101 +102 +103 +1 +1.5 +2 +2.5 +%Test error (MNIST) +Sandwich γ = 0.1 +Sandwich γ = 0.5 +Sandwich γ = 1.0 +Orthogon γ = 0.1 +Orthogon γ = 0.5 +Orthogon γ = 1.0 +MLP +0 +1 +2 +3 +4 +0 +20 +40 +60 +80 +100 +Sandwich γ = 0.1 +Sandwich γ = 0.5 +Sandwich γ = 1.0 +Orthogon γ = 0.1 +Orthogon γ = 0.5 +Orthogon γ = 1.0 +MLP +100 +101 +102 +103 +10 +15 +20 +25 +30 +Lipschitz +%Test error (CIFAR-10) +Sandwich γ = 1 +Sandwich γ = 10 +Sandwich γ = 100 +Orthogon γ = 1 +Orthogon γ = 10 +Orthogon γ = 100 +CNN +0 +0.5 +1 +1.5 +2 +20 +40 +60 +80 +100 +Perturbation size +Sandwich γ = 1 +Sandwich γ = 10 +Sandwich γ = 100 +Orthogon γ = 1 +Orthogon γ = 10 +Orthogon γ = 100 +CNN +Figure 4: Image classification: test error vs Lipschitz constants (left) and robustness to adversarial perturbations +(right), for MNIST with fully-connected layers (top) and CIFAR-10 with convolutional layers (bottom). Note +that the proposed sandwich layer achieves both better test error and greater robustness than alternatives. +of [14]. Again we see that Lipschitz bounds improve test error compared to a vanilla CNN, while also improving +robustness dramatically. And again, the sandwich layer with γ = 1 achieves similar test error to the orthogonal +layer with γ = 10, while offering much better robustness, see additional results in Table 2. +A major aim of our paper is to make the Lipschitz bounds of [14] tractable for training of larger models, +compared to previous methods that depended on semidefinite programming. In Figure 5 we plot the training +curves (test-error vs epoch) and the computational time per epoch for the sandwich, orthogonal, and MLP/CNN +models for MNIST (fully-connected models) and CIFAR-10 (convolutional models). Firstly, we observe that +for both fully-connected and convolutional models, the training time per epoch of the sandwich model is only +around double that of a vanilla model. Furthermore, this is offset by the fact that in both cases the sandwich +model surpasses the final error of the MLP/CNN in less than half as many epochs. In fact, for the fully +connected case it does so in around a third as many epochs. +An interesting observation from Figure 5 is that both the MLP and CNN models seem to exhibit the +epoch-wide double descent phenomenon (see, e.g., [32]), whereas neither of the Lipschitz bounded models +(sandwich and orthogonal) do, they simply improve test error monotonically with epochs. Weight regularization +has been suggested as a mitigating factor for other forms of double descent [33], however we are not aware of +this specific phenomenon having been observed before. +5 +Conclusions and Future Work +In this paper we have introduced a new parameterization of neural networks that automatically satisfy the +tightest currently-known computationally-tractable Lipschitz bounds. This enables learning of Lipschitz- +10 + +0 +20 +40 +60 +80 +100 +0 +2 +4 +6 +8 +10 +Test error (%) +MNIST +Sandwich +Orthogon +MLP +0 +20 +40 +60 +80 +100 +20 +40 +60 +CIFAR-10 +Sandwich +Orthogon +CNN +0 +20 +40 +60 +80 +100 +1.6 +1.8 +2 +2.2 +2.4 +Epochs +Training time (s) +0 +20 +40 +60 +80 +100 +2 +4 +6 +8 +10 +12 +Epochs +Figure 5: Learning curves and training time per epoch for image classification tasks, obtained from 5 experiments. +The Lipschitz bounds are 0.5 for MNIST and 10 for CIFAR-10. Note that the “double-descent” phenomenon is +avoided with the Lipschitz-bounded models. +Table 2: Test error, empirical adversarial robustness and empirical lower Lipschitz bound on image classification +benchmarks, mean and standard deviation from 5 experiments. Sandwich layers outputs other methods in both +test and ℓ2 certifiable robust accuarcy. +MNIST +Sandwich +Orthogon +MLP +Cert. γ +0.1 +0.5 +1.0 +0.1 +0.5 +1.0 +– +Emp. γ +9.32±.06E-2 +0.41±.01 +0.78±.01 +9.15±.07E-2 +0.41±.01 +0.78±.01 +1.0±.3E3 +Err. ϵ = 0.0 +1.37±.05 +1.11±.05 +1.19±.05 +2.42±.04 +1.38±.03 +1.42±.05 +2.05±.10 +Err. ϵ = 0.2 +1.85±.05 +1.75±.05 +1.87±.07 +3.12±.08 +2.21±.05 +2.49±.10 +3.47±.24 +CIFAR +Sandwich +Orthogon +CNN +Cert. γ +1 +10 +100 +1 +10 +100 +– +EMP. γ +0.64±.00 +3.51±.07 +14.80±.67 +0.53±.01 +2.48±.07 +9.45±.26 +0.56±.12E4 +Err. ϵ = 0.0 +18.24±.25 +14.34±.26 +13.39±.52 +22.96±.26 +17.73±.33 +17.07±.45 +19.89±.59 +Err. ϵ = 0.1 +24.08±.28 +23.72±.39 +28.05±70 +29.34±.16 +28.90±.11 +34.71±.62 +49.85±1.07 +bounded networks with standard first-order gradient methods, avoiding the need for complex projections or +barrier evaluations. 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Bethge, “Foolbox: A python toolbox to benchmark the robustness of +machine learning models,” arXiv preprint arXiv:1707.04131, 2017. +A +Preliminaries on LMI-based Lipschitz bound estimation +Here we review the theoretical work of LMI-based Lipschitz bound estimation for neural networks from [14, 23]. +Consider an L-layer forward network y = f(x) described by the following recursive equation: +z0 = x, +zk+1 = σ(Wkzk + bk), +k = 0, . . . , L − 1, +y = WLzL + bL, +(15) +where x ∈ Rn0, zk ∈ Rnk, y ∈ RnL+1 are the network input, hidden unit of the kth layer and network output, +respectively. We stack all hidden unit z1, . . . , zL together and obtain a compact form of (15) as follows: +z +� �� � +� +���� +z1 +z2 +... +zL +� +���� = σ +� +� +� +� +� +� +� +� +� +� +� +W +� +�� +� +� +����� +0 +W1 +... +... +... +0 +0 +· · · +WL−1 +0 +� +����� +� +���� +z1 +z2 +... +zL +� +���� + +U +� �� � +� +���� +W0 +0 +... +0 +� +����x + +bz +� �� � +� +���� +b0 +b1 +... +bL−1 +� +���� +� +� +� +� +� +� +� +� +� +� +� +, +y = +Y +� +�� +� +�0 +· · · +0 +WL +� +� +���� +z1 +z2 +... +zL +� +���� + +by +���� +bL . +(16) +By introducing an intermediate variable v ∈ Rn with n = �L +k=1 nk, we can rewrite the above equation by +v = Wz + Ux + bz, +z = σ(v), +y = Y z + by. +(17) +Given two different solutions sa = (xa, va, za, ya) and sb = (xb, vb, zb, yb), their difference ∆s = sb − sa satisfies +∆v = W∆z + U∆x, +∆z = σ(vb) − σ(va) := Jab∆v, +∆y = Y ∆z +(18) +where Jab ∈ Jq ++. For any Lipschitz bound γ > 0 we have γ2∥∆x∥2 − |∆y∥2 ≥ 0 holds for all nonzero ∆s +satisfying (18). The main challenge is how to handle the varying weight Jab. One way is to replace it with a +quadratic inequality constraint, which might be conserve but easy to analyze. +14 + +Lemma A.1. If Assumption 2.1 holds, then for any Λ ∈ Dn +++ the following incremental quadratic constraint +(iQC) holds for any pair of (va, za) and (vb, zb) satisfying z = σ(v): +�∆v⊤ +∆z⊤ +�⊤ �0 +Λ +Λ +−2Λ +� �∆v +∆z +� +≥ 0 +(19) +where ∆v = vb − va and ∆z = zb − za. +Remark A.2. Assumption 2.1 implies that each channel satisfies 2∆zi(∆vi − ∆zi) ≥ 0, which can be leads to +(19) by a linear conic combination of each channel with multiplier Λ ∈ Dn +++. In [14] it was claimed that iQC (19) +holds with a richer (more powerful) class of multipliers (i.e. Λ is a symmetric matrix), which were previously +introduced for robust stability analysis of systems with repeated nonlinearities [34, 35, 36]. However this is not +true: a counterexample was given in [25], and here we give a brief explanation: even if the nonlinearities σ(vi) +are repeated when considered as functions of vi, their increments ∆zi = σ(va +i + ∆vi) − σ(va +i ) are not repeated +when considered as functions of ∆vi, since the diagonal elements of Jab depend on the particular va +i which +generally differs between units. +Theorem A.3. The feedforward neural network (15) is γ-Lipschitz if Assumption 2.1 holds, and there exist +an Λ ∈ Dn +++ satisfying the following LMI: +H := +� +� +γI +−U ⊤Λ +0 +−ΛU +2Λ − ΛW − W ⊤Λ +−Y ⊤ +0 +−Y +γI +� +� ⪰ 0. +(20) +Remark A.4. In [23], the above LMI condition also applies to more general network structures with full weight +matrix W. An equivalent form of (20) was applied in [14] for a tight Lipschitz bound estimation: +min +γ,Λ γ +s.t. +(20) +(21) +which can be solved by convex programming for moderate models, e.g., n < 10K in [14]. +B +Derivation of direct parameterization +First, by substituting W, U, Y and Λ into (20), its left-hand side can be rewritten as +H = +� +������������ +γI +−W ⊤ +0 Λ0 +−Λ0W0 +2Λ0 +−W ⊤ +1 Λ1 +−Λ1W1 +2Λ1 +... +... +... +... +... +2ΛL−2 +−W ⊤ +L−1ΛL−1 +−ΛL−1WL−1 +2ΛL−1 +−W ⊤ +L +−WL +γI +� +������������ +. +(22) +Note that H ⪰ 0 has a chordal structure, which means that it can be factorized as H = PP ⊤ with +P = +� +���� +D−1 +−V0 +D0 +... +... +−VL +DL +� +���� . +15 + +By substituting it back into (22) we have +D−1D⊤ +−1 = γI, +VkV ⊤ +k + DkD⊤ +k = 2Λk, 0 ≤ k < L, +VLV ⊤ +L + DLD⊤ +L = γI, +(23) +Wk = Λ−1 +k VkD⊤ +k−1. +(24) +By defining Ψk = Λ +1 +2 +k , Ak := +√ +2ΨkDk and Bk := +√ +2ΨkV ⊤ +k +with k = 0, . . . , L − 1 we have AkA⊤ +k + BkB⊤ +k = I. +Then, we can easily parameterize (Ψk, Ak, Bk) via simple exponential mapping of diagonal matrices and Cayley +transformation on full matrices, see (6). Finally, we can obtain the weight matrices as follows +Wk = Λ−1 +k VkD⊤ +k−1 = Ψ−2 +k +× ( +√ +2ΨkBk) × ( +√ +2A⊤ +k−1Ψk−1) = 2Ψ−1 +k BkA⊤ +k−1Ψk−1 +(25) +with k = 0, . . . , L − 1, where A−1 = I and Ψ−1 = +� +γ/2I. The above formula can also be applied to k = L by +choosing ΨL = +� +2/γI. +C +Proofs +C.1 +Proof of Lemma A.1 +By substituting ∆z = Jab∆v into (19) we have +�∆v⊤ +∆z⊤ +�⊤ �0 +Λ +Λ +−2Λ +� �∆v +∆z +� += 2∆z⊤Λ(∆v − ∆z) = 2∆v⊤JabΛ(I − Jab)∆v ≥ 0 +where the last inequality follows as Jab ∈ Jq ++. +C.2 +Proof of Theorem A.3 +We first apply Schur complement to (20), which yields +� γI +−U ⊤Λ +−ΛU +2Λ − ΛW − W ⊤Λ − 1 +γ Y ⊤Y +� +≻ 0. +Then, by left-multiplying the above equation by +� +∆x⊤ +∆z⊤� +and right-multiplying +� +∆x⊤ +∆z⊤�⊤ we can +obtain +γ∥∆x∥2 − 1 +γ ∥∆y∥2 − 2∆z⊤Λ∆z − 2∆z⊤Λ(W∆z + U∆x) = γ∥∆x∥2 − 1 +γ ∥∆y∥2 − 2∆z⊤Λ(∆z − ∆v) ≥ 0, (26) +which further implies that (15) is γ-Lipschitz since +γ∥∆x∥2 − 1 +γ ∥∆y∥2 ≥ 2∆z⊤Λ(∆v − ∆z) ≥ 0 +where the last inequality follows by Lemma A.1. +C.3 +Proof of Theorem 3.2 +Sufficient. +We show that (20) holds with Λ = diag(Λ0, . . . , ΛL−1) where Λk = Ψ2 +k. Since the structure of H +is a chordal graph, H ⪰ 0 is equivalent to the existence of a chordal decomposition [37]: +H = +L +� +k=0 +EkHkE⊤ +k +(27) +16 + +where 0 ⪯ Hk ∈ R(nk+nk+1)×(nk+nk+1) and Ek = +�0a,k +Ib,k +0c,k +� +with Ib,k being the identity matrix the +same size as Hk, and 0a,k, 0c,k being zero matrices of appropriate dimension. We then construct Hk as follows. +For k = 0, we take +H0 = +� +γI +−√2γB⊤ +0 Ψ0 +−√2γΨ0B0 +2Ψ0(I − A0A⊤ +0 )Ψ0 +� +. +(28) +Note that H0 ⪰ 0 since [H0]11 = γI ≻ 0, and the Schur complement to [H0]11 yields +2Ψ0(I − A0A⊤ +0 )Ψ0 − +� +2γΨ0B0 +1 +γ I +� +2γB⊤ +0 Ψ0 = 2Ψ0(I − A0A⊤ +0 − B0B⊤ +0 )Ψ0 = 0. +For k = 1, . . . , L − 1 we take +Hk = +�2Ψk−1Ak−1A⊤ +k−1Ψk−1 +−2Ψk−1Ak−1B⊤ +k Ψk +−2ΨkBkA⊤ +k−1Ψk−1 +2Ψk(I − AkA⊤ +k )Ψk +� +. +(29) +If Ak−1 is zero, then it is trival to have Hk ⪰ 0. For nonzero Ak−1, we can verify that Hk ⪰ 0 since the Schur +complement to [Hk]11 shows +2Ψk(I − AkA⊤ +k )Ψk − 2ΨkBkA⊤ +k−1Ψk−1 +� +2Ψk−1Ak−1A⊤ +k−1Ψk−1 +�+ 2Ψk−1Ak−1B⊤ +k Ψk +=2Ψk(I − AkA⊤ +k − BkB⊤ +k )Ψk + 2ΨkBk(I − A+ +k−1Ak−1)B⊤ +k Ψk +=2ΨkBk(I − A+ +k−1Ak−1)B⊤ +k Ψk ⪰ 0 +where X+ denotes the Moore–Penrose inverse of the matrix X, and it satisfies I − X+X ⪰ 0. +For k = L we take +HL = +� +2ΨL−1AL−1A⊤ +L−1ΨL−1 +−√2γAL−1B⊤ +L ΨL−1 +−√2γΨL−1BLA⊤ +L−1 +γI +� +. +(30) +Similarly, we can conclude HL ⪰ 0 using Schur complement +γI− +� +2γΨL−1BLA⊤ +L−1 +� +2ΨL−1AL−1A⊤ +L−1ΨL−1 +�+ � +2γAL−1B⊤ +L ΨL−1 = γΨL−1BL(I−A+ +L−1AL−1)B⊤ +L ΨL−1 ⪰ 0. +We now show that Hk with k = 0, . . . L satisfy the chordal decomposition (27) holds since +[Hk]21 = −2ΨkBkA⊤ +k−1Ψk−1 = −Ψ2 +k(2Ψ−1 +k BkA⊤ +k−1Ψk−1) = −ΛkWk, +[Hk]22 + [Hk+1]11 = 2Ψk(I − AkA⊤ +k )Ψk + 2ΨkAkA⊤ +k Ψk = 2Ψ2 +k = 2Λk. +Finally, we conclude that H ⪰ 0 from [37][Theorem 2.1]. +Necessary. +For any Wk and Λk satisfying (20), we will find set of free variables dk, Xk, Yk such that (7) +holds. We take Ψk = Λ +1 +2 which further leads to dk = diag(log Ψk). By letting A−1 = I, Ψ−1 = +� +γ/2I and +ΨL = +� +2/γI we then construct Ak, Bk recursively via +Bk = 1 +2ΨkWkΨ−1 +k−1A−⊤ +k−1, +Ak = chol(I − BkB⊤ +k )Qk +(31) +where chol(·) denotes the Cholesky factorization, Qk is an arbitrary orthogonal matrix such that Ak does not +have eigenvalue of −1. If Ak−1 is non-invertible but non-zero, we replace A−⊤ +k−1 with +� +A+ +k−1 +�⊤. If Ak−1 = 0 (i.e. +Wk = 0), we simply reset Ak−1 = I. It is easy to verify that Ψk, Ak and Bk satisfy the model parameterization +(7). Finally, we can construct Xk, Yk using (36), which is well-defined as Ak does not have eigenvalue of −1. +17 + +C.4 +Proof of Theorem 3.3 +The proposed layer (9) can be rewritten as a compact network (17) with W = 0, Y = +√ +2A⊤Ψ and U = +√ +2Ψ−1B, +i.e., +v = Uhin + b, +z = σ(v), +hout = Y z. +From the model parameterization (6) we have AA⊤ + BB⊤ = I, which further implies +2Ψ2 − Y ⊤Y − Ψ2UU ⊤Ψ2 = 2Ψ2 − 2ΨAA⊤Ψ − 2ΨBB⊤Ψ = 2Ψ(I − AA⊤ − BB⊤)Ψ = 0 +By applying Schur complement twice to the above equation we have +� +� +I +−U ⊤Ψ2 +0 +−Ψ2U +2Ψ2 +−Y ⊤ +0 +−Y +I +� +� ⪰ 0. +Then, the 1-Lipschitzness of (9) is obtained by Theorem A.3. +C.5 +Proof of Proposition 3.4 +Sufficient. +It is a direct corollary of Theorem 3.3 by taking the identity operator as the nonlinear activation. +Necessary. +Here we give a constructive proof. That is, given a weight matrix W with ∥W∥ ≤ 1, we will find +a (generally non-unique) pair of (X, Y ) such that 2A⊤B = W with A, B given by (6). +We first construct A, B from W. Since it is obvious for W = 0, we consider the case with nonzero W. First, +we take a singular value decomposition (SVD) of W, i.e. W = UwΣwV ⊤ +w where Uw is a q × q orthogonal matrix, +Σw is an q × p rectangular diagonal matrix with Σw,ii ≥ 0 non-increasing, Vw is a p × p orthogonal matrix. +Then, we consider the candidates for A and B as follows: +A = UΣaU ⊤ +w , +B = UΣbV ⊤ +w +(32) +where Σa is a diagonal matrix, Σb a rectangular diagonal matrix U ∈ Rq×q an orthogonal matrix. By substituting +(32) into the equalities AA⊤ + BB⊤ = Iq and W = 2A⊤B we have +Σ2 +a + Σ2 +b′ = Iq, +2ΣaΣb′ = Σw′ +(33) +where Σb′, Σw′ ∈ Rq×q are obtained by either removing the extra columns of zeros on the right or adding extra +rows of zeros at the bottom to Σb and Σw, respectively. The solution to (33) is +Σa,ii = 1 +2 +�� +1 + Σw′,ii + +� +1 − Σw′,ii +� +, +Σb′,ii = 1 +2 +�� +1 + Σw′,ii − +� +1 − Σw′,ii +� +(34) +where are well-defined as ∥W∥ ≤ 1. Now we can obtain Σb from Σb′ by removing extra rows of zeros at the +bottom or adding extra columns of zeros on the right. At last, we pick up any orthogonal matrix U such that +A = UΣaU ⊤ +w does not have eigenvalue of −1. +The next step is to find a pair of (X, Y ) such that +A⊤ = (I + Z)−1(I − Z), +B⊤ = −2Y (I + Z)−1, +Z = X − X⊤ + Y ⊤Y. +(35) +One solution to the above equation is +Z = (I − A⊤)(I + A⊤)−1, +Y = −1 +2B⊤(I + Z), +X = 1 +2tril(Z − Z⊤) +(36) +where tril(W) denotes the strictly lower triangle part of W. Note that the above solution is well-defined since +A does not has eigenvalue of −1. +18 + +C.6 +Proof of Proposition 3.5 +From (28) we have +H0 = +� +γI +−W ⊤ +0 Ψ2 +0 +−Ψ2 +0W0 +2Ψ0B0B⊤ +0 Ψ0 +� +⪰ 0. +Applying the Schur complement yields γI − 1/2W ⊤ +0 Ψ0(B0B⊤ +0 )+Ψ0W0 ⪰ 0, which implies ∥B+ +0 Ψ0W0∥ ≤ √2γ. +From (29) we obtain +Hk = +�2Ψk−1Ak−1A⊤ +k−1Ψk−1 +−W ⊤ +k Ψ2 +k +−Ψ2 +kWk +2ΨkBkB⊤ +k Ψk +� +⪰ 0 +⇒Ψk−1Ak−1A⊤ +k−1Ψk−1 − 1 +4W ⊤ +k Ψk(BkB⊤ +k )+ΨkWk ⪰ 0 +⇒I − 1 +4A+ +k−1Ψ−⊤ +k−1W ⊤ +k Ψk(BkB⊤ +k )+ΨkWkΨ−1 +k−1 +� +A⊤ +k−1 +�+ ⪰ 0 +⇒ +���� +1 +2B+ +k ΨkWkΨ−1 +k−1 +� +A⊤ +k−1 +�+ +���� ≤ 1. +Similarly, from (30) we have +HL = +� +2ΨL−1AL−1A⊤ +L−1ΨL−1 +−W ⊤ +L +−WL +γI +� +⪰ 0 ⇒ +���WLΨ−1 +L−1 +� +A⊤ +L−1 +�+��� ≤ +� +2γ. +The bound of Jcf is then obtained by +∥Jcf∥ = ∥WLJL−1WL−1 · · · J0W0∥ += +����� +1 +2WLΨ−1 +L−1 +� +A⊤ +L−1 +�+(2A⊤ +L−1JL−1BL−1) +1 +� +k=L−1 +�1 +2B+ +k ΨkWkΨ−1 +k−1 +� +A⊤ +k−1 +�+ +� +(2A⊤ +k−1Jk−1Bk−1)(B+ +0 Ψ0W0) +����� +≤ +�� +2γ +�2/2 = γ +where the inequality follows as 2A⊤ +k JkBk is the Clake Jacobian of a nonexpansive layer (9), i.e. ∥2A⊤ +k JkBk∥ ≤ 1. +D +Experiments +Dataset. +For the square wave experiment, we take 300 and 600 samples (xi, yi) with xi ∼ U([−2, 2]) for +training and testing, respectively. For the sine wave experiment, we take 50 and 200 points (xi, yi) for training +and testing, where xi = 2i/Nπ with N as the size of dataset. We use batch size of 50 for those two tasks. For +the CIFAR-10 dataset, we applied standard augmentation,i.e., random cropping and flipping. Inputs to all +image classification models are normalized. +Model architectures for image classification benchmarks. +We trained small fully-connected model on +MNIST and the KWLarge network from [38] on CIFAR-10, with detailed architecture listed in Table 3. To +make the different models have similar number of parameters in the same experiment, we slightly reduce the +hidden layer width of sandwich model in the MNIST experiment and increases width of the first fully-connected +layer of CNN and orthogonal models. We also applied the emulated 2-stride from [30] to the second and fourth +convolution layers. For each experiment, we also trained a conventional MLP/CNN model with similar amount +of parameters as baselines. +Training details. +For all experiments, we used a piecewise triangular learning rate [39] with maximum rate +in {0.05, 0.01, 0.005, 0.001}, choosing the one with the best test performance. We found that 0.01 often works +19 + +Table 3: Model architectures for image classification benchmarks. +MNIST +Sandwich +Orthogon +MLP +Fc +(190, 190, 128) +(256,256,128) +(256,256,128) +Num. params +300,374 +300,942 +300,938 +KWLarge (CIFAR-10) +Sandwich +Orthogon +CNN +Conv +(32,32,64,64) +(32,32,64,64) +(32,32,64,64) +Fc +(512, 512) +(640,512) +(640,512) +Num. params +2,982,225 +2,995,373 +3,020,970 +well and took it as a default rate. We use Adam [40] and ReLU as our default optimizaer and activation, +respectively. For the image classification tasks, we used similar loss function and initialization method as [30]. +Because the Cayley transform in (6) involves both linear and quadratic terms, we implemented the weight +normalization method from [41]. That is, we reparameterize X, Y in Z = X −X⊤ +Y ⊤Y by g +X +∥X∥F and h +Y +∥Y ∥F +with learable scalars g, h. For all γ-LBDNs, we choose hidden layer to be nonexpansive (i.e., 1-Lipschitz) and +put a scaling factor √γ on the input and output, respectively. On the image classification tasks, we did not +observe significant improvement by uniformly distributing the Lipschitz capacity, i.e., each nonexpansive hidden +layer is scaled by γ +1 +L . We choose Lipschitz bounds of 0.1, 0.5, 1.0 for MNIST and 1,10,100 for CIFAR-10. We +search for the empirical lower Lipschitz bound γ of a network fθ by a PGD-like method, i.e., updating the input +x and its deviation δx based on the gradient of ∥fθ(x + δx) − fθ(x)∥/∥δx∥. As we are interested in the global +lower Lipschitz bound, we do not project x and x + δx into any compact region. For robustness verification, we +used the L2FastGradientAttack method from the package foolbox [42]. All image classification experiments +were performed on a Nvidia A5000. +20 + diff --git a/kdFJT4oBgHgl3EQfYiwU/content/tmp_files/load_file.txt b/kdFJT4oBgHgl3EQfYiwU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e400edea22f943082c0a748b3b3e3763d211de5c --- /dev/null +++ b/kdFJT4oBgHgl3EQfYiwU/content/tmp_files/load_file.txt @@ -0,0 +1,989 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf,len=988 +page_content='Direct Parameterization of Lipschitz-Bounded Deep Networks Ruigang Wang, Ian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Manchester∗ Abstract This paper introduces a new parameterization of deep neural networks (both fully-connected and convolutional) with guaranteed Lipschitz bounds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' limited sensitivity to perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The Lipschitz guarantees are equivalent to the tightest-known bounds based on certification via a semidefinite program (SDP), which does not scale to large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In contrast to the SDP approach, we provide a “direct” parameterization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' a smooth mapping from RN onto the set of weights of Lipschitz-bounded networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This enables training via standard gradient methods, without any computationally intensive projections or barrier terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The new parameterization can equivalently be thought of as either a new layer type (the sandwich layer), or a novel parameterization of standard feedforward networks with parameter sharing between neighbouring layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We illustrate the method with some applications in image classification (MNIST and CIFAR-10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 1 Introduction Neural networks have enjoyed wide application due to their many favourable properties, including highly accurate fits to training data, surprising generalisation performance within a distribution, and well as scalability to very large models and data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Nevertheless, it has also been observed that they can be highly sensitive to small input perturbations [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This is a critical limitation in applications in which certifiable robustness is required, or the smoothness of a learned function is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A standard way to quantify sensitivity of a models is via a Lipschitz bound, which generalises the notion of a slope-restricted scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A learned function x �→ f(x) between normed spaces satisfies a Lipschitz bound γ if ∥f(x1) − f(x2)∥ ≤ γ∥x1 − x2∥ for all x1, x2 in its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The (true) Lipschitz constant of a function is the smallest such γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A natural application of Lipschitz-bounds is to control a model’s sensitivity adversarial (worst-case) inputs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [2, 3], but Lipschitz constants also appear in bounds on statistical generalisation performance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Lipschitz- bounds have been applied to help stabilise the learning of generative adversarial networks [5, 6], and more recently in implicit geometry mechanisms for computer graphics [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Lipschitz-bounded networks have also been investigated in the context of reinforcement learning and control problems, for controlling sensitivity to measurement noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [8]) and ensuring robust stability of feedback loops during training [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In robotics applications, several learning-based planning and control algorithms require known Lipschitz bounds in learned stability certificates, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' the recent surveys [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Unfortunately, even for two-layer perceptions with ReLU activations, exact calculation of the true Lipschitz constant for ℓ2 (Euclidean) norms is NP-hard [12], so attention has focused on approximations that balance accuracy with computational tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For l1 and l∞ norms, simple Lipschitz bounds can be expressed in terms of row and column sums of layer weights [13], but for l2 it is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Crude bounds can be found via the product of spectral norms of layer weights [1], however to date the most accurate bounds require solution of a semidefinite program (SDP) [14], which is computationally tractable only for relatively small fully-connected networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' ∗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Wang and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Manchester are with Australian Centre for Field Robotics, The University of Sydney, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' E-mail: ruigang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='wang, ian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='manchester@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='au 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='11526v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='LG] 27 Jan 2023 Furthermore, while certification of a Lipschitz bound of a fixed network is a (convex) SDP with this method, the set of weights satisfying a prescribed Lipschitz bound is highly non-convex, complicating training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Both [15] and [11] specifically highlight the computationally-intensive nature of these bounds as limitations for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We will continue to discuss related work below, but first we state the main contributions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In this paper we introduce a new parameterization of neural networks, both fully-connected multi-layer perceptions (MLP) and deep convolutional neural networks (CNN), which has built-in guarantees on the network’s Lipschitz bound, equivalent the best-known bounds provided by the SDP method [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Roughly speaking, we construct a smooth surjective mapping from an unconstrained parameter space RN onto the (non-convex) set of network weights satisfying these bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This enables learning of lipschitz-bounded networks via standard unconstrained optimization methods such as stochastic gradient methods or ADAM [16], avoiding the complex projection steps or barrier function computations that have previously been required and limited scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In particular, to the authors knowledge this paper represents the first time that convolutional models for image classification have been successfully trained that satisfy the state-of-the-art Lipschitz bounds of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This work sits in the broader context of methods to certify safety and robustness of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This includes interval propagation bounds (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [17]) which are simple but can be conservative, to methods based on convex relaxation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [18] and [19]) and mixed-integer programming [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Specific to computation of Lipschitz bounds, [1] already suggested analysis via layer-wise spectral bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [3] proposed a computationally tractable approach for convolutional models, based one layer-wise estimates of spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [21] proposed a novel activation function and associated weight constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [13] presented training methods incorporating Lipschitz bounds for multiple norms, using a power iteration method to approximate spectral norms for the ℓ2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [22] bounded Lipschitz constants via incremental dissipativity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Beyond standard feedforward networks, [23] proposed a class of Lipschitz-bounded equilibrium networks and [24] extended this to recurrent (dynamic) equilibrium networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Since the SDP-based bounds of [14] appeared, several papers have proposed methods to allow training of Lipschitz models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [25], which proposed an alternating direction method of multipliers (ADMM) approach, which required solving an SDP at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [26] and [27] improve computational tractability by exploiting chordal sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Moving beyond the fully-connected case, [28] proposed a method based on 2D systems theory to certify 1D convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We let R, C be the set of real and complex numbers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The identity matrix is denoted by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Throughout the paper we have Jn + := {diagonal J ∈ Rn×n | Jii ∈ [0, 1], ∀1 ≤ i ≤ n}, Dn ++ := {diagonal D ∈ Rn×n | Dii > 0, ∀1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A ⪰ 0 means that A is positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For a vector x ∈ Rn, its 2-norm is denoted by ∥x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Given a matrix A ∈ Rm×n, ∥A∥ is defined as its the largest singular value and A+ is its generalized inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 2 Problem Setup and Preliminaries Consider an L-layer feed-forward neural network y = f(x) described by the following recursive equations: z0 = x, zk+1 = σ(Wkzk + bk), k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1 y = WLzL + bL, (1) 2 where x ∈ Rn0, zk ∈ Rnk, y ∈ RnL+1 are the network input, hidden unit of the kth layer and network output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Here Wk ∈ Rnk+1×nk and bk ∈ Rnk+1 are the weight matrix and bias vector for the kth layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We make the following assumption on σ, which holds for commonly-used activation functions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The nonlinear activation σ : R → R is piecewise differentiable and sloped restricted in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note also that if different channels have different activation functions, then we simply require that they all satisfy the above assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A feed-forward neural network f of the form (1) is said to be globally Lipschitz bounded by γ > 0 (or simply γ-Lipschitz) if ∥f(x1) − f(x2)∥ ≤ γ∥x1 − x2∥, ∀x1, x2 ∈ Rn0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (2) Moreover, f is nonexpansive if it is 1-Lipschitz in ℓ2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The main goal of this work is to learn feed-forward networks (1) with certificated Lipschitz bound of γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', min θ L(fθ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' fθ is γ-Lipschitz (3) where L(·) is a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Since it is NP-hard to compute the Lipschitz constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' the smallest Lipschitz bound) of fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We need an accurate Lipschitz bound estimation so that the constraint in (3) does not lead to a significant restriction on the model expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In [14], integral quadratic constraint (IQC) theory was applied to capture both monotonicity and 1- Lipschitzness properties of σ, leading to a state-of-art tight Lipschitz bound estimation based on the following linear matrix inequality (LMI), see details in Appendix A: H := � � γI −U ⊤Λ 0 −ΛU 2Λ − ΛW − W ⊤Λ −Y ⊤ 0 −Y γI � � ⪰ 0 (4) where Λ ∈ Dn ++ with n = �L k=1 nk, and W = � ����� 0 W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 0 0 · · WL−1 0 � ����� , U = � ���� W0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 0 � ���� , Y = �0 · · 0 WL � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Although (4) can be converted into a convex constraint for Lipschitz bound estimation of a network with fixed W, U, Y , the learning problem in (3) is highly nonconvex due to changeable W, U, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For even relatively small- scale networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' ∼ 1000 neurons), the associate barrier terms or projections become a major computational bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The published paper [14] claimed that even tighter Lipschitz bounds could be achieved with a less restrictive class of multipliers Λ than diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However, this claim was false: a counterexample was presented in [25], and an explanation of the error was presented in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 3 Model parameterization In this section we will present a model parameterization (see Figure 1) such that the learning problem (3) with complicated matrix inequality constraint (4) can be transformed into an unconstrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 3 σ Ψ−1 k Wk+1 σ Wk Ψk−1 2BkA⊤ k−1 Ψk 2Bk+1A⊤ k Ψ−1 k+1 · · x f(x) RN ∋ θ = {dk, Xk, Yk} −→ Lip(fθ) ≤ γ σ dk ∈ Rk+1 edk e−dk � Xk Yk � ∈ R(nk+1+nk)×nk+1 Cayley(·) · · Figure 1: Direct parameterization for Lipschitz-bounded deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A mapping MΘ : θ ∈ Θ ⊆ RN �→ fθ is called a parameterization of DNNs with Lipschitz bound of γ if fθ is γ-Lipschitz for any θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Furthermore, such mapping is called a direct parameterization if Θ = RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The free parameter θ in the proposed direct parameterization consists of dj ∈ Rnj+1, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1, Xk ∈ Rnk+1×nk+1, Yk ∈ Rnk×nk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note that bias terms are dropped for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Based these parameters, we first construct Ψj = diag � edj� , �Ak Bk �⊤ = Cayley ��Xk Yk �� (5) where the Cayley transform is defined as Cayley ��X Y �� := �(I + Z)−1(I − Z) −2Y (I + Z)−1 � (6) with Z = X − X⊤ + Y ⊤Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, the weight matrices of (1) are given by Wk = 2Ψ−1 k BkA⊤ k−1Ψk−1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L (7) where A−1 = I, Ψ−1 = � γ/2I and ΨL = � 2/γI with γ as the prescribed Lipschitz bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Notice that weight k depends on parameters of index k and k − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' there is an “interlacing” coupling between parameters and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The proposed approach is mainly based on the observation that the structure of H in (4) is a chordal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Thus, any semi-definite matrix with such structure can be parameterized by H = PP ⊤ where P = � ���� D−1 −V0 D0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' −VL DL � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Substituting the above parameterization into (4) yields (6), see detailed derivation in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The main theoretical results is that our parameterization is complete (necessary and sufficient) for the set of DNNs satisfying the LMI constraint (4) of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 4 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The forward network (1) satisfies the LMI condition (4) iff its weight matrices Wk can be parameterized via (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The proof of this and all other theorems can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Nonexpansive sandwich layer The proposed parameterization can also be interpreted as a new layer type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' By introducing new hidden units hk = √ 2A⊤ k Ψkzk for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' L, we can rewrite the proposed γ-LBDN as h0 =√γx hk+1 = √ 2A⊤ k Ψkσ( √ 2Ψ−1 k Bkhk + bk) y =√γBLhL + bL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (8) The core component of the above model is a sandwich-structured layer of the form: hout = √ 2A⊤Ψσ �√ 2Ψ−1Bhin + b � (9) where hin ∈ Rp, hout ∈ Rq are the layer input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Unlike the parameterization in (7), consecutive layers in (8) does not have coupled free parameters, which allows for modular implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Another advantage is that such representation can reveal some fundamental insights on the roles of Ψ, A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The layer (9) with Ψ, A, B constructed by (5) is nonexpansive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' To understand the role of Ψ, we look at a simple nonlinear activation layer which is obtained simply by placing Ψ ∈ Dq ++ and its inverse after and before σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', u = Ψσ(Ψ−1v + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (10) Here Ψ can change the shape and shift the position of individual activation channel while keeping their slopes within [0, 1], allowing the optimizer to search over a rich set of activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the roles of A and B, we need to look at another special case of (9) where σ is the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, (9) becomes a linear layer hout = 2A⊤Bhin + ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (11) As a direct corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3, the above linear layer is nonexpansive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', ∥2A⊤B∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We show that such parameterization is complete for nonexpansive linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A linear layer is nonexpansive iff its weight W satisfies W = 2A⊤B with A, B given by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 Nonexpansive convolutional layer Our proposed layer parameterization can also incorporate more structured linear operators such as convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Let hin ∈ Rp×s×s be a p-channel image tensor with s × s spatial domain and hout ∈ Rq×s×s be q-channel output tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We also let A ∈ Rq×q×s×s denote a multi-channel convolution operator and similarly for B ∈ Rq×p×s×s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the sake of simplicity, we assume that the convolutional operators A, B are circular and unstrided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Such assumption can be easily related to plain and/or 2-strided convolutions, see [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Similar to (9), the proposed convolutional layer can be rewritten as Vec(hout) = √ 2C⊤ AΨsσ �√ 2Ψ−1 s CB Vec(hin) + b � (12) where CA ∈ Rqs2×qs2, CB ∈ Rqs2×ps2 are the doubly-circular matrix representations of A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For instance, Vec(B ∗ hin) = CB Vec(hin) where ∗ is the convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We choose Ψs = Ψ ⊗ Is with Ψ = diag(ed) so that individual channel has a constant scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' To ensure that (12) is nonexpansive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 5 Algorithm 1 Nonexpansive convolutional layer Require: hin ∈ Rp×s×s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' P ∈ R(p+q)×q×s×s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' d ∈ Rq 1: ˜hin ← FFT(hin) 2: Ψ ← diag(ed),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' � ˜A ˜B �∗ ← Cayley(FFT(P)) 3: ˜h[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j] ← √ 2Ψ−1 ˜B[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' :,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j]˜hin[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j] 4: ˜h ← FFT � σ(FFT−1(˜h) + b) � 5: ˜hout[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j] ← √ 2A[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' :,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j]∗Ψ˜h[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' j] 6: hout ← FFT−1(˜hout) we need to construct CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' CB using the Cayley transformation (6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' which involves inverting a highly-structured large matrix I + CZ ∈ Rqs2×qs2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Thanks to the doubly-circular structure, we can perform efficient computation on the Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Taking a 2D case for example, circular convolution of two matrices is simply the elementwise product of their representations in the Fourier domain [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In [30], the 2D convolution theorem was extended to multi-channel circular convolutions of tensors, which are reduced to a batch of complex matrix-vector products in the Fourier domain rather than elementwise products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For example, the Fourier-domain output related to the (i, j)th pixel is a matrix-vector product: FFT(B ∗ hin)[:, i, j] = ˜B[:, :, i, j]˜hin[:, i, j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' where ˜B[:, :, i, j] ∈ Cq×p and ˜hin[:, i, j] ∈ Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Here ˜x = FFT(x) is the fast Fourier transformation (FFT) of a multi-channel tensor x ∈ Rc1×···×cr×s×s: FFT(x)[i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , ir, :, :] = Fsx[i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , ir, :, :]F∗ s where Fs[i, j] = 1 se−2π(i−1)(j−1)ι/s with ι = √−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Moreover, transposing or inverting a convolution is equivalent to applying the complex version of the same operation to its Fourier domain representation – a batch of small complex matrices: FFT(A⊤)[:, :, i, j] = ˜A[:, :, i, j]∗, FFT((I + Z)−1)[:, :, i, j] = (I + ˜Z[:, :, i, j])−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Since the FFT of a real tensor is Hermitian-symmetric, the batch size can be reduced to s × (⌊s/2⌋ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We now give both model parameterization and forward computation of a nonexpansive convolutional layer in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In line 1 and 6, we use the (inverse) FFT on the input/output tensor, which can be either/both removed for multiple consecutive convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In line 2, we perform the Cayley transformation of convolutions in the Fourier domain, which involves s × (⌊s/2⌋ + 1) parallel complex matrix inverse of size q × q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In line 3-5, all operations related to the (i, j)th term can be done in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3 Comparison to Semi-Orthogonal Layers In this section we will compare the proposed approach to a closely related method developed in [30], which also applies the Cayley transform to construct non-expansive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We first compare from the layer point of view as both approaches provide parameterization for nonexpansive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We show that our layer parameterization is more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Second, we compare the networks constructed from those two layers, in terms of layerwise spectral bound �L k=0 ∥Wk∥, which is a loose Lipschitz upper bound of the network Jacobian operator Jcfθ = WL L � k=1 JL−kWL−k ∈ RnL+1×n0 where Jk = Jcσ(Wkzk + bk) ∈ Jnk+1 + with Jc as the generalized Clake Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It often leads to conservative results when training a 1-Lipschitz network subject to the naive bound �L k=0 ∥Wk∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We present some 6 theoretical analysis to show that our parameterization allows for both the individual layer and network spectral bound to be larger than 1, while the network Lipschitz constant is still bounded by a weighted layerwise spectral bound of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In the next section, we will illustrate this result using a toy example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Layer-level comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The core component in [30] is the parameterization of (semi)-orthogonal layers via Cayley transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It takes a free variable P ∈ Rq×p and then produce (semi)-orthogonal weight matrix via W = Cayley(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However, for the case p = q, such parameterization is incomplete as it is limited to the orthogonal matrices without −1 eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' As shown in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4 our parameterization is complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', it includes any weight matrix whose eigenvalues within the unit disk, at the the cost of extra q×q parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Taking p = q = 2 for example, Cayley(P) only contains rotation matrices while 2A⊤B can also include reflection matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', we obtain W = diag(−1, 1) via (6) with X = 1 3 � 0 0 2 √ 2 0 � , Y = 1 3 � 1 √ 2 √ 2 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Network-level comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The γ-LBDN constructed by semi-orthogonal layers has weight matrices of Wk = �√γ Cayley(Pk), k = 0, L Cayley(Pk), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1, (13) where Pk ∈ Rnk+1×nk is a free variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It is easy to verify ∥W0∥ = √γ, ∥Wk∥ = 1, 0 ≤ k < L, ∥WL∥ = √γ, which further provides the Lipschitz certification by ∥Jcf∥ ≤ �L k=0 ∥Wk∥ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However, such model set is quite restrictive as the bound estimation is quite loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Our parameterization uses more sophisticated weighting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' full matrices) for the spectral bounds, which leads to a more expressive model set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The forward network with (7) satisfy the following weighted spectral bounds: � � � � � � � ∥B+ 0 Ψ0W0∥ ≤ √2γ, ���B⊤ k ΨkWkΨ−1 k−1 � A⊤ k−1 �+��� ≤ 2, 1 ≤ k < L, ∥WLΨ−1 L−1 � A⊤ L−1 �+∥ ≤ √2γ, which further implies that ∥Jcf∥ ≤ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 4 Experiments Our experiments have two goals: First, to show that our model parameterization can provide a tight Lipschitz bounds, which we illustrate with simple curve-fitting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Second, to examine the performance and scalability of the proposed new layer parameterization on adversarially robustness image classification tasks, illustrated with the MNIST and CIFAR-10 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Throughout this section we use sandwich and orthogon to denote the LBDNs constructed from the proposed sandwich layer and orthogonal layer from [30], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Further results and training details can be found in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Tightness of Lipschitz Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We illustrate the tight Lipschitz bound of our model parameterization some toy examples of curve fitting: square wave: f(x) = � 0, x ∈ [−1, 0) ∪ [1, 2] 1, x ∈ [−2, −1) ∪ [0, 1) multi-sine: f(x) = sin(x) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 sin(5x), x ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (14) 7 −2 −1 0 1 2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 x y True curve Sandwich γ = 2 Sandwich γ = 5 MLP Orthogon γ = 2 Orthogon γ = 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5π π 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5π 2π −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 x y Train data True curve Sandwich Orthogon MLP (a) square wave (b) multi-sine wave Figure 2: Simple 1D curve-fitting tasks with Lipschitz bound constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Table 1 includes details of the model structures and numerical results of the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note that we used wider layers for the orthogonal and MLP layers in order to approximately match the total number of parameters with the sandwich layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the square wave, the true function has no global Lipschitz bound due to the points of discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Thus a function approximator will naturally try to find models with large local Lipschitz constant near those singular points, and if a global γ-Lipschitz constraint is imposed this is a useful test of its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='We evaluate the tightness of Lipschitz bound using γ/γ where γ is an empirical lower Lipschitz bound obtained by a PGD-like method and γ is the imposed upper bound, which was 1, 2, and 5 in the cases we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In Table 1 it can be seen that our approach achieves a much tighter Lipschitz bounds for all three cases, roughly 99% versus 75% for orthogonal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In Figure 3 we break down the Lipschitz bounds and spectral norms over layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It can be seen that both the orthogonal layer zk → zk+1 and sandwich layer hk → hk+1 have quite tight Lipschitz bounds on a per-layer basis of around 98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However, for the complete network the sandwich layer achieves a much tighter bound of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='6% vs 75%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='This illustrates the benefits of taking into account coupling between neighborhood layers, thus allowing individual layers to have spectral norm greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We note that, for the sandwich model, the layer-wise product of spectral norms reaches 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2, illustrating how poor this commonly-used bound is compared to our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the multi-sine, we have a sum of a low-frequency sine wave and high-frequency sine-waves, each of which individually have a maximum slope (Lipschitz bound) of 1, and the true Lipschitz constant is 2 (achieved at x = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We fit Lipschitz-bounded MLP models using the proposed sandwich layers and orthogonal layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It can be seen in Table 1 that the tightness of our Lipschitz bound is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4%, whereas for the orthogonal layers it is only 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The MLP model has a maximum slope of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='051, despite the actual curve being fit having a maximum slope of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Adversarial Robust Training of Image Classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We next examine an application in image classifica- tion, using the Lipschitz bounds to enhance adversarial robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The first case we considered was the MNIST data set with fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In Figure 4 we observe that the sandwich layer had lower test error than the orthogonal layer in all cases, illustrating the improved flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Both layers achieved have similarly tight Lipschitz bounds, however the were not nearly as tight as in the curve fitting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We note that with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 both models offer significant robustness advantages to adversarial perturbations, while Sandwich offers better nominal test error that MLP, but Orthogonal degrades test error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In fact, the sandwich layer with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 achieves similar test error to the orthogonal layer γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 and 1, while offering much better robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For CIFAR10 we fit multi-layer convolutional models, and see similar trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We note that this is, to our knowledge, the first experimental result of multi-layer 2D convolutional models satisfying the Lipschitz bounds 8 Table 1: Hyperparameters and Lipschitz bounds for the 1D curve fitting tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Square wave Sine wave Sandwich Orthogon MLP Sandwich Orthogon MLP Fc: depth×width 8×90 8×128 8×128 8×90 8×128 8×128 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' params 139,512 132,491 132,481 139,512 132,491 132,481 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' lr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 Epochs 200 200 200 400 400 400 Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ / Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='998/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='738/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='7/– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='984/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='613/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='051/– Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ / Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='998/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='472/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 – – – – Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ / Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='950/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='355/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 – – – – 75 80 90 100 Input layer Hidden layer 1 Hidden layer 2 Hidden layer 3 Hidden layer 4 Hidden layer 5 Hidden layer 6 Hidden layer 7 Hidden layer 8 Output layer Full network γ/γ (%) Sandwich Orthogon 1 2 3 ∥W ∥ Figure 3: Left: empirical Lipschitz bound for curve fitting of a square wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The lower bound γ is obtained using PGD-like method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We observed tight layer Lipschitz bound for both orthogonal and sandwich layers (≥ 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However, the propose sandwich layer has a much tighter Lipschitz bound for the entire network (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='6% versus 75%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Right: the spectral norm of weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Our approach admits weight matrices with spectral norm larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The layerwise product �L k=0 ∥Wk∥ is about 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2, which is much larger than that of orthogonal layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 9 10−1 100 101 102 103 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 %Test error (MNIST) Sandwich γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Sandwich γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 Sandwich γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 Orthogon γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Orthogon γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 Orthogon γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 MLP 0 1 2 3 4 0 20 40 60 80 100 Sandwich γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Sandwich γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 Sandwich γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 Orthogon γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Orthogon γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 Orthogon γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 MLP 100 101 102 103 10 15 20 25 30 Lipschitz %Test error (CIFAR-10) Sandwich γ = 1 Sandwich γ = 10 Sandwich γ = 100 Orthogon γ = 1 Orthogon γ = 10 Orthogon γ = 100 CNN 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 2 20 40 60 80 100 Perturbation size Sandwich γ = 1 Sandwich γ = 10 Sandwich γ = 100 Orthogon γ = 1 Orthogon γ = 10 Orthogon γ = 100 CNN Figure 4: Image classification: test error vs Lipschitz constants (left) and robustness to adversarial perturbations (right), for MNIST with fully-connected layers (top) and CIFAR-10 with convolutional layers (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note that the proposed sandwich layer achieves both better test error and greater robustness than alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Again we see that Lipschitz bounds improve test error compared to a vanilla CNN, while also improving robustness dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' And again, the sandwich layer with γ = 1 achieves similar test error to the orthogonal layer with γ = 10, while offering much better robustness, see additional results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A major aim of our paper is to make the Lipschitz bounds of [14] tractable for training of larger models, compared to previous methods that depended on semidefinite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In Figure 5 we plot the training curves (test-error vs epoch) and the computational time per epoch for the sandwich, orthogonal, and MLP/CNN models for MNIST (fully-connected models) and CIFAR-10 (convolutional models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Firstly, we observe that for both fully-connected and convolutional models, the training time per epoch of the sandwich model is only around double that of a vanilla model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Furthermore, this is offset by the fact that in both cases the sandwich model surpasses the final error of the MLP/CNN in less than half as many epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In fact, for the fully connected case it does so in around a third as many epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' An interesting observation from Figure 5 is that both the MLP and CNN models seem to exhibit the epoch-wide double descent phenomenon (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', [32]), whereas neither of the Lipschitz bounded models (sandwich and orthogonal) do, they simply improve test error monotonically with epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Weight regularization has been suggested as a mitigating factor for other forms of double descent [33], however we are not aware of this specific phenomenon having been observed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 5 Conclusions and Future Work In this paper we have introduced a new parameterization of neural networks that automatically satisfy the tightest currently-known computationally-tractable Lipschitz bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' This enables learning of Lipschitz- 10 0 20 40 60 80 100 0 2 4 6 8 10 Test error (%) MNIST Sandwich Orthogon MLP 0 20 40 60 80 100 20 40 60 CIFAR-10 Sandwich Orthogon CNN 0 20 40 60 80 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='8 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4 Epochs Training time (s) 0 20 40 60 80 100 2 4 6 8 10 12 Epochs Figure 5: Learning curves and training time per epoch for image classification tasks, obtained from 5 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The Lipschitz bounds are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 for MNIST and 10 for CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note that the “double-descent” phenomenon is avoided with the Lipschitz-bounded models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Table 2: Test error, empirical adversarial robustness and empirical lower Lipschitz bound on image classification benchmarks, mean and standard deviation from 5 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Sandwich layers outputs other methods in both test and ℓ2 certifiable robust accuarcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' MNIST Sandwich Orthogon MLP Cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 – Emp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' γ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='32±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='06E-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='41±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='78±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 9.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='62 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='85±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='07 bounded networks with standard first-order gradient methods, avoiding the need for complex projections or barrier evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We have illustrated the approach on simple curve-fitting tasks and adversarially-robust image classification with both fully-connected and convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Our future work will include exploring applications in other settings including robust reinforcement learning and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We also note that the current approach to convolutional models (Sec 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2) is limited to doubly-circular convolutions, which we will seek to address in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 11 References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Szegedy, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Zaremba, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Sutskever, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Bruna, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Erhan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Goodfellow, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Fergus, “Intriguing properties of neural networks,” in ICLR: International Conference on Learning Representations, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [2] A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Winston and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Kolter, “Monotone operator equilibrium networks,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 10718–10728, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Rauber, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Brendel, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Bethge, “Foolbox: A python toolbox to benchmark the robustness of machine learning models,” arXiv preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='04131, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' A Preliminaries on LMI-based Lipschitz bound estimation Here we review the theoretical work of LMI-based Lipschitz bound estimation for neural networks from [14, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Consider an L-layer forward network y = f(x) described by the following recursive equation: z0 = x, zk+1 = σ(Wkzk + bk), k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1, y = WLzL + bL, (15) where x ∈ Rn0, zk ∈ Rnk, y ∈ RnL+1 are the network input, hidden unit of the kth layer and network output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We stack all hidden unit z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , zL together and obtain a compact form of (15) as follows: z � �� � � ���� z1 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' zL � ���� = σ � � � � � � � � � � � W � �� � � ����� 0 W1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 0 0 · · WL−1 0 � ����� � ���� z1 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' zL � ���� + U � �� � � ���� W0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 0 � ����x + bz � �� � � ���� b0 b1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' bL−1 � ���� � � � � � � � � � � � , y = Y � �� � �0 · · 0 WL � � ���� z1 z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' zL � ���� + by ���� bL .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (16) By introducing an intermediate variable v ∈ Rn with n = �L k=1 nk, we can rewrite the above equation by v = Wz + Ux + bz, z = σ(v), y = Y z + by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (17) Given two different solutions sa = (xa, va, za, ya) and sb = (xb, vb, zb, yb), their difference ∆s = sb − sa satisfies ∆v = W∆z + U∆x, ∆z = σ(vb) − σ(va) := Jab∆v, ∆y = Y ∆z (18) where Jab ∈ Jq +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For any Lipschitz bound γ > 0 we have γ2∥∆x∥2 − |∆y∥2 ≥ 0 holds for all nonzero ∆s satisfying (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The main challenge is how to handle the varying weight Jab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' One way is to replace it with a quadratic inequality constraint, which might be conserve but easy to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 14 Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' If Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 holds, then for any Λ ∈ Dn ++ the following incremental quadratic constraint (iQC) holds for any pair of (va, za) and (vb, zb) satisfying z = σ(v): �∆v⊤ ∆z⊤ �⊤ �0 Λ Λ −2Λ � �∆v ∆z � ≥ 0 (19) where ∆v = vb − va and ∆z = zb − za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 implies that each channel satisfies 2∆zi(∆vi − ∆zi) ≥ 0, which can be leads to (19) by a linear conic combination of each channel with multiplier Λ ∈ Dn ++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In [14] it was claimed that iQC (19) holds with a richer (more powerful) class of multipliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Λ is a symmetric matrix), which were previously introduced for robust stability analysis of systems with repeated nonlinearities [34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' However this is not true: a counterexample was given in [25], and here we give a brief explanation: even if the nonlinearities σ(vi) are repeated when considered as functions of vi, their increments ∆zi = σ(va i + ∆vi) − σ(va i ) are not repeated when considered as functions of ∆vi, since the diagonal elements of Jab depend on the particular va i which generally differs between units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The feedforward neural network (15) is γ-Lipschitz if Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 holds, and there exist an Λ ∈ Dn ++ satisfying the following LMI: H := � � γI −U ⊤Λ 0 −ΛU 2Λ − ΛW − W ⊤Λ −Y ⊤ 0 −Y γI � � ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (20) Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' In [23], the above LMI condition also applies to more general network structures with full weight matrix W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' An equivalent form of (20) was applied in [14] for a tight Lipschitz bound estimation: min γ,Λ γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (20) (21) which can be solved by convex programming for moderate models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', n < 10K in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' B Derivation of direct parameterization First, by substituting W, U, Y and Λ into (20), its left-hand side can be rewritten as H = � ������������ γI −W ⊤ 0 Λ0 −Λ0W0 2Λ0 −W ⊤ 1 Λ1 −Λ1W1 2Λ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 2ΛL−2 −W ⊤ L−1ΛL−1 −ΛL−1WL−1 2ΛL−1 −W ⊤ L −WL γI � ������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (22) Note that H ⪰ 0 has a chordal structure, which means that it can be factorized as H = PP ⊤ with P = � ���� D−1 −V0 D0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' −VL DL � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 15 By substituting it back into (22) we have D−1D⊤ −1 = γI, VkV ⊤ k + DkD⊤ k = 2Λk, 0 ≤ k < L, VLV ⊤ L + DLD⊤ L = γI, (23) Wk = Λ−1 k VkD⊤ k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (24) By defining Ψk = Λ 1 2 k , Ak := √ 2ΨkDk and Bk := √ 2ΨkV ⊤ k with k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1 we have AkA⊤ k + BkB⊤ k = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, we can easily parameterize (Ψk, Ak, Bk) via simple exponential mapping of diagonal matrices and Cayley transformation on full matrices, see (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Finally, we can obtain the weight matrices as follows Wk = Λ−1 k VkD⊤ k−1 = Ψ−2 k × ( √ 2ΨkBk) × ( √ 2A⊤ k−1Ψk−1) = 2Ψ−1 k BkA⊤ k−1Ψk−1 (25) with k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1, where A−1 = I and Ψ−1 = � γ/2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The above formula can also be applied to k = L by choosing ΨL = � 2/γI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' C Proofs C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1 By substituting ∆z = Jab∆v into (19) we have �∆v⊤ ∆z⊤ �⊤ �0 Λ Λ −2Λ � �∆v ∆z � = 2∆z⊤Λ(∆v − ∆z) = 2∆v⊤JabΛ(I − Jab)∆v ≥ 0 where the last inequality follows as Jab ∈ Jq +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3 We first apply Schur complement to (20), which yields � γI −U ⊤Λ −ΛU 2Λ − ΛW − W ⊤Λ − 1 γ Y ⊤Y � ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, by left-multiplying the above equation by � ∆x⊤ ∆z⊤� and right-multiplying � ∆x⊤ ∆z⊤�⊤ we can obtain γ∥∆x∥2 − 1 γ ∥∆y∥2 − 2∆z⊤Λ∆z − 2∆z⊤Λ(W∆z + U∆x) = γ∥∆x∥2 − 1 γ ∥∆y∥2 − 2∆z⊤Λ(∆z − ∆v) ≥ 0, (26) which further implies that (15) is γ-Lipschitz since γ∥∆x∥2 − 1 γ ∥∆y∥2 ≥ 2∆z⊤Λ(∆v − ∆z) ≥ 0 where the last inequality follows by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='2 Sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We show that (20) holds with Λ = diag(Λ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , ΛL−1) where Λk = Ψ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Since the structure of H is a chordal graph, H ⪰ 0 is equivalent to the existence of a chordal decomposition [37]: H = L � k=0 EkHkE⊤ k (27) 16 where 0 ⪯ Hk ∈ R(nk+nk+1)×(nk+nk+1) and Ek = �0a,k Ib,k 0c,k � with Ib,k being the identity matrix the same size as Hk, and 0a,k, 0c,k being zero matrices of appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We then construct Hk as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For k = 0, we take H0 = � γI −√2γB⊤ 0 Ψ0 −√2γΨ0B0 2Ψ0(I − A0A⊤ 0 )Ψ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (28) Note that H0 ⪰ 0 since [H0]11 = γI ≻ 0, and the Schur complement to [H0]11 yields 2Ψ0(I − A0A⊤ 0 )Ψ0 − � 2γΨ0B0 1 γ I � 2γB⊤ 0 Ψ0 = 2Ψ0(I − A0A⊤ 0 − B0B⊤ 0 )Ψ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' , L − 1 we take Hk = �2Ψk−1Ak−1A⊤ k−1Ψk−1 −2Ψk−1Ak−1B⊤ k Ψk −2ΨkBkA⊤ k−1Ψk−1 2Ψk(I − AkA⊤ k )Ψk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (29) If Ak−1 is zero, then it is trival to have Hk ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For nonzero Ak−1, we can verify that Hk ⪰ 0 since the Schur complement to [Hk]11 shows 2Ψk(I − AkA⊤ k )Ψk − 2ΨkBkA⊤ k−1Ψk−1 � 2Ψk−1Ak−1A⊤ k−1Ψk−1 �+ 2Ψk−1Ak−1B⊤ k Ψk =2Ψk(I − AkA⊤ k − BkB⊤ k )Ψk + 2ΨkBk(I − A+ k−1Ak−1)B⊤ k Ψk =2ΨkBk(I − A+ k−1Ak−1)B⊤ k Ψk ⪰ 0 where X+ denotes the Moore–Penrose inverse of the matrix X, and it satisfies I − X+X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For k = L we take HL = � 2ΨL−1AL−1A⊤ L−1ΨL−1 −√2γAL−1B⊤ L ΨL−1 −√2γΨL−1BLA⊤ L−1 γI � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (30) Similarly, we can conclude HL ⪰ 0 using Schur complement γI− � 2γΨL−1BLA⊤ L−1 � 2ΨL−1AL−1A⊤ L−1ΨL−1 �+ � 2γAL−1B⊤ L ΨL−1 = γΨL−1BL(I−A+ L−1AL−1)B⊤ L ΨL−1 ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We now show that Hk with k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' L satisfy the chordal decomposition (27) holds since [Hk]21 = −2ΨkBkA⊤ k−1Ψk−1 = −Ψ2 k(2Ψ−1 k BkA⊤ k−1Ψk−1) = −ΛkWk, [Hk]22 + [Hk+1]11 = 2Ψk(I − AkA⊤ k )Ψk + 2ΨkAkA⊤ k Ψk = 2Ψ2 k = 2Λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Finally, we conclude that H ⪰ 0 from [37][Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For any Wk and Λk satisfying (20), we will find set of free variables dk, Xk, Yk such that (7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We take Ψk = Λ 1 2 which further leads to dk = diag(log Ψk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' By letting A−1 = I, Ψ−1 = � γ/2I and ΨL = � 2/γI we then construct Ak, Bk recursively via Bk = 1 2ΨkWkΨ−1 k−1A−⊤ k−1, Ak = chol(I − BkB⊤ k )Qk (31) where chol(·) denotes the Cholesky factorization, Qk is an arbitrary orthogonal matrix such that Ak does not have eigenvalue of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' If Ak−1 is non-invertible but non-zero, we replace A−⊤ k−1 with � A+ k−1 �⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' If Ak−1 = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Wk = 0), we simply reset Ak−1 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It is easy to verify that Ψk, Ak and Bk satisfy the model parameterization (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Finally, we can construct Xk, Yk using (36), which is well-defined as Ak does not have eigenvalue of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 17 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3 The proposed layer (9) can be rewritten as a compact network (17) with W = 0, Y = √ 2A⊤Ψ and U = √ 2Ψ−1B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', v = Uhin + b, z = σ(v), hout = Y z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' From the model parameterization (6) we have AA⊤ + BB⊤ = I, which further implies 2Ψ2 − Y ⊤Y − Ψ2UU ⊤Ψ2 = 2Ψ2 − 2ΨAA⊤Ψ − 2ΨBB⊤Ψ = 2Ψ(I − AA⊤ − BB⊤)Ψ = 0 By applying Schur complement twice to the above equation we have � � I −U ⊤Ψ2 0 −Ψ2U 2Ψ2 −Y ⊤ 0 −Y I � � ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, the 1-Lipschitzness of (9) is obtained by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='4 Sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' It is a direct corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='3 by taking the identity operator as the nonlinear activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Here we give a constructive proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' That is, given a weight matrix W with ∥W∥ ≤ 1, we will find a (generally non-unique) pair of (X, Y ) such that 2A⊤B = W with A, B given by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We first construct A, B from W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Since it is obvious for W = 0, we consider the case with nonzero W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' First, we take a singular value decomposition (SVD) of W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' W = UwΣwV ⊤ w where Uw is a q × q orthogonal matrix, Σw is an q × p rectangular diagonal matrix with Σw,ii ≥ 0 non-increasing, Vw is a p × p orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Then, we consider the candidates for A and B as follows: A = UΣaU ⊤ w , B = UΣbV ⊤ w (32) where Σa is a diagonal matrix, Σb a rectangular diagonal matrix U ∈ Rq×q an orthogonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' By substituting (32) into the equalities AA⊤ + BB⊤ = Iq and W = 2A⊤B we have Σ2 a + Σ2 b′ = Iq, 2ΣaΣb′ = Σw′ (33) where Σb′, Σw′ ∈ Rq×q are obtained by either removing the extra columns of zeros on the right or adding extra rows of zeros at the bottom to Σb and Σw, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The solution to (33) is Σa,ii = 1 2 �� 1 + Σw′,ii + � 1 − Σw′,ii � , Σb′,ii = 1 2 �� 1 + Σw′,ii − � 1 − Σw′,ii � (34) where are well-defined as ∥W∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Now we can obtain Σb from Σb′ by removing extra rows of zeros at the bottom or adding extra columns of zeros on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' At last, we pick up any orthogonal matrix U such that A = UΣaU ⊤ w does not have eigenvalue of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The next step is to find a pair of (X, Y ) such that A⊤ = (I + Z)−1(I − Z), B⊤ = −2Y (I + Z)−1, Z = X − X⊤ + Y ⊤Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' (35) One solution to the above equation is Z = (I − A⊤)(I + A⊤)−1, Y = −1 2B⊤(I + Z), X = 1 2tril(Z − Z⊤) (36) where tril(W) denotes the strictly lower triangle part of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Note that the above solution is well-defined since A does not has eigenvalue of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='6 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5 From (28) we have H0 = � γI −W ⊤ 0 Ψ2 0 −Ψ2 0W0 2Ψ0B0B⊤ 0 Ψ0 � ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Applying the Schur complement yields γI − 1/2W ⊤ 0 Ψ0(B0B⊤ 0 )+Ψ0W0 ⪰ 0, which implies ∥B+ 0 Ψ0W0∥ ≤ √2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' From (29) we obtain Hk = �2Ψk−1Ak−1A⊤ k−1Ψk−1 −W ⊤ k Ψ2 k −Ψ2 kWk 2ΨkBkB⊤ k Ψk � ⪰ 0 ⇒Ψk−1Ak−1A⊤ k−1Ψk−1 − 1 4W ⊤ k Ψk(BkB⊤ k )+ΨkWk ⪰ 0 ⇒I − 1 4A+ k−1Ψ−⊤ k−1W ⊤ k Ψk(BkB⊤ k )+ΨkWkΨ−1 k−1 � A⊤ k−1 �+ ⪰ 0 ⇒ ���� 1 2B+ k ΨkWkΨ−1 k−1 � A⊤ k−1 �+ ���� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Similarly, from (30) we have HL = � 2ΨL−1AL−1A⊤ L−1ΨL−1 −W ⊤ L −WL γI � ⪰ 0 ⇒ ���WLΨ−1 L−1 � A⊤ L−1 �+��� ≤ � 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' The bound of Jcf is then obtained by ∥Jcf∥ = ∥WLJL−1WL−1 · · · J0W0∥ = ����� 1 2WLΨ−1 L−1 � A⊤ L−1 �+(2A⊤ L−1JL−1BL−1) 1 � k=L−1 �1 2B+ k ΨkWkΨ−1 k−1 � A⊤ k−1 �+ � (2A⊤ k−1Jk−1Bk−1)(B+ 0 Ψ0W0) ����� ≤ �� 2γ �2/2 = γ where the inequality follows as 2A⊤ k JkBk is the Clake Jacobian of a nonexpansive layer (9), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' ∥2A⊤ k JkBk∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' D Experiments Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the square wave experiment, we take 300 and 600 samples (xi, yi) with xi ∼ U([−2, 2]) for training and testing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the sine wave experiment, we take 50 and 200 points (xi, yi) for training and testing, where xi = 2i/Nπ with N as the size of dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We use batch size of 50 for those two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the CIFAR-10 dataset, we applied standard augmentation,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', random cropping and flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Inputs to all image classification models are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Model architectures for image classification benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We trained small fully-connected model on MNIST and the KWLarge network from [38] on CIFAR-10, with detailed architecture listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' To make the different models have similar number of parameters in the same experiment, we slightly reduce the hidden layer width of sandwich model in the MNIST experiment and increases width of the first fully-connected layer of CNN and orthogonal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We also applied the emulated 2-stride from [30] to the second and fourth convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For each experiment, we also trained a conventional MLP/CNN model with similar amount of parameters as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For all experiments, we used a piecewise triangular learning rate [39] with maximum rate in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='001}, choosing the one with the best test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We found that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='01 often works 19 Table 3: Model architectures for image classification benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' MNIST Sandwich Orthogon MLP Fc (190, 190, 128) (256,256,128) (256,256,128) Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' params 300,374 300,942 300,938 KWLarge (CIFAR-10) Sandwich Orthogon CNN Conv (32,32,64,64) (32,32,64,64) (32,32,64,64) Fc (512, 512) (640,512) (640,512) Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' params 2,982,225 2,995,373 3,020,970 well and took it as a default rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We use Adam [40] and ReLU as our default optimizaer and activation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For the image classification tasks, we used similar loss function and initialization method as [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' Because the Cayley transform in (6) involves both linear and quadratic terms, we implemented the weight normalization method from [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' That is, we reparameterize X, Y in Z = X −X⊤ +Y ⊤Y by g X ∥X∥F and h Y ∥Y ∥F with learable scalars g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For all γ-LBDNs, we choose hidden layer to be nonexpansive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', 1-Lipschitz) and put a scaling factor √γ on the input and output, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' On the image classification tasks, we did not observe significant improvement by uniformly distributing the Lipschitz capacity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', each nonexpansive hidden layer is scaled by γ 1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We choose Lipschitz bounds of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='0 for MNIST and 1,10,100 for CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' We search for the empirical lower Lipschitz bound γ of a network fθ by a PGD-like method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=', updating the input x and its deviation δx based on the gradient of ∥fθ(x + δx) − fθ(x)∥/∥δx∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' As we are interested in the global lower Lipschitz bound, we do not project x and x + δx into any compact region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' For robustness verification, we used the L2FastGradientAttack method from the package foolbox [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFJT4oBgHgl3EQfYiwU/content/2301.11526v1.pdf'} +page_content=' All image classification experiments were performed on a Nvidia A5000.' metadata={'source': 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+1,2179 @@ +arXiv:2301.01157v1 [math.AT] 3 Jan 2023 +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS +D’HOMOTOPIE ET D’HOMOLOGIE SINGULI`ERE +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +R´esum´e. La cat´egorie des espaces topologiques avec deux points marqu´es est munie de deux +familles Fn et Hn, index´ees par un entier n ě 0, de foncteurs vers la cat´egorie des groupes +ab´eliens, la premi`ere associant `a l’objet pX, x, yq le quotient de Zπ1pX, x, yq par un sous- +groupe ab´elien associ´e `a la n ` 1-i`eme puissance de d’id´eal d’augmentation de l’alg`ebre de +groupe Zπ1pX, xq, la seconde associant au mˆeme objet le n-i`eme groupe d’homologie sin- +guli`ere relative de Xn par rapport `a un sous-espace d´efini en termes de diagonales partielles. +Nous construisons une famille de transformations naturelles νn : Fn Ñ Hn. Nous identifions +la transformation naturelle obtenue par restriction de νn `a la sous-cat´egorie des vari´et´es +alg´ebriques et tensorisation avec Q avec l’´equivalence naturelle due `a Beilinson. +Table des mati`eres +Introduction +2 +1. +Rappels +3 +1.1. +Espaces topologiques et homologie singuli`ere +3 +1.2. +Paires d’espaces topologiques et (co)homologie singuli`ere relative +3 +2. +Une identit´e en homologie relative +3 +2.1. +Mat´eriel de base et r´esultat principal +4 +2.2. +D´emonstration de (a) du th´eor`eme 2.3 +5 +2.3. +Constructions combinatoires +5 +2.3.1. +Diagramme commutatif impliquant une involution de Zn ˆ Sn ˆ rr0, nss +5 +2.3.2. +Constructions et r´esultats relatifs aux permutations +7 +2.3.3. +Diagramme commutatif impliquant les applications pf, sgnq, p ˜f, Ą +sgnq et bij +11 +2.3.4. +Battages et transpositions +12 +2.3.5. +La bijection bij +13 +2.3.6. +Diagramme commutatif impliquant Affp∆n´1, ∆nq ˆ t˘1u +17 +2.4. +Construction d’endomorphismes de groupes de chaˆınes +19 +2.4.1. +Morphismes dans une cat´egorie C +19 +2.4.2. +D´emonstration de divk +n ˝ Bn´1,n “ Bn´1,n ˝ divk +n´1 +20 +2.4.3. +Relation dans C entre divk +‚ et id‚ +21 +2.4.4. +Endomorphismes de groupes de chaˆınes singuli`eres +21 +2.5. +D´emonstration de (b) du th´eor`eme 2.3 +22 +2.5.1. +Composition de chemins +22 +2.5.2. +Calcul de pdivk +nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq +22 +2.5.3. +Une ´egalit´e dans CnpXnq +24 +2.5.4. +D´emonstration de (b) du th´eor`eme 2.3 +26 +3. +Lien avec l’isomorphisme de Beilinson +26 +3.1. +Rappels sur l’isomorphisme de Beilinson +27 +3.2. +Relation du th´eor`eme 2.3 avec l’isomorphisme de Beilinson +27 +4. +Construction de transformations naturelles +30 +Bibliographie +31 +Date: 3 janvier 2023. +1 + +2 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +Introduction +Pour X un espace topologique connexe et a, b P X, on note π1pa, bq l’ensemble des classes +de chemins reliant a `a b. Le Z-module Zπ1pa, bq est alors un module `a droite sous l’action de +l’alg`ebre du groupe π1paq :“ π1pa, aq, et on d´efinit, pour n ě 0, le Z-module FnpX, a, bq comme +son quotient par le sous-module Zπ1pa, bq ¨ pZπ1paqqn`1 +` +engendr´e par l’action de la n ` 1-`eme +puissance de l’id´eal d’augmentation de l’alg`ebre de groupe de π1paq. +Si X est une vari´et´e diff´erentiable connexe, ayant le type d’homotopie d’un CW-complexe +fini, on dispose d’une interpr´etation cohomologique de HomZpFnpX, a, bq, Qq, sous la forme d’un +isomorphisme de Q-espaces vectoriels +(0.0.1) +HnpXn, Y pnq +ba ; Qq „ +Ñ HomZpFnpX, a, bq, Qq, +o`u Y pnq +ba +est la partie de Xn d´efinie par +(0.0.2) +Y pnq +ba +:“ Yn +i“0Y pnq +ba,i +avec Y pnq +ba,i “ tpx1, . . . , xnq P Xn|xi “ xi`1u avec x0 “ b et xn`1 “ a, et H‚p´, ´; Qq +d´esigne la cohomologie singuli`ere relative `a coefficients dans Q (travail de Beilinson, r´edig´e +dans [DG, BGFr]). La construction de cet isomorphisme repose sur des techniques faisceau- +tiques : pr´ecis´ement, on construit un morphisme b ˜Ka Ñb Ka de complexes de faisceaux sur +X et un isomorphisme isoba +BGF : HnpXn,b ˜Kaxnyq Ñ HnpXn, Y pnq +ba ; Qq, o`u Hp´, ´q d´esigne +l’hypercohomologie des complexes de faisceaux ([BGFr], lemme 3.281) ; (0.0.1) est alors con- +struit comme une composition +HnpXn, Y pnq +ba ; Qq +pisoba +BGFq´1 +Ñ +HnpXn,b ˜Kaxnyq Ñ HnpXn,b Kaxnyq Ñ HomZpFnpX, a, bq, Qq. +La nature topologique de la source et du but de l’application (0.0.1) sugg`ere la possibilit´e +d’une construction topologique de cette application. Le but de ce travail est de fournir une telle +construction, dans le cadre plus g´en´eral o`u X est un espace topologique, et en travaillant sur +Z. Plus pr´ecis´ement, nous contruisons un morphisme de Z-modules +FnpX, a, bq Ñ HnpXn, Y pnq +ab q, +o`u Hnp´, ´q est l’homologie singuli`ere relative (`a coefficients dans Z), cf. th´eor`eme 2.3. Nous +montrons, dans le cas o`u X est une vari´et´e diff´erentiable comme ci-dessus, la compatibilit´e +de ce morphisme avec (0.0.1) et l’isomorphisme HnpXn, Y pnq +ab q Ñ HnpXn, Y pnq +ba q induit par +l’automorphisme de Xn donn´e par px1, . . . , xnq ÞÑ pxn, . . . , x1q (cf. proposition 3.1). +Ce travail est organis´e comme suit : la section 1 contient des rappels sur l’homologie sin- +guli`ere ; la section principale est la section 2, qui a pour objectif la d´emonstration du th´eor`eme +2.3 ; la section 3 ´etablit le lien de ce r´esultat avec l’isomorphisme (0.0.1) de Beilinson (proposi- +tion 3.1) ; en section 4, on ´etudie l’aspect fonctoriel de l’application construite dans le th´eor`eme +2.3. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +3 +1. Rappels +En section 1.1, on rappelle la construction de l’homologie singuli`ere, et en section 1.2, celles +de l’homologie et la cohomologie relatives. +1.1. Espaces topologiques et homologie singuli`ere. On note Top la cat´egorie des espaces +topologiques ; si X, Y sont deux espaces topologiques, on note ainsi ToppX, Y q l’ensemble des +applications continues X Ñ Y . +Pour n ě 0, on note ∆n le simplexe donn´e par ∆n :“ tpt1, . . . , tnq P Rn|0 ď t1 ď . . . ď tn ď +1u. Si X est un espace topologique, on pose CnpXq :“ ZTopp∆n, Xq pour n ě 0, ainsi que +C´1pXq “ 0. Un sous-ensemble Y de X est naturellement muni de la topologie induite, on note +YX (ou simplement Y s’il n’y a pas de risque de confusion) l’espace topologique correspondant. +On a alors Topp∆n, YXq “ tf P Topp∆n, Xq|fp∆nq Ă Y u et CnpYXq :“ ZTopp∆n, YXq Ă +CnpXq pour n ě 0. +Pour X un espace topologique et n ě 0, on note B˚ +n,n´1 : CnpXq Ñ Cn´1pXq la diff´erentielle +singuli`ere. L’homologie du complexe pC‚pXq, B˚q est alors l’homologie singuli`ere H‚pXq. +On note AbGr celle des groupes ab´eliens Z-gradu´es. +L’homologie singuli`ere d´efinit un +foncteur H‚ : Top Ñ AbGr. +1.2. Paires d’espaces topologiques et (co)homologie singuli`ere relative. Si X est un +espace topologique et Y est un sous-ensemble de X, alors pC‚pYXq, B˚q est un sous-complexe +de pC‚pXq, B˚q, et l’homologie relative H‚pX, Y q est l’homologie du complexe quotient +pC‚pXq{C‚pYXq, B˚q; +c’est un groupe ab´elien gradu´e. La cohomologie relative H‚pX, Y q est celle du sous-complexe +C‚pYXqK du complexe HomZpC‚pXq, Zq, muni de la diff´erentielle duale de B˚ ; on dispose d’un +couplage H‚pX, Y q b H‚pX, Y q Ñ Z. Alors H‚pX, Y ; Qq “ H‚pX, Y q b Q est la cohomologie du +sous-complexe C‚pYXqK b Q de HomZpC‚pXq, Qq. +Soit Paires la cat´egorie des paires d’espaces topologiques. Les objets de Paires sont les cou- +ples pX, Y q, avec X espace topologique et Y sous-ensemble de X. On a PairesppA, Bq, pA1, B1qq :“ +tf P EnspA, A1q|f est continue et fpBq Ă B1u. Alors pX, Y q ÞÑ H‚pX, Y q d´efinit un foncteur +H‚ : Paires Ñ AbGr. On note f˚ le morphisme H‚pA, Bq Ñ H‚pA1, B1q dans AbGr associ´e `a +un morphisme f : pA, Bq Ñ pA1, B1q dans Paires (not´e H‚pfq dans [Ha], pp. 108, 124). +L’application identit´e id∆n d´efinit un ´el´ement de Cnp∆nq, dont l’image dans Cnp∆nq{CnpB∆nq +est un cycle pour le bord relatif, et d´efinit donc un ´el´ement du groupe d’homologie relative +rid∆ns P Hnp∆n, B∆nq. +2. Une identit´e en homologie relative +Le but de cette section est la d´emonstration du th´eor`eme 2.3. Ce r´esultat est formul´e en +section 2.1, et sa premi`ere partie (th´eor`eme 2.3(a)) est d´emontr´ee en section 2.2. +Le reste + +4 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +de la section est consacr´e `a la d´emonstration de sa deuxi`eme partie (th´eor`eme 2.3(b)). +La +section 2.3 contient des r´esultats combinatoires, et la section 2.4 l’application de ses r´esultats +`a la construction d’endomorphismes divk +‚ des complexes de chaˆınes singuli`eres satisfaisant la +relation d’homotopie (2.4.5). Cette relation est appliqu´ee en section 2.5 `a la d´emonstration du +th´eor`eme 2.3(b). +2.1. Mat´eriel de base et r´esultat principal. Dans la section 2.1, on fixe un espace topologique +X, des ´el´ements a, b P X, et n ě 1. On d´efinit la partie Y pnq +ab +Ă Xn par (0.0.2). On note +Chempa, bq l’ensemble des applications continues ˜γ : r0, 1s Ñ X telles que ˜γp0q “ a et ˜γp1q “ b. +D´efinition 2.1. Pour ˜γ P Chempa, bq, on note ˜γpnq P Topp∆n, Xnq l’application compos´ee +∆n cann +Ñ r0, 1sn ˜γn +Ñ Xn o`u cann : ∆n Ñ r0, 1sn est l’injection canonique. +Lemme 2.2. B˚ +n,n´1p˜γpnqq P Cn´1pY pnq +ab q. +D´emonstration. On a B˚ +n,n´1p˜γpnqq “ řn +i“0p´1qi˜γpnq ˝ Bn +i , o`u pour i P rr0, nss l’application Bn +i : +∆n´1 Ñ ∆n est donn´ee par pt1, . . . , tn´1q ÞÑ pt1, . . . , ti, ti, . . . , tn´1q (avec par convention t0 “ 0, +tn “ 1). On v´erifie que ˜γpnq ˝ Bn +i “ Bn,X +i +˝ ˜γpn´1q, avec Bn,X +i +: Xn´1 Ñ Xn l’application donn´ee +par px1, . . . , xn´1q ÞÑ px1, . . . , xi, xi, . . . , xn´1q (avec par convention x0 “ a, xn “ b), donc +B˚ +n,n´1p˜γpnqq “ řn +i“0p´1qiBn,X +i +˝ ˜γpn´1q. On a pour tout i P rr0, nss les relations Bn,X +i +pXn´1q “ +Y pnq +ab,i Ă Y pnq +ab +qui impliquent la relation annonc´ee. +□ +On rappelle que C‚pY pnq +ab q est un sous-complexe de C‚pXnq, et que l’homologie du complexe +quotient C‚pXnq{C‚pY pnq +ab q est l’homologie relative H‚pXn, Y pnq +ab q. +Il suit du lemme 2.2 que la classe de ˜γpnq dans CnpXnq{CnpY pnq +ab q est un cycle du complexe +quotient, et d´efinit donc une classe r˜γpnqs P HnpXn, Y pnq +ab q. +On note π1pa, bq le quotient de +Chempa, bq par la relation d’´equivalence donn´ee par l’homotopie entre deux chemins. +Th´eor`eme 2.3. (a) Il existe une unique application π1pa, bq Ñ HnpXn, Y pnq +ab q, γ ÞÑ Fnpγq +telle que l’application Chempa, bq Ñ HnpXn, Y pnq +ab q, ˜γ ÞÑ r˜γpnqs admette une factorisation +Chempa, bq Ñ π1pa, bq Ñ HnpXn, Y pnq +ab q. On a donc pour ˜γ P Chempa, bq et on notant ˜γ Ñ r˜γs +l’application canonique Chempa, bq Ñ π1pa, bq, +(2.1.1) +r˜γpnqs “ Fnpr˜γsq. +(b) Pour α0, . . . , αn P π1paq, et I Ă rr0, nss, on pose ś +iPI αi le produit αip1q ¨ ¨ ¨ αip|I|q, o`u i +est l’unique bijection croissante rr1, |I|ss Ñ I. Alors +ÿ +IĂrr0,nss +p´1q|I|Fnpγ ¨ +ź +iPI +αiq “ 0 +(´egalit´e dans HnpXn, Y pnq +ab q). On a donc factorisation de Fn en une application lin´eaire +F +pnq +ab : Zπ1pa, bq{pZπ1pa, bqpZπ1paqqn`1 +` +q Ñ HnpXn, Y pnq +ab q, +o`u pZπ1paqq` est l’id´eal d’augmentation de Zπ1paq. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +5 +2.2. D´emonstration de (a) du th´eor`eme 2.3. +Lemme 2.4. (a) Pour ˜γ P Chempa, bq, ˜γpnq induit un morphisme +˜γpnq,Paires : p∆n, B∆nq Ñ pXn, Y pnq +ab q +dans Paires. +(b) Pour ˜γ P Chempa, bq, on a r˜γpnqs “ p˜γpnq,Pairesq˚prid∆nsq P HnpXn, Y pnq +ab q. +D´emonstration. (a) suit de la d´emonstration du lemme 2.2. (b) suit de ce que l’´el´ement ˜γpnq P +CnpXnq est l’image par le morphisme ∆n Ñ Xn dans Top induit par ˜γpnq de id∆n P Cnp∆nq. +□ +Le th´eor`eme 2.3,(a) suit alors du lemme suivant : +Lemme 2.5. L’application ChempXq Ñ HnpXn, Y pnq +ab q, ˜γ ÞÑ r˜γpnqs est invariante par homo- +topie. +D´emonstration. Soit ˜γ, ˜γ1 P ChempXq. Une homotopie entre ˜γ et ˜γ1 produit une homotopie +entre les morphismes de paires p∆n, B∆nq Ñ pXn, Y pnq +ab q donn´es par ˜γpnq,Paires et p˜γ1qpnq,Paires. +Par l’invariance homotopique de l’homologie relative (Proposition 13.14 dans [GrH]), les mor- +phismes associ´es Hnp∆n, B∆nq Ñ HnpXn, Y pnq +ab q induits en homologie, `a savoir Hnp˜γpnq,Pairesq +et Hnpp˜γ1qpnq,Pairesq sont ´egaux. Les images qu’ils donnent `a rid∆ns sont donc ´egales, et le +lemme 2.4(b) implique alors r˜γpnqs “ r˜γ1pnqs. +□ +2.3. Constructions combinatoires. Le but de cette sous-section est la construction, pour +tout couple pn, kq avec n, k ě 1 : (a) d’un ensemble AffpRn´1, Rnq, d’une application pf, sgnq : +Zn ˆ Sn ˆ rr0, nss Ñ AffpRn´1, Rn´1q ˆ t˘1u et d’une involution invol de Zn ˆ Sn ˆ rr0, nss, +telle que le diagramme (2.3.1) commute ; (b) d’applications p ˜f, ˜ +sgnq : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ +AffpRn´1, Rn´1q ˆ t˘1u et bij : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ Zn ˆ Sn ˆ rr0, nss telles que le +diagramme (2.3.17) commute (c) d’un sous-ensemble Ensk +n Ă Zn ˆ Sn et d’une bijection bij : +Ensk +n´1 ˆ rr0, nss Ñ tx P Ensk +n ˆ rr0, nss|involpxq R Ensk +n ˆ rr0, nssu, telle que le diagramme +(2.3.19) commute (d) d’un sous-ensemble Affp∆n´1, ∆nq de AffpRn´1, Rnq et la construction +d’applications f : Ensk +n ˆrr0, nss Ñ Affp∆n´1, ∆nq et ˜f : Ensk +n´1 ˆrr0, nss Ñ Affp∆n´1, ∆nq telle +que le diagramme (2.3.22) commute. +2.3.1. Diagramme commutatif impliquant une involution de Zn ˆ Sn ˆ rr0, nss. On note Aff la +cat´egorie des espaces affines, dont les morphismes sont les applications affines. +Pour n ‰ 0, on note pen +1, . . . , en +nq la base canonique de Rn et pour i P rr0, nss, on pose +En +i :“ en +n ` en +n´1 ` ¨ ¨ ¨ ` en +n´i`1 P Rn (on a en particulier En +0 “ 0). +D´efinition 2.6. Soit n, m ě 0. Pour P0, . . . , Pn P Rm, on note rP0, . . . , Pns P AffpRn, Rmq +l’unique application affine Rn Ñ Rm telle que En +i ÞÑ Pi pour i “ 0, . . . , n. + +6 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +D´efinition 2.7. Pour i P rr0, nss, on pose Bn +i :“ rEn +0 , . . . , En +n´i´1, En +n´i`1, . . . , En +ns P AffpRn´1, Rnq. +Soit Sn le groupe des permutations de rr1, nss. Pour σ P Sn, on note σ˚ la permutation de Rn +donn´ee par σ˚pt1, . . . , tnq :“ ptσp1q, . . . , tσpnqq. On a alors σ˚pen +i q “ en +σ´1piq, et pστq˚ “ τ ˚ ˝ σ˚ +pour σ, τ P Sn. +D´efinition 2.8. Pour pv, σq P Zn ˆ Sn, on d´efinit ckpv, σq P AffpRn, Rnq comme l’application +de Rn dans lui-mˆeme donn´ee par x ÞÑ p1{kqpv ` σ˚pxqq. +On a alors ckpv, σq “ rp1{kqpv ` σ˚En +0 q, . . . , p1{kqpv ` σ˚En +nqs pour pv, σq P Zn ˆ Sn. +D´efinition 2.9. On note +f : Zn ˆ Sn ˆ rr0, nss Ñ AffpRn´1, Rnq +et +sgn : Zn ˆ Sn ˆ rr0, nss Ñ t˘1u +les applications donn´ees par +fpv, σ, iq :“ ckpv, σq ˝ Bn +i , +sgnpv, σ, iq :“ p´1qiǫpσq. +D´efinition 2.10. On note invol l’application de Zn ˆ Sn ˆ rr0, nss dans lui-mˆeme donn´ee par +involpv, σ, iq :“ pv, si,i`1 ˝ σ, iq +pour +pv, σ, iq P Zn ˆ Sn ˆ rr1, n ´ 1ss, +o`u si,i`1 P Sn est la permutation de i et i ` 1, +involpv, σ, nq :“ pv ` σ˚pen +nq, c ˝ σ, 0q +pour +pv, σq P Zn ˆ Sn, +o`u c P Sn est le n-cycle donn´e par cpiq :“ i ` 1 pour i ‰ n, cpnq “ 1, et +involpv, σ, 0q :“ pv ´ σ˚pen +1q, c´1 ˝ σ, nq +pour +pv, σq P Zn ˆ Sn. +Lemme 2.11. (a) invol est une involution de Zn ˆ Sn ˆ rr0, nss. +(b) Le diagramme suivant commute +(2.3.1) +Zn ˆ Sn ˆ rr0, nss +invol +� +pf,sgnq +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +Zn ˆ Sn ˆ rr0, nss +pf,´sgnq +�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ +AffpRn´1, Rn´1q ˆ t˘1u +D´emonstration. (a) Soit pv, σ, iq P Zn ˆ Sn ˆ rr0, nss. Si i ‰ 0, n, alors s2 +i,i`1 “ id implique +invol ˝ involpv, σ, iq “ pv, σ, iq. Si i “ n, alors invol ˝ involpv, σ, nq “ involpv ` σ˚pen +nq, c ˝ σ, 0q “ +pv ` σ˚pen +nq ´ pc ˝ σq˚pen +1q, c´1 ˝ c ˝ σ, nq “ pv, σ, nq car pc ˝ σq˚pen +1q “ σ˚ ˝ c˚pen +1q “ σ˚pen +nq et si +i “ 0, on a invol ˝ involpv, σ, 0q “ involpv ´ σ˚pen +1q, c´1 ˝ σ, nq “ pv ´ σ˚pen +1q ` pc´1 ˝ σq˚pen +nq, c ˝ +c´1 ˝ σ, 0q “ pv, σ, 0q car pc´1 ˝ σq˚pen +nq “ σ˚ ˝ pc´1q˚pen +nq “ σ˚pen +1q. On a donc dans tous les +cas invol ˝ involpv, σ, iq “ pv, σ, iq. +(b) Soit pv, σ, iq P Zn ˆ Sn ˆ rr0, nss. +Si i ‰ 0, n, alors sgn ˝ involpv, σ, iq “ ´sgnpv, σ, iq du fait de ǫpsi,i`1q “ ´1. Si i “ n, +alors sgn ˝ involpv, σ, nq “ sgnpv ` σ˚pen +1q, c´1 ˝ σ, 0q “ p´1qnǫpc´1 ˝ σq “ ´p´1q0ǫpσq “ + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +7 +´sgnpv, σ, nq du fait de ǫpcq “ p´1qn´1 et sgn ˝ involpv, σ, 0q “ sgnpv ´ σ˚pen +nq, c ˝ σ, nq “ +p´1q0ǫpc ˝ σq “ ´p´1qnǫpσq “ ´sgnpv, σ, 0q pour la mˆeme raison. On a donc dans tous les cas +sgn ˝ involpv, σ, iq “ ´sgnpv, σ, iq. +Si i ‰ 0, n, alors +f ˝ involpv, σ, iq “ fpv, si,i`1 ˝ σ, iq “ rpv ` psi,i`1 ˝ σq˚pEn +0 q{k, . . . , pv ` psi,i`1 ˝ σq˚pEn +nq{ks +˝ rEn +0 , . . . , En +n´i´1, En +n´i`1, . . . , En +ns “ rpv ` psi,i`1 ˝ σq˚pEn +0 q{k, . . . , pv ` psi,i`1 ˝ σq˚pEn +n´i´1q{k, +pv ` psi,i`1 ˝ σq˚pEn +n´i`1q{k, . . . , pv ` psi,i`1 ˝ σq˚pEn +nq{ks +“ rpv ` σ˚pEn +0 q{k, . . . , pv ` σ˚pEn +n´i´1q{k, pv ` σ˚pEn +n´i`1q{k, . . . , pv ` σ˚pEn +nq{ks “ fpv, σ, iq +en utilisant pσ ¨ τq˚ “ τ ˚ ˝ σ˚ et s˚ +i,i`1pEn +j q “ En +j pour j P rr0, nss et j ‰ n ´ i. +Si i “ n, alors +f ˝ involpv, σ, nq “ fpv ` σ˚pen +nq, c ˝ σ, 0q +“ rpv ` σ˚pen +nq ` pc ˝ σq˚pEn +0 qq{k, . . . , pv ` σ˚pen +nq ` pc ˝ σq˚pEn +nqq{ks ˝ rEn +0 , . . . , En +n´1s +“ rpv ` σ˚pen +nq ` pc ˝ σq˚pEn +0 qq{k, . . . , pv ` σ˚pen +nq ` pc ˝ σq˚pEn +n´1qq{ks +“ rpv ` σ˚pen +n ` c˚pEn +0 qqq{k, . . . , pv ` σ˚pen +n ` c˚pEn +n´1qqq{ks +“ rpv ` σ˚pEn +1 qq{k, . . . , pv ` σ˚pEn +nqq{ks +“ rpv ` σ˚pEn +0 qq{k, . . . , pv ` σ˚pEn +nqq{ks ˝ rEn +1 , . . . , En +ns “ fpv, σ, nq +du fait de c˚pEn +i q ` en +n “ En +i`1 pour i P rr0, n ´ 1ss. +Si i “ 0, alors +f ˝ involpv, σ, 0q “ fpv ´ σ˚pen +1q, c´1 ˝ σ, nq +“ rpv ´ σ˚pen +1q ` pc´1 ˝ σq˚pEn +0 qq{k, . . . , pv ´ σ˚pen +1q ` pc´1 ˝ σq˚pEn +nqq{ks ˝ rEn +1 , . . . , En +ns +“ rpv ´ σ˚pen +1q ` pc´1 ˝ σq˚pEn +1 qq{k, . . . , pv ´ σ˚pen +1q ` pc´1 ˝ σq˚pEn +nqq{ks +“ rpv ` σ˚p´en +1 ` pc´1q˚pEn +1 qqq{k, . . . , pv ` σ˚p´en +1 ` pc´1q˚pEn +nqqq{ks +“ rpv ` σ˚pEn +0 qq{k, . . . , pv ` σ˚pEn +n´1qq{ks +“ rpv ` σ˚pEn +0 qq{k, . . . , pv ` σ˚pEn +nqq{ks ˝ rEn +0 , . . . , En +n´1s “ fpv, σ, 0q +du fait de pc´1q˚pEn +i q´en +1 “ En +i´1 pour i P rr1, nss. On a donc dans tous les cas f ˝involpv, σ, iq “ +fpv, σ, iq. +□ +Remarque 2.12. On v´erifie directement que invol est sans point fixe; cela r´esulte aussi de +sgn ˝ invol “ ´sgn et du fait que t˘1u n’a pas de point fixe sous le changement de signe. +2.3.2. Constructions et r´esultats relatifs aux permutations. +Lemme 2.13. Soit τ P Sn´1. Pour i P rr1, n´1ss, soit invpτ, iq :“ tj P rr1, n´1ss|pj ´iqpτpjq´ +τpiqq ă 0u. Alors |invpτ, iq| ” τpiq ´ i (mod 2). + +8 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +D´emonstration. Montrons l’´enonc´e par r´ecurrence sur i. Si i “ 1, invpτ, 1q “ tj P rr1, n ´ +1ss|τpjq ă τp1qu “ τ ´1prr1, τp1q ´ 1ssq, donc |invpτ, 1q| “ |τ ´1prr1, τp1q ´ 1ssq| “ |rr1, τp1q ´ 1ss| “ +τp1q ´ 1 o`u la deuxi`eme ´egalit´e suit de la bijectivit´e de τ, ce qui implique l’´enonc´e pour i “ 1. +Soit 1 ď i ă n ´ 1 : supposons l’´enonc´e vrai pour i et montrons-le pour i ` 1. Pour cela, on +note A0 :“ tj|j ă i et τpjq ą τpiqu, A1 :“ tj|j ă i et τpjq ą τpi ` 1qu, B0 :“ tj|j ą i ` 1 et +τpjq ă τpiqu, B1 :“ tj|j ą i ` 1 et τpjq ă τpi ` 1qu ; on observe que Aα X Bβ “ H pour tous +α, β P t0, 1u. +Deux cas se pr´esentent : +‚ on a τpiq ă τpi ` 1q. On a alors i ` 1 R invpτ, iq et i R invpτ, i ` 1q, ce qui implique +invpτ, iq “ A0 Y B0 et invpτ, i ` 1q “ A1 Y B1. +En notant △ l’op´eration de diff´erence +sym´etrique, on obtient +(2.3.2) +invpτ, iq△invpτ, i ` 1q “ pA0△A1q Y pB0△B1q, +compte tenu de pX0YY0q△pX1YY1q “ pX0△X1qYpY0△Y1q pour tous ensembles X0, X1, Y0, Y1 +tels que Xα X Yβ “ H. Du fait que τpiq ă τpi ` 1q, on a A1 Ă A0 et B1 Ą B0, ce qui implique +(2.3.3) +pA0△A1q Y pB0△B1q “ pA0 ´ A1q Y pB1 ´ B0q. +Alors +A0´A1 “ τ ´1prrτpiq`1, τpi`1qssqXtj|j ă iu, +B1´B0 “ τ ´1prrτpiq, τpi`1q´1ssqXtj|j ą i`1u. +Comme τpi ` 1q R τptj|j ă iuq et τpiq R τptj|j ą i ` 1uq on en d´eduit +A0´A1 “ τ ´1prrτpiq`1, τpi`1q´1ssqXtj|j ă iu, +B1´B0 “ τ´1prrτpiq`1, τpi`1q´1ssqXtj|j ą i`1u, +donc pA0 ´ A1q Y pB1 ´ B0q “ τ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq X tj|j ‰ i, i ` 1u. Compte tenu de +ti, i ` 1u X τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq “ H, on en d´eduit +(2.3.4) +pA0 ´ A1q Y pB1 ´ B0q “ τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq. +Alors +|invpτ, i ` 1q| ´ |invpτ, iq| ” |invpτ, iq△invpτ, i ` 1q| “ |pA0 ´ A1q Y pB1 ´ B0q| +“ |τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq| “ |rrτpiq ` 1, τpi ` 1q ´ 1ss| “ τpi ` 1q ´ τpiq ´ 1 +(2.3.5) +mod 2, o`u la premi`ere ´egalit´e suit de +(2.3.6) +|A△B| ” |B| ´ |A| mod 2 pour A, B ensembles finis, +la deuxi`eme ´egalit´e suit de la combinaison de (2.3.2) et (2.3.3), la troisi`eme ´egalit´e suit de +(2.3.4), la quatri`eme ´egalit´e suit de la bijectivit´e de τ ; +‚ on a τpi ` 1q ă τpiq. On a alors A1 Ą A0 et B1 Ă B0, ce qui implique +(2.3.7) +pA0△A1q Y pB0△B1q “ pA1 ´ A0q Y pB0 ´ B1q. +Alors +A1´A0 “ τ ´1prrτpi`1q`1, τpiqssqXtj|j ă iu, +B0´B1 “ τ ´1prrτpi`1q, τpiq´1ssqXtj|j ą i`1u. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +9 +Or τpiq R τptj|j ă iuq, τpi ` 1q R τptj|j ą i ` 1uq, donc +A1´A0 “ τ ´1prrτpi`1q`1, τpiq´1ssqXtj|j ă iu, +B0´B1 “ τ ´1prrτpi`1q`1, τpiq´1ssqXtj|j ą i`1u. +Donc pA1 ´ A0q Y pB0 ´ B1q “ τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq X tj|j ‰ i, i ` 1u. Compte tenu +de ti, i ` 1u X τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq “ H, on en d´eduit +(2.3.8) +pA1 ´ A0q Y pB0 ´ B1q “ τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq. +De plus, comme τpi`1q ă τpiq, on a invpτ, iq “ A0YB0Yti`1u et invpτ, i`1q “ A1YB1Ytiu. +On a i R invpτ, iq et i ` 1 R invpτ, i ` 1q, donc +(2.3.9) +invpτ, iq△invpτ, i ` 1q “ ppA0 Y B0q△pA1 Y B1qq Y ti, i ` 1u. +Alors +|invpτ, iq| ´ |invpτ, i ` 1q| ” |invpτ, iq△invpτ, i ` 1q| “ |pA0 Y B0q△pA1 Y B1q| ´ 2 +” |pA0 Y B0q△pA1 Y B1q| “ |τ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq| “ |rrτpi ` 1q ` 1, τpiq ´ 1ss| “ τpiq ´ τpi ` 1q ´ 1. +(2.3.10) +o`u la premi`ere ´egalit´e suit de (2.3.6) et x ” ´x mod 2, la deuxi`eme ´egalit´e suit de (2.3.9) +combin´e `a ti, i ` 1u X ppA0 Y B0q△pA1 Y B1qq Ă ti, i ` 1u X tj|j ‰ i, i ` 1u “ H, la quatri`eme +´egalit´e suit de la combinaison de (2.3.2) et (2.3.7), la cinqui`eme ´egalit´e suit de la bijectivit´e de +τ. +On d´eduit des ´egalit´es (2.3.5) dans le premier cas et (2.3.10) dans le second l’´egalit´e |invpi` +1q| ´ τpi ` 1q ` i ` 1 ” |invpiq| ´ τpiq ` i mod 2. On a |invpτ, iq| ´ τpiq ` i ” 0 mod 2 d’apr`es +l’hypoth`ese de r´ecurrence, ce qui implique |invpτ, i ` 1q| ´ τpi ` 1q ` i ` 1 ” 0 mod 2. +□ +D´efinition 2.14. Soit i P rr1, n ´ 1ss. +(a) On note pi : rr1, nss Ñ rr1, n´1ss l’application donn´ee par pipxq “ x si x ď i et pipxq “ x´1 +si x ě i ` 1. +(b) On note sti : rr1, n ´ 1ss Ñ rr1, nss l’application donn´ee par x ÞÑ x si x ď i et x ÞÑ x ` 1 si +x ě i ` 1. +Lemme-D´efinition 2.15. Soit τ P Sn´1 et i P rr0, nss. Il existe un unique ´el´ement τ piq P Sn +satisfaisant les conditions suivantes : +(a) pτpiq ˝ τ piq “ τ ˝ pi et τ piqpiq “ τpiq, τ piqpi ` 1q “ τpiq ` 1 si i ‰ 0, n ; +(b) τ p0qp1q “ 1 et τ p0qpxq “ τpx ´ 1q ` 1 pour tout x P rr2, nss si i “ 0 ; +(c) τ pnqpnq “ n et τ pnqpxq “ τpxq pour tout x P rr1, n ´ 1ss si i “ n. +D´emonstration. (a) Pour j P rr0, nss, stj ˝ pj est l’application de rr1, nss dans lui-mˆeme telle que +x ÞÑ x pour x ‰ j ` 1 et j ` 1 ÞÑ j. Si τpiq est une application de rr1, nss dans lui-mˆeme +satisfaisant les conditions dites, on a alors stτpiq ˝ ˝pτpiq ˝ τ piq “ stτpiq ˝ τ ˝ pi ce qui implique +τ piqpxq “ stτpiq ˝τ ˝pipxq pour tout x ‰ i, i`1 ainsi que τpiqpiq “ τpiq, τ piqpi`1q “ τpiq`1. Les +conditions dites d´eterminent donc uniquement τpiq comme application de rr1, nss dans lui-mˆeme. + +10 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +On v´erifie alors que τ piq P Sn. (b,c) Pour i “ 0, n, l’application τ piq est la juxtaposition de +deux permutations, donc est une permutation. +□ +Lemme 2.16. Si τ P Sn´1 et i P rr1, n ´ 1ss, alors ǫpτ piqq “ ǫpτqp´1qτpiq´i. +D´emonstration. Si p ě 1 et σ P Sp, notons invpσq :“ tpa, bq P rr1, pss2|a ă b et σpaq ą σpbqu. +On a alors +(2.3.11) +ǫpσq “ p´1q|invpσq|. +Si τ P Sn´1 et i P rr1, n ´ 1ss, on a une partition +(2.3.12) +invpτq “ A \ B, +avec A :“ tpa, bq P invpτq|a ‰ i et b ‰ iu et B :“ tpa, bq P invpτq|a “ i ou b “ iu. On a de +mˆeme une partition +(2.3.13) +invpτ piqq “ A1 \ B1 \ B2, +avec +A1 :“ tpa, bq P invpτ piqq|a R ti, i ` 1u et b R ti, i ` 1uu, +B1 :“ tpa, bq P invpτ piqq|pa “ i et b R ti, i ` 1uq ou pb “ i et a R ti, i ` 1uqu +B2 :“ tpa, bq P invpτ piqq|pa “ i ` 1 et b R ti, i ` 1uq ou pb “ i ` 1 et a R ti, i ` 1uqu +ceci du fait que pi, i ` 1q R invpτ piqq car τ piqpi ` 1q “ 1 ` τ piqpiq. +On a encore des bijections +(2.3.14) +A „ +Ñ A1, +B „ +Ñ B1, +B „ +Ñ B2 +induites respectivement par pa, bq ÞÑ pstipaq, stipbqq (application A Ñ A1), pa, iq ÞÑ pa, iq et +pi, bq ÞÑ pi, b ` 1q (application B Ñ B1), pa, iq ÞÑ pa, i ` 1q et pi, bq ÞÑ pi ` 1, b ` 1q (application +B Ñ B2). Enfin on a une bijection +(2.3.15) +invpτ, iq „ +Ñ B +donn´ee par a ÞÑ pa, iq si a ă i et a ÞÑ pi, aq si i ă a. +On a alors +(2.3.16) +|invpτ piqq| “ |A1| \ |B1| \ |B2| “ p|A| \ |B|q \ |B| “ |invpτq| ` |invpτ, iq| ” |invpτq| ` τpiq ´ i +mod 2, o`u la premi`ere (resp. deuxi`eme, troisi`eme, quatri`eme) ´egalit´e suit de (2.3.13) (resp. +(2.3.14), (2.3.15), lemme 2.13). On a alors +ǫpτ piqq “ p´1q|invpτ piqq| “ p´1q|invpτq|`τpiq´i “ p´1q|invpτq|p´1qτpiq´i “ ǫpτqp´1qτpiq´i. +o`u la premi`ere (resp. deuxi`eme, derni`ere) ´egalit´e suit de (2.3.11) (resp. (2.3.16), (2.3.11)). +□ + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +11 +2.3.3. Diagramme commutatif impliquant les applications pf, sgnq, p ˜f, Ą +sgnq et bij. +D´efinition 2.17. On note +˜f : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ AffpRn´1, Rnq +et +Ą +sgn : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ t˘1u +les applications donn´ees par +˜fp˜v, ˜σ, iq :“ Bn +i ˝ ckp˜v, ˜σq, +Ą +sgnp˜v, ˜σ, iq :“ p´1qiǫp˜σq. +D´efinition 2.18. On note w ÞÑ wp0q et w ÞÑ wpnq les applications Zn´1 Ñ Zn donn´ees par +wp0q :“ p0, wq et wpnq :“ pw, k ´ 1q. +D´efinition 2.19. On note bij : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ Zn ˆ Sn ˆ rr0, nss l’application +donn´ee par pw, τ, iq ÞÑ pw ˝ pi, τ piq, τpiqq si i ‰ 0, n, par pw, τ, 0q ÞÑ pwp0q, τ p0q, 0q, et par +pw, τ, nq ÞÑ pwpnq, τ pnq, nq. +Lemme 2.20. Le diagramme suivant commute +(2.3.17) +Zn´1 ˆ Sn´1 ˆ rr0, nss +bij +� +p ˜ +f,Ą +sgnq +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +Zn ˆ Sn ˆ rr0, nss +pf,sgnq +�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ +AffpRn´1, Rn´1q ˆ t˘1u +D´emonstration. Soit pw, τ, iq P Zn´1 ˆ Sn´1 ˆ rr0, nss. Supposons i ‰ 0, n. Alors +sgn ˝ bijpw, τ, iq “ sgnpw ˝ pi, τ piq, τpiqq “ ǫpτ piqqp´1qτpiq “ ǫpτqp´1qi “ Ą +sgnpw, τ, iq +d’apr`es le lemme 2.16. De plus +f ˝ bijpw, τ, iq “ fpw ˝ pi, τ piq, τpiqq “ ckpw ˝ pi, τ piqq ˝ Bn +τpiq +“ rpt1, . . . , tn´1q ÞÑ pw ˝ pi ` pτ piqq˚pt1, . . . , tτpiq, tτpiq, . . . , tn´1qq{ks +“ rpt1, . . . , tn´1q ÞÑ pw ˝ pi ` pτ piqq˚ptpτpiqp1q, . . . , tpτpiqpnqqq{ks +“ rpt1, . . . , tn´1q ÞÑ pw ˝ pi ` ptpτpiq˝τ piqp1q, . . . , tpτpiq˝τ piqpnqqq{ks +“ rpt1, . . . , tn´1q ÞÑ pw ˝ pi ` ptτ˝pip1q, . . . , tτ˝pipnqqq{ks +“ Bn +i ˝ rpt1, . . . , tn´1q ÞÑ pw ` ptτp1q, . . . , tτpn´1qqq{ks +“ Bn +i ˝ rpt1, . . . , tn´1q ÞÑ pw ` τ ˚pt1, . . . , tn´1qq{ks “ Bn +i ˝ ckpw, τq “ ˜fpw, τ, iq +o`u la sixi`eme ´egalit´e provient de τ ˝ pi “ pτpiq ˝ τ piq (´egalit´e d’applications rr1, nss Ñ rr1, n ´ 1ss) +et la septi`eme ´egalit´e suit de Bn +i pxq “ x ˝ pi pour x P Rn´1 “ Applprr1, n ´ 1ss, Rq. +Si i “ 0, alors +f ˝ bijpw, τ, 0q “ fpwp0q, τ p0q, 0q “ ckpwp0q, τ p0qq ˝ Bn +0 +“ rpt1, . . . , tn´1q ÞÑ pwp0q ` pτ p0qq˚p0, t1, . . . , tn´1qq{ks +“ rpt1, . . . , tn´1q ÞÑ pp0, wq ` p0, τ ˚pt1, . . . , tn´1qqq{ks +“ rpt1, . . . , tn´1q ÞÑ p0, w ` τ ˚pt1, . . . , tn´1qq{ks “ Bn +0 ˝ ckpw, τq “ ˜fpw, τ, 0q. + +12 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +Si i “ n, alors +f ˝ bijpw, τ, nq “ fpwpnq, τ pnq, nq “ ckpwpnq, τ pnqq ˝ Bn +n +“ rpt1, . . . , tn´1q ÞÑ pwpnq ` pτ pnqq˚pt1, . . . , tn´1, 1qq{ks +“ rpt1, . . . , tn´1q ÞÑ ppw, k ´ 1q ` pτ ˚pt1, . . . , tn´1q, 1qq{ks +“ rpt1, . . . , tn´1q ÞÑ pw ` τ ˚pt1, . . . , tn´1q, kq{ks +“ rpt1, . . . , tn´1q ÞÑ ppw ` τ ˚pt1, . . . , tn´1qq{k, 1qs +“ Bn +n ˝ rpt1, . . . , tn´1q ÞÑ pw ` τ ˚pt1, . . . , tn´1qq{ks “ Bn +n ˝ ckpw, τq “ ˜fpw, τ, nq. +□ +2.3.4. Battages et transpositions. +D´efinition 2.21. Si n ě 1 et n1, . . . , nk sont des entiers positifs ou nuls avec n1 `¨ ¨ ¨`nk “ n, +on note Sn1,...,nk l’ensemble des ´el´ements σ P Sn tel que pour tout i “ 1, . . . , k, la restriction +de σ `a n1 ` ¨ ¨ ¨ ` ni´1 ` rr1, niss est croissante. +Si ni “ 0, l’ensemble rr1, niss est vide, la condition relative `a i est alors automatiquement +satisfaite. +Lemme 2.22. Soit n1, . . . , nk des entiers ě 1 et σ P Sn1,...,nk et i P rr1, n ´ 1ss avec n :“ +n1`¨ ¨ ¨`nk. Alors si,i`1˝σ R Sn1,...,nk si et seulement si σ´1piq R tn1, n1`n2, . . . , n1`¨ ¨ ¨`nku +et σ´1pi ` 1q “ σ´1piq ` 1. +D´emonstration. Soit σ P Sn1,...,nk et i P rr1, n ´ 1ss. Soit j :“ σ´1piq, j1 :“ σ´1pi ` 1q. Pour +α P rr1, kss, notons Iα :“ n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nαss. Alors on a une partition rr1, nss “ \k +α“1Iα. +Soit α, α1 P rr1, kss les indices tels que j P Iα, j1 P Iα1. Montrer l’´equivalence annonc´ee, on +montre d’abord l’´equivalence pα “ α1q ðñ psi,i`1 ˝ σ R Sn1,...,nkq, puis l’´equivalence pα “ +α1q ðñ pσ´1piq R tn1, n1 ` n2, . . . , n1 ` ¨ ¨ ¨ ` nku et σ´1pi ` 1q “ σ´1piq ` 1q. +Premi`ere ´etape : ´equivalence pα “ α1q ðñ psi,i`1 ˝ σ R Sn1,...,nkq. (a) Supposons α “ α1. +Alors on a j, j1 P Iα. La restriction σ|Iα est strictement croissante, et σpjq “ i, σpj1q “ i ` 1, +donc j ă j1. De plus, si on avait j1 ą j ` 1, alors j ` 1 P Iα et σpjq ă σpj ` 1q ă σpj1q donc +i ă σpj ` 1q ă i ` 1 ce qui est impossible, σpj ` 1q ´etant entier. Donc j1 “ j ` 1. Comme +j ` 1 P Iα, on a n´ecessairement j ‰ n1 ` ¨ ¨ ¨ ` nα. De plus, la restriction de si,i`1 ˝ σ `a Iα est +telle que j ÞÑ i ` 1 et j ` 1 ÞÑ i ; cette restriction n’est donc pas strictement croissante, donc +si,i`1 ˝ σ R Sn1,...,nk. +Supposons α ‰ α1. Comme σpIαq S i ` 1, la restriction de si,i`1 ˝ σ `a Iα est ´egale `a a ˝ σ|Iα, +o`u a : σpIαq Ñ rr1, nss est donn´ee par x ÞÑ x si x ‰ i et i ÞÑ i`1. L’application a est croissante, +donc il est de mˆeme de psi,i`1 ˝ σq|Iα. De mˆeme, σpIα1q S i, donc la restriction de si,i`1 ˝ σ `a +Iα1 est ´egale `a a1 ˝ σ|Iα, o`u a1 : σpIα1q Ñ rr1, nss est donn´ee par x ÞÑ x si x ‰ i ` 1 et i ` 1 ÞÑ i. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +13 +L’application a1 est croissante, donc il est de mˆeme de psi,i`1 ˝σq|Iα1. Enfin pour tout β ‰ α, α1, +on a psi,i`1 ˝ σq|Iβ “ σ|Iβ donc psi,i`1 ˝ σq|Iβ est croissante. Donc si,i`1 ˝ σ P Sn1,...,nk. +On a donc ´equivalence entre si,i`1 ˝ σ R Sn1,...,nk et α “ α1. +Seconde ´etape : pα “ α1q ðñ pσ´1piq R tn1, n1 ` n2, . . . , n1 ` ¨ ¨ ¨ ` nku et σ´1pi ` 1q “ +σ´1piq ` 1q. On a vu que si α “ α1, alors j1 “ j ` 1 et j ‰ n1 ` ¨ ¨ ¨ ` nα. Comme j P Iα, on a +aussi j ‰ n1 ` ¨ ¨ ¨ ` nβ pour tout β ‰ α, donc j R tn1, . . . , n1 ` ¨ ¨ ¨ ` nku. +Inversement, si j1 “ j ` 1 et j R tn1, . . . , n1 ` ¨ ¨ ¨ ` nku, on a j ‰ n1 ` ¨ ¨ ¨ ` nα donc +j P n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nα ´ 1ss donc j1 “ j ` 1 P n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nαss “ Iα, donc +α “ α1. +□ +2.3.5. La bijection bij. +D´efinition 2.23. (a) On note rr0, k ´ 1ssn +ď :“ tpv1, . . . , vnq P Zn|0 ď v1 ď . . . ď vn ď k ´ 1u. +(b) On note Ensk +n l’ensemble des couples pv, σq P rr0, k´1ssn +ďˆSn, tels que σ P S|v´1p0q|,...,|v´1pk´1q|. +Si pv, σq P Ensk +n, on a pour tout i P rr1, n ´ 1ss l’implication +(2.3.18) +pvi “ vi`1q ùñ pσpiq ă σpi ` 1qq. +Lemme 2.24. Soit pv, σ, iq P Ensk +n ˆ rr0, nss. +(a) Si i ‰ 0, n, la condition involpv, σ, iq R Ensk +n ˆ rr0, nss est ´equivalente `a la conjonction de +σ´1piq R t|v´1p0q|, . . . , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u et σ´1pi ` 1q “ σ´1piq ` 1. +(b) Si i “ n, la condition involpv, σ, iq R Ensk +n ˆ rr0, nss est ´equivalente `a la conjonction +σpnq “ n et vpnq “ k ´ 1 ; +(c) Si i “ 0, la condition involpv, σ, iq R Ensk +n ˆ rr0, nss est ´equivalente `a la conjonction +σp1q “ 1 et vp1q “ 0. +D´emonstration. (a) On a σ P S|v´1p0q|,...,|v´1p0q|`¨¨¨`|v´1pk´1q| et involpv, σ, iq “ pv, si,i`1 ˝ σ, iq. +On a v P rr0, k ´ 1ssn +ď donc on a ´equivalence entre involpv, σ, iq R Ensk +n ˆ rr0, nss et si,i`1 ˝ σ R +S|v´1p0q|,...,|v´1p0q|`¨¨¨`|v´1pk´1q|. Le r´esultat est alors cons´equence du lemme 2.22. +(b) Rappelons que involpv, σ, nq “ pv ` en +σ´1pnq, c ˝ σ, 0q. +Notons nα :“ |v´1pα´1q| pour α “ 1, . . . , k. Alors on a une partition rr1, nss “ \k +α“1Iα, avec +Iα “ n1`¨ ¨ ¨`nα´1`rr1, nαss (avec la convention que Iα “ H si nα “ 0), et v prend la valeur α´1 +sur Iα, pour tout α. Rappelons l’identification de rr0, k´1ssn +ď `a l’ensemble Applďprr1, nss, rr0, k´ +1ssq. On a σ´1pnq P tn1 ` ¨ ¨ ¨ ` nβ|nβ ‰ 0u donc v ` en +σ´1pnq P Applďprr1, nss, rr0, kssq. De plus, +v`en +σ´1pnq atteint la valeur k si et seulement si nk ‰ 0 et σ´1pnq “ n1`¨ ¨ ¨`nk. La conjonction +de ces conditions est ´equivalente `a celle de vpnq “ k ´ 1 et σpnq “ n. Ceci montre l’´equivalence +entre v ` en +σ´1pnq R rr0, k ´ 1ssn +ď et la conjonction des conditions vpnq “ k ´ 1 et σpnq “ n. +Alors si vpnq “ k ´ 1 et σpnq “ n, on a v ` en +σ´1pnq R rr0, k ´ 1ssn +ď donc involpv, σ, 0q R +Ensk +n ˆ rr0, nss. + +14 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +Si vpnq ă k ´ 1 ou σpnq ‰ n, montrons que involpv, σ, 0q P Ensk +n ˆ rr0, nss. D’une part, on +a w :“ v ` en +σ´1pnq P rr0, k ´ 1ssn +ď. Montrons qu’on a d’autre part c ˝ σ P S|w´1p0q|,...,|w´1pk´1q|, +ceci dans chacun des cas σpnq ‰ n et (σpnq “ n et vpnq ă k ´ 1). +Premier cas : σpnq ‰ n. Soit α l’indice tel que σ´1pnq P Iα. On a n´ecessairement α ă k et +nα ą 0 ; de plus σ´1pnq “ maxpIαq. Rappelons que p|v´1p0q|, . . . , |v´1pk ´ 1q|q “ pn1, . . . , nkq, +alors p|w´1p0q|, . . . , |w´1pk ´ 1q|q “ pn1, . . . , nα ´ 1, nα`1 ` 1, . . . , nkq, et la partition de rr1, nss +associ´ee `a w est pJ1, . . . , Jkq avec Jβ “ Iβ pour β P rr1, kss ´ tα, α ` 1u, Jα “ Iα ´ tmaxpIαqu, +Jα`1 “ tmaxpIαqu Y Iα`1. Montrons que c ˝ σ P Sn1,...,nα´1,nα`1`1,...,nk. Si β P rr1, kss ´ tαu, +alors σpIβq S n. La restriction de c `a σpIβq est donc x ÞÑ x ` 1, qui est strictement croissante. +Donc la restriction de c˝σ `a Iβ est strictement croissante. En particulier, si β P rr1, kss´tα, α`1u, +la restriction de c ˝ σ `a Jβ “ Iβ est strictement croissante. +Rappelons que la restriction de c˝σ `a Iα`1 est strictement croissante. On a Jα “ tmaxpIαquY +Iα`1 avec maxpIαq ď Iα`1 et c ˝ σpmaxpIαqq “ 0, donc la restriction de c ˝ σ `a Jα`1 est +strictement croissante. +Comme Jα “ Iα ´ tmaxpIαqu et que maxpIαq “ σ´1pnq, on a σpJαq S n. La restriction de c +`a σpJβq est donc x ÞÑ x ` 1, qui est strictement croissante. Donc la restriction de c ˝ σ `a Jα est +strictement croissante. +Donc on a c ˝ σ P S|w´1p0q|,...,|w´1pk´1q|. +Deuxi`eme cas : σpnq “ n et vpnq ă k ´ 1. Soit p :“ 1 ` maxti P rr0, k ´ 1ss||v´1piq| ‰ 0u; +alors p ă k. Alors p|v´1p0q|, . . . , |v´1pk ´ 1q|q “ pn1, . . . , np, 0, . . . , 0q. Comme σpnq “ n, on a +σ´1pnq “ n “ n1 ` ¨ ¨ ¨ ` np. Donc p|w´1p0q|, . . . , |w´1pk ´ 1q|q “ pn1, . . . , np ´ 1, 1, 0, . . ., 0q, +la partition correspondant `a w ´etant donn´ee par pJ1, . . . , Jkq avec Jα “ Iα pour α ‰ p, p ` 1, +Jp :“ Ip ´ tnu, et Jp`1 “ tnu. Si α ‰ p, p ` 1, σpIαq S n, donc la restriction de c `a σpIαq est +x ÞÑ x ` 1 qui est strictement croissante, donc la restriction de c ˝ σ `a Jα “ Iα est strictement +croissante. On a Jp “ Ip ´ tnu et σpnq “ n, donc σpJpq S n, donc la restriction de c `a σpJpq +est x ÞÑ x ` 1 qui est strictement croissante, donc la restriction de c ˝ σ `a Jp est strictement +croissante. Enfin, Jp`1 est un singleton, donc la restriction de c˝σ `a cet ensemble est strictement +croissante. +Donc c ˝ σ P Sn1,...,np´1,1,0,...,0 “ S|w´1p0q|,...,|w´1pk´1q|. +(c) D´emonstration semblable `a celle de (b). +□ +Lemme 2.25. bij induit une bijection bij entre Ensk +n´1ˆrr0, nss et tx P Ensk +nˆrr0, nss|involpxq R +Ensk +n ˆ rr0, nssu. On a alors le diagramme commutatif +(2.3.19) +Ensk +n´1 ˆ rr0, nss +� � +� +bij +„ +� tx P Ensk +n ˆ rr0, nss|involpxq R Ensk +n ˆ rr0, nssuu +� � +� +Zn´1 ˆ Sn´1 ˆ rr0, nss +bij +� Zn ˆ Sn ˆ rr0, nss + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +15 +D´emonstration. D’apr`es le lemme 1.29, il suffit de montrer que bij induit une bijection entre +Ensk +n´1 ˆ rr0, nss et +E :“ En \ E0 \ Err1,n´1ss +o`u +En :“ tpv, σ, nq|pv, σq P Ensk +n et σpnq “ n et vpnq “ k ´ 1u, +E0 :“ tpv, σ, 0q|pv, σq P Ensk +n et σp1q “ 1 et vp1q “ 0u, +Err1,n´1ss :“tpv, σ, iq P Ensk +n ˆ rr1, n ´ 1ss|σ´1piq R t|v´1p0q|, . . . , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u +et σ´1pi ` 1q “ σ´1piq ` 1u. +Pour cela, on montre s´epar´ement que bij induit une bijection entre (a) Ensk +n´1 ˆ tnu et En, +(b) Ensk +n´1 ˆ t0u et E0, et (c) Ensk +n´1 ˆ rr1, n ´ 1ss et Err1,n´1ss. +(a) Montrons que bij envoie Ensk +n´1 ˆ tnu dans En. +Soit pw, τq P Ensk +n´1. +On a w P +rr0, k ´ 1ssn´1 +ď +, ce qui implique wpnq “ pw, k ´ 1q P rr0, k ´ 1ssn +ď. Posons v :“ wpnq, alors pour +tout i P rr0, k ´ 1ss, on a v´1piq “ w´1piq si i ‰ k ´ 1 et v´1pk ´ 1q “ w´1pk ´ 1q \ tnu. +Si i ‰ k ´ 1, la restriction de τ pnq `a v´1piq co¨ıncide avec la restriction de τ `a w´1piq, qui +est croissante car τ P S|w´1p0q|,...,|w´1pk´1q|. La restriction de τ pnq `a v´1pk ´ 1q est l’union +disjointe de la restriction de τ `a w´1pk ´ 1q, qui est croissante et `a valeurs dans rr1, n ´ 1ss et de +l’application n ÞÑ n. Comme on a w´1pk ´ 1q ă n, cette union disjointe est croissante. Donc +τ pnq P S|v´1p0q|,...,|v´1pk´1q|, ce qui implique pwpnq, τ pnqq P Ensk +n. De plus, on a wpnqpnq “ k ´ 1 +et τ pnqpnq “ n, donc bijpw, τ, nq “ pwpnq, τ pnq, nq P En. +Donc bijpEnsk +n´1 ˆ tnuq Ă En, notons +bijn : Ensk +n´1 ˆ tnu Ñ En +l’application induite. Soit invbijn : En Ñ Zn´1ˆSn´1ˆtnu l’application donn´ee par pv, σ, nq ÞÑ +pv|rr1,n´1ss, σ|rr1,n´1ss, nq ; comme σ P Sn et σpnq “ n, on a bien σ|rr1,n´1ss P Sn´1. +Montrons que invbijn envoie En dans Ensk +n´1 ˆ tnu. Soit pv, σ, nq P En, et posons w :“ +v|rr1,n´1ss, τ :“ σ|rr1,n´1ss. Alors v P rr0, k ´ 1ssn +ď ce qui implique w P rr0, k ´ 1ssn +ď. De plus, pour +chaque i P r0, k ´ 1s, w´1piq est contenu dans w´1piq (on a mˆeme ´egalit´e si i ‰ k ´ 1). La +restriction de τ `a w´1piq co¨ıncide avec la restriction de σ au mˆeme ensemble, qui est croissante +par la croissance de σ en restriction `a v´1piq, qui contient w´1piq. Donc pw, τq P Ensk +n´1. +Donc invbijn envoie En dans Ensk +n´1 ˆ tnu. Notons +invbijn : En Ñ Ensk +n´1 ˆ tnu +l’application induite. On v´erifie que les compositions invbijn˝bijn et bijn˝invbijn sont l’identit´e. +On en d´eduit que bijn est une bijection. +(b) D´emonstration semblable `a celle de (a). +(c) Montrons que bij envoie Ensk +n´1 ˆ rr1, n ´ 1ss dans Err1,n´1ss. Soit pw, τ, iq P Ensk +n´1 ˆ +rr1, n ´ 1ss et posons v :“ w ˝ pi, σ :“ τ piq. Comme pi : rr1, nss Ñ rr1, n ´ 1ss est croissante et + +16 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +w P rr0, k ´ 1ssn´1 +ď +, on a v P rr0, k ´ 1ssn +ď. On a alors pour j P rr0, k ´ 1ss, v´1pjq “ w´1pjq si +j ă wpiq, l’´egalit´e v´1pjq “ w´1pjq`1 si j ą wpiq, et v´1pwpiqq “ w´1pwpiqqYpw´1pwpiqq`1q. +Si j ă wpiq, la restriction de σ `a v´1pjq “ w´1pjq est ´egale `a la composition stτpiq ˝ τ|w´1pjq, +qui est croissante par croissance de chacun des termes. +Si j ą wpiq, la restriction de σ `a +v´1pjq “ w´1pjq ` 1 est ´egale `a la composition stτpiq´1 ˝ τ|w´1pjq ˝ px ÞÑ x ´ 1q, qui est +croissante par croissance de chacun des termes. Notons α, β les ´el´ements minimaux et maximaux +de w´1pwpiqq ; comme cet ensemble est un intervalle, on a w´1pwpiqq “ rrα, βss Q i. Alors +v´1pwpiqq “ rrα, β ` 1ss. La restriction de σ `a v´1pwpiqq “ rrα, β ` 1ss est donn´ee par x ÞÑ τpxq +si x P rrα, iss et x ÞÑ τpx ´ 1q ` 1 si x P rri ` 1, β ` 1ss ; les restrictions de τ `a rrα, iss et rri, βss sont +croissantes, ce qui implique que les restrictions de σ `a rrα, iss et rri ` 1, β ` 1ss le sont aussi. On +a σpi ` 1q “ τpiq ` 1 “ σpiq ` 1 ce qui, combin´e `a la croissance de σ sur rrα, iss et rri ` 1, β ` 1ss +implique la croissance de σ sur rrα, β ` 1ss “ v´1pwpiqq. Donc σ P S|v´1p0q|,...,|v´1pk´1q| donc +pv, σq “ pw ˝ pi, τ piqq P Ensk +n. +On sait que τ piqpiq “ τpiq et τ piqpi ` 1q “ τpiq ` 1. Alors pτ piqq´1pτpiqq “ i. L’ensemble +t|v´1p0q|, . . . , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u est celui des maxima des intervalles de la partition +rr1, nss “ v´1p0q \ . . . \ v´1pk ´ 1q. Celui de ces intervalles auquel appartient i est v´1pvpiqq “ +rrα, β ` 1ss, on a donc pour j ‰ vpiq, i ‰ maxpvpjqq. +Comme i ď β, on a i ‰ β ` 1 “ +maxpv´1pvpiqqq. On a donc pour tout j P rr0, k ´ 1ss, i ‰ maxpv´1pjqq, donc pτ piqq´1pτpiqq “ +i R t|v´1p0q|, . . . , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u. De plus, on a pτ piqq´1pτpiq ` 1q “ i ` 1 “ +pτ piqq´1pτpiqq ` 1. On en d´eduit pw ˝ pi, τ piq, τpiqq P Err1,n´1ss. +Donc bijpEnsk +n´1 ˆ rr1, n ´ 1ssq Ă Err1,n´1ss. Notons +bijrr1,n´1ss : Ensk +n´1 ˆ rr1, n ´ 1ss Ñ Err1,n´1ss +l’application induite. +Pour pv, σ, jq P Err1,n´1ss posons +(2.3.20) +invbijrr1,n´1sspv, σ, jq :“ pv ˝ stσ´1pjq, pj ˝ σ ˝ stσ´1pjq, σ´1pjqq “ pw, τ, iq. +Alors stσ´1pjq est une application rr1, n ´ 1ss Ñ rr1, nss, donc w P Zn´1. On a aussi i P rr1, nss. +Enfin τ “ pj ˝ σ ˝ stσ´1pjq est une application de rr1, n ´ 1ss dans lui-mˆeme. Montrons son +injectivit´e. Supposons x ‰ x1 P rr1, n ´ 1ss et τpxq “ τ 1pxq. Par injectivit´e de σ et stσ´1pjq on +a σ ˝ stσ´1pjqpxq ‰ σ ˝ stσ´1pjqpx1q donc on a (quitte `a ´echanger x et x1) σ ˝ stσ´1pjqpxq “ j et +σ ˝ stσ´1pjqpx1q “ j ` 1. Donc stσ´1pjqpxq “ σ´1pjq et stσ´1pjqpxq “ σ´1pjq ` 1. Or σ´1pjq ` 1 +n’est pas dans l’image de stσ´1pjq, contradiction. On a donc l’injectivit´e de τ, donc τ P Sn´1. +Ceci montre que (2.3.20) d´efinit une application +invbijrr1,n´1ss : Err1,n´1ss Ñ Zn ˆ Sn ˆ rr1, n ´ 1ss. +Montrons que invbijrr1,n´1ss envoie Err1,n´1ss dans Ensk +n ˆ rr1, n ´ 1ss. Dans (2.3.20), on a w “ +v ˝stσ´1pjq. On a v P rr0, k ´1ssn´1 +ď +et stσ´1pjq est une application croissante rr1, n´1ss Ñ rr1, nss, +ce qui implique w P rr1, k ´ 1ssn +ď. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +17 +Montrons que τ P S|w´1p0q|,...,|w´1pk´1q|. Ceci revient `a montrer que la restriction de τ `a +w´1pℓq est croissante pour tout ℓ P rr0, k ´ 1ss. Rappelons que v´1pvpσ´1piqqq est un intervalle +de rr1, nss dont σ´1piq n’est pas le plus grand ´el´ement. Notons α, β les minimum et maximum +de cet intervalle, alors v´1pvpσ´1piqqq “ rrα, βss avec α ď σ´1piq ă β. +Si ℓ ă vpσ´1piqq, alors w´1pℓq “ v´1pℓq, et la restriction de τ `a w´1pℓq co¨ıncide avec +la restriction de pσ´1pjq ˝ σ au mˆeme intervalle. Comme σ|v´1pℓq et pσ´1pjq sont croissantes, +τ|w´1pℓq est donc croissante. +Si ℓ ą vpσ´1piqq, alors w´1pℓq “ v´1pℓq ´ 1, et la restriction +de τ `a w´1pℓq co¨ıncide avec la restriction de pσ´1pjq ˝ σ ˝ px ÞÑ x ` 1q au mˆeme intervalle. +Comme pσ ˝ px ÞÑ x ` 1qq|v´1pℓq´1 et pσ´1pjq sont croissantes, τ|w´1pℓq est donc croissante. On +a w´1pvpσ´1piqqq “ rrα, β ´ 1ss. La restriction de τ `a rrα, σ´1piqss co¨ıncide avec celle de σ `a cet +intervalle qui est croissante, cet intervalle ´etant contenu dans v´1pvpσ´1piqqq, donc τ|rrα,σ´1piqss +est croissante. La restriction de τ `a rrσ´1piq, β ´ 1ss co¨ıncide avec celle de σ ˝ px ÞÑ x ` 1q `a +cet intervalle qui est croissante par croissance de σ sur v´1pvpσ´1piqqq, donc τ|rrσ´1piq,β´1ss est +croissante. Il suit que la restriction de τ `a rrα, β ´ 1ss “ w´1pvpσ´1piqqq est croissante. Tout +ceci implique τ P S|w´1p0q|,...,|w´1pk´1q|. +Comme τ P S|w´1p0q|,...,|w´1pk´1q| et w P rr1, k ´ 1ssn +ď, on a pw, τq P Ensk +n´1. +Donc invbijrr1,n´1sspErr1,n´1ssq Ă Ensk +n ˆ rr1, n ´ 1ss. Notons +invbijrr1,n´1ss : Err1,n´1ss Ñ Ensk +n ˆ rr1, n ´ 1ss +l’application induite. On v´erifie que les compositions invbijn˝bijn et bijn˝invbijn sont l’identit´e. +On en d´eduit que bijrr1,n´1ss est une bijection. +□ +2.3.6. Diagramme commutatif impliquant Affp∆n´1, ∆nq ˆ t˘1u. +D´efinition 2.26. Pour n, m ě 0, on note Affp∆n, ∆mq l’ensemble des applications φ : ∆n Ñ +∆m telles qu’il existe une application affine φ : Rn Ñ Rm telle que φ ˝ cann “ canm ˝ φ. (avec +cann, canm comme en d´ef. 2.1) +Lemme 2.27. Soit n, m ě 0. +(a) Si φ P Affp∆n, ∆mq, une application affine φ : Rn Ñ Rm comme en d´ef. 2.26 est unique. +(b) L’application Affp∆n, ∆mq Ñ AffpRn, Rmq, φ ÞÑ φ est injective. +D´emonstration. (a) est une cons´equence de ce que ∆n contient une base affine de Rn, `a savoir +pEn +0 , . . . , En +nq. +(b) provient de ce qu’une application affine est uniquement d´etermin´ee par +l’image d’une base affine. +□ +On d´eduit du lemme 2.27 une famille d’inclusions +(2.3.21) +Affp∆n, ∆mq Ă AffpRn, Rmq +pour n, m ě 0, compatible avec les compositions d’applications. + +18 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +Lemme-D´efinition 2.28. Dans la situation de la d´ef. 2.6, si P0, . . . , Pn P ∆m, l’application +rP0, . . . , Pns envoie ∆n dans ∆m, donc appartient `a l’image de l’injection Affp∆n, ∆mq ãÑ +AffpRn, Rmq. On note rP0, . . . , Pns l’´el´ement de Affp∆n, ∆mq correspondant. +D´emonstration. Suit de la convexit´e de ∆n et ∆m, et de ce que pEn +0 , . . . , En +nq forme une base +du convexe ∆n. +□ +Lemme-D´efinition 2.29. Si pv, σq P Ensk +n, alors ckpv, σq appartient `a l’image de l’injection +Affp∆n, ∆nq ãÑ AffpRn, Rnq. On note ckpv, σq P Affp∆n, ∆nq la pr´eimage de ckpv, σq. +D´emonstration. L’application associ´ee `a pv, σq P Ensk +n est donn´ee par +px1, . . . , xnq ÞÑ ppv1 ` xσp1qq{k, . . . , pvn ` xσpnqq{kq “: py1, . . . , ynq; +c’est une endo-application continue de Rn. +Supposons px1, . . . , xnq P ∆n. +Comme v1 ě 0 +et xσp1q ě 0, on a y1 ě 0 ; de mˆeme, vn ď k ´ 1 et xσpnq ď 1 implique yn ď 1. +Enfin +pour i P rr1, n ´ 1ss, la d´ef. 2.23 implique que vi “ vi`1 ou vi ă vi`1. Dans le premier cas +(vi “ vi`1), (2.3.18) implique σpiq ă σpi`1q, ce qui implique par croissance de i ÞÑ xi l’in´egalit´e +xσpiq ď xσpi`1q, ce qui combin´e avec vi “ vi`1 implique l’in´egalit´e dans yi “ pvi ` xσpiqq{k ď +pvi`1 ` xσpi`1qq{k “ yi`1, o`u les ´egalit´es extrˆemes proviennent des d´efinitions. Dans le second +cas (vi ă vi`1), on a yi “ pvi ` xσpiqq{k ď pvi ` 1q{k ď vi`1{k ď pvi`1 ` xσpi`1qq{k “ yi`1, o`u +la premi`ere et derni`ere ´egalit´e proviennent des d´efinitions, o`u la premi`ere et derni`ere in´egalit´e +proviennent respectivement de xσpiq ě 0 et xσpi`1q ď 1, et o`u l’in´egalit´e centrale provient de +vi ă vi`1 et du caract`ere entier des composantes de v. On a donc dans tous les cas yi ď yi`1, +ce qui ach`eve de montrer que py1, . . . , ynq P ∆n. Ceci montre l’´enonc´e. +□ +On a ckpv, σq “ rp1{kqpv ` σ˚En +0 q, . . . , p1{kqpv ` σ˚En +nqs pour pv, σq P Ensk +n. +D´efinition 2.30. On note pour i P rr0, nss, +Bn +i :“ rEn +0 , . . . , En +n´i´1, En +n´i`1, . . . , En +ns P Affp∆n´1, ∆nq. +Lemme 2.31. Bn +i est l’image de Bn +i par l’application Affp∆n´1, ∆nq Ñ AffpRn´1, Rnq. +D´emonstration. Imm´ediat. +□ +D´efinition 2.32. On note +f : Ensk +n ˆ rr0, nss Ñ Affp∆n´1, ∆nq, +˜f : Ensk +n´1 ˆ rr0, nss Ñ Affp∆n´1, ∆nq +les applications donn´ees par +fpv, σ, iq :“ ckpv, σq ˝ Bn +i , +˜fpw, τ, iq :“ Bn +i ˝ ckpw, τq. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +19 +Lemme 2.33. Les diagrammes suivants commutent +Ensk +n ˆ rr0, nss +f +� +� � +� +Affp∆n´1, ∆nq +� � +� +Zn ˆ Sn ˆ ˆrr0, nss +f +� AffpRn´1, Rnq +Ensk +n´1 ˆ rr0, nss +˜ +f +� +� � +� +Affp∆n´1, ∆nq +� � +� +Zn´1 ˆ Sn´1 ˆ ˆrr0, nss ˜ +f +� AffpRn´1, Rnq +D´emonstration. Cela provient de la compatibilit´e des applications (2.3.21) avec la composition, +du lemme 2.31, et ce que pour pv, σq P Ensk +n (resp. pw, τq P Ensk +n´1), l’image de ckpv, σq (resp. +ckpw, τq) sous Affp∆n, ∆nq Ñ AffpRn, Rnq (resp. Affp∆n´1, ∆n´1q Ñ AffpRn´1, Rn´1q) est +ckpv, σq (resp. ckpw, τq) (cf. lemme 2.29). +□ +Lemme 2.34. Le diagramme suivant commute +(2.3.22) +Ensk +n´1 ˆ rr0, nss +bij +„ +� +p ˜ +f,Ą +sgnq +�❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +tx P Ensk +n ˆ rr0, nss|involpxq R Ensk +n ˆ rr0, nssuu +pf,sgnq +�❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ +Affp∆n´1, ∆nq ˆ t˘1u +D´emonstration. Ceci provient de la commutativit´e des diagrammes (2.3.17) et (2.3.19) ainsi +que du lemme 2.34. +□ +2.4. Construction d’endomorphismes de groupes de chaˆınes. Les r´esultats de cette +section seront utilis´es en section 2.5 afin de montrer le (b) du th´eor`eme 2.3. +2.4.1. Morphismes dans une cat´egorie C. +D´efinition 2.35. Soit C la petite cat´egorie dont l’ensemble d’objets est Zě0, avec Cpn, mq :“ +ZTopp∆n, ∆mq, et dont la composition est donn´ee par la lin´earisation de la composition dans +Top. +Toute application affine ´etant continue, on a une famille de diagrammes +Affp∆n, ∆mq Ă Cpn, mq +pour n, m ě 0, compatible avec les compositions d’applications. +D´efinition 2.36. On pose +divk +n :“ +ÿ +pv,σqPEnsk +n +ǫpσqckpv, σq P ZAffp∆n, ∆nq Ă Cpn, nq. +D´efinition 2.37. On note +Bn´1,n :“ +nÿ +i“0 +p´1qiBn +i P ZAffp∆n´1, ∆nq Ă Cpn ´ 1, nq. + +20 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +On sait que Bn,n`1 ˝ Bn´1,n “ 0. De plus, pour tout p ě 0, le complexe ¨ ¨ ¨ Ñ Cpk, pq +´˝Bk´1,k +Ñ +Cpk ´ 1, pq Ñ ¨ ¨ ¨ Ñ Cp0, pq Ñ Z Ñ 0 est acyclique (l’homologie de ∆p ´etant donn´ee par +Hkp∆pq “ 0 si k ą 0 et “ Z si k “ 0). +2.4.2. D´emonstration de divk +n ˝Bn´1,n “ Bn´1,n ˝divk +n´1. Dans la section 2.4.2, on fixe n, k ě 1. +On a +(2.4.1) +divk +n ˝ Bn´1,n “ +ÿ +xPEnsk +nˆrr0,nss +sgnpxqfpxq. +en combinant les d´efs. 2.9, 2.32, 2.36, 2.37, et +(2.4.2) +Bn´1,n ˝ divk +n´1 “ +ÿ +˜xPEnsk +n´1ˆrr0,nss +Ą +sgnp˜xq ˜fp˜xq. +en combinant les d´efs. 2.17 2.17, 2.32, 2.36, 2.37 (´egalit´es dans ZAffp∆n´1, ∆nq). +Proposition 2.38. On a pour tous n, k ě 1 +divk +n ˝ Bn´1,n “ Bn´1,n ˝ divk +n´1 +(´egalit´e dans Cpn ´ 1, nq). +D´emonstration. On a +ÿ +xPEnsk +nˆrr0,nss| +involpxqPEnsk +nˆrr0,nss +sgnpxqfpxq “ +ÿ +xPEnsk +nˆrr0,nss| +involpxqPEnsk +nˆrr0,nss +sgnpinvolpxqqfpinvolpxqq +“ ´ +ÿ +xPEnsk +nˆrr0,nss| +involpxqPEnsk +nˆrr0,nss +sgnpxqfpxq +o`u la premi`ere ´egalit´e suit de ce que invol est une involution de tx P Ensk +n ˆ rr0, nss|involpxq P +Ensk +nˆrr0, nssu (cf. lemme 2.11(a)) et la deuxi`eme ´egalit´e suit du lemme 2.11(b) et de la premi`ere +´egalit´e du lemme 2.33. On en d´eduit +(2.4.3) +ÿ +xPEnsk +nˆrr0,nss| +involpxqPEnsk +nˆrr0,nss +sgnpxqfpxq “ 0. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +21 +Alors +divk +n ˝ Bn´1,n “ +ÿ +xPEnsk +nˆrr0,nss +sgnpxqfpxq +“ +ÿ +xPEnsk +nˆrr0,nss| +involpxqPEnsk +nˆrr0,nss +sgnpxqfpxq ` +ÿ +xPEnsk +nˆrr0,nss| +involpxqREnsk +nˆrr0,nss +sgnpxqfpxq +“ +ÿ +xPEnsk +nˆrr0,nss| +involpxqREnsk +nˆrr0,nss +sgnpxqfpxq “ +ÿ +yPEnsk +n´1ˆrr0,nss +sgnpbijpyqqfpbijpyqq +“ +ÿ +yPEnsk +n´1ˆrr0,nss +Ą +sgnpyq ˜fpyq “ Bn´1,n ˝ divk +n´1 +(´egalit´e dans ZAffp∆n´1, ∆nq) o`u la premi`ere ´egalit´e suit de (2.4.1), la troisi`eme ´egalit´e suit de +(2.4.3), la quatri`eme ´egalit´e suit du lemme 2.25, la cinqui`eme ´egalit´e suit du lemme 2.20 et des +deux ´egalit´es du lemme 2.33, la sixi`eme ´egalit´e suit de (2.4.2). +On en d´eduit l’´egalit´e annonc´ee, les applications Affp∆n, ∆mq Ñ Cpn, mq ´etant compatibles +aux compositions. +□ +2.4.3. Relation dans C entre divk +‚ et id‚. +Lemme 2.39. Pour tout k ě 0, il existe une famille pLk +n`1,nqně0 avec Lk +n`1,n P Cpn ` 1, nq, +telle que pour tout n ě 0, on a +(2.4.4) +idn ´ divk +n “ Lk +n`1,n ˝ Bn,n`1 ` Bn´1,n ˝ Lk +n,n´1. +D´emonstration. Montrons par r´ecurrence sur n ě 0 l’existence d’une famille pLk +m`1,mqmďn telle +que pour tout m ď n, on a idn ´ divk +n “ Lk +n`1,n ˝ Bn,n`1 ` Bn´1,n ˝ Lk +n,n´1 (´enonc´e Epnq). +Posons Lk +1,0 :“ 0, alors on a id0 ´ divk +0 “ Lk +1,0 ˝ B0,1 d’o`u l’´enonc´e Ep0q. +Soit n ě 1, et supposons Epn ´ 1q v´erifi´e avec une famille pLk +m`1,mqmďn´1. Alors +pidn ´ divk +n ´ Bn´1,n ˝ Lk +n,n´1q ˝ Bn´1,n “ Bn´1,n ´ divk +n ˝ Bn´1,n ´ Bn´1,n ˝ Lk +n,n´1 ˝ Bn´1,n +“ Bn´1,n ´ Bn´1,n ˝ divk +n´1 ´ Bn´1,n ˝ Lk +n,n´1 ˝ Bn´1,n “ Bn´1,n ˝ pidn´1 ´ divk +n´1 ´ Lk +n,n´1 ˝ Bn´1,nq +“ Bn´1,n ˝ pidn´1 ´ divk +n´1 ´ Lk +n,n´1 ˝ Bn´1,n ´ Bn´2,n´1 ˝ Lk +n´1,n´2q “ Bn´1,n ˝ 0 “ 0 +o`u la seconde ´egalit´e suit de la proposition 2.38, la quatri`eme ´egalit´e suit de Bn´1,n ˝Bn´2,n´1 “ +0, la cinqui`eme ´egalit´e suit de Epn ´ 1q. +Donc idn ´divk +n ´Bn´1,n ˝Lk +n,n´1 appartient au noyau de l’application ´˝Bn´1,n : Cpn, nq Ñ +Cpn´1, nq, qui par acyclicit´e est ´egal `a l’image de l’application ´˝Bn,n`1 : Cpn`1, nq Ñ Cpn, nq. +Il existe donc Lk +n`1,n P Cpn ` 1, nq tel que idn ´ divk +n ´ Bn´1,n ˝ Lk +n,n´1 “ Lk +n`1,n ˝ Bn,n`1, ce +qui implique Epnq. +□ +2.4.4. Endomorphismes de groupes de chaˆınes singuli`eres. Soit X un espace topologique. Pour +n ě 0, on note CnpXq :“ ZTopp∆n, Xq. +Pour n, m ě 0, on a une application CnpXq ˆ +Cpm, nq Ñ CmpXq induite par la composition pc, xq ÞÑ c ˝ x. + +22 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +D´efinition 2.40. Pour x P Cpm, nq, on note x˚ : CnpXq Ñ CmpXq l’application c ÞÑ c ˝ x. +Alors px ˝ yq˚ “ y˚ ˝ x˚. +Lemme 2.41. Si Y est un sous-ensemble de X, on a x˚pCnpY qq Ă CmpY q. +D´emonstration. Provient de ce que x˚ est une composition `a la source. +□ +De plus pour n ě 1, B˚ +n´1,n : CnpXq Ñ Cn´1pXq co¨ıncide avec la diff´erentielle singuli`ere Bn. +Alors (2.4.4) implique +(2.4.5) +idCnpXq ´ pdivk +nq˚ “ B˚ +n,n`1 ˝ pLk +n`1,nq˚ ` pLk +n,n´1q˚ ˝ B˚ +n´1,n +(´egalit´e d’endomorphismes de CnpXq) pour tout n ě 0. +2.5. D´emonstration de (b) du th´eor`eme 2.3. On se place dans le cadre de la section 2.1: +X est un espace topologique et a, b P X. +2.5.1. Composition de chemins. +D´efinition 2.42. Si s, t, u, v P R avec s ‰ t, on note au,v +s,t l’unique application affine de R dans +lui-mˆeme telle que s ÞÑ u et t ÞÑ v. +Soit X un espace topologique, soit a0, . . . , am P X et ˜γi P Chempai, ai`1q pour i P rr0, m´1ss. +D´efinition 2.43. ˜γm´1 ˚. . .˚ ˜γ0 P Chempa0, amq est le chemin tel que pour pour i P rr0, m´1ss, +la restriction p˜γm´1 ˚ . . . ˚ ˜γ0q|ri{m,pi`1q{ms `a ri{m, pi ` 1q{ms co¨ıncide avec ˜γi ˝ a0,1 +i{m,pi`1q{m +(conditions coh´erentes car ˜γip1q “ ˜γi`1p0q pour i P rr0, m ´ 2ss). +On a alors, en notant r´s l’application canonique Chempa, bq Ñ π1pX; a, bq pour a, b P X +quelconques, l’´egalit´e +(2.5.1) +r˜γn´1 ˚ . . . ˚ ˜γ0s “ r˜γn´1s ¨ ¨ ¨ r˜γ0s +(´egalit´e dans π1pX; a0, amq, le produit dans le membre de droite ´etant celui dans le groupo¨ıde +π1pXq). +2.5.2. Calcul de pdivk +nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq. +Lemme 2.44. Il existe une unique application +(2.5.2) +tpn1, . . . , nkq|n1 ě 0, . . . , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ nu Ñ rr0, k ´ 1ssn +qui envoie pn1, . . . , nkq vers l’´el´ement v P rr0, k ´ 1ssn tel que pour i “ 0, . . . , k ´ 1 on a +v|n1`¨¨¨`ni`rr1,ni`1ss “ i. Cette application est bijective, et la bijection r´eciproque envoie v P +rr0, k ´ 1ssn vers pn1, . . . , nkq donn´e par ni “ |v´1pi ´ 1q| pour i P rr1, kss. +D´emonstration. Imm´ediate. +□ +D´efinition 2.45. Pour v P rr0, k ´ 1ssn, on pose Spvq :“ tσ P Sn|pv, σq P Ensk +nu (cf. d´ef. 2.23). + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +23 +Lemme 2.46. Si pn1, . . . , nkq P tpn1, . . . , nkq|n1 ě 0, . . . , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ nu et si v +est l’image de pn1, . . . , nkq par la bijection (2.5.2), alors on a Sn1,...,nk “ Spvq. +D´emonstration. L’application v : rr1, nss Ñ rr0, k ´ 1ss ´etant constante sur chaque sous-ensemble +n1`¨ ¨ ¨`ni`rr1, ni`1ss et les valeurs prises sur des sous-ensembles cons´ecutifs ´etant strictement +croissantes, on a pour i P rr1, nss l’´equivalence +(2.5.3) +pvi ă vi`1q ðñ pi P tn1, n1 ` n2, . . . , n1 ` ¨ ¨ ¨ ` nk´1uq. +Soit alors σ P Sn. On a σ P Spvq si et seulement si +@i P rr1, nss, +pσpiq ą σpi ` 1qq ùñ pvi ă vi`1q; +d’apr`es (2.5.3) une condition ´equivalente est +@i P rr1, nss, +pσpiq ą σpi ` 1qq ùñ pi P tn1, n1 ` n2, . . . , n1 ` ¨ ¨ ¨ ` nk´1uq. +Donc σ P Spvq si et seulement si σ est croissante sur les sous-ensembles rr1, n1ss, n1 ` rr1, n2ss, +etc., n1 ` ¨ ¨ ¨ ` nk´1 ` rr1, nkss, c’est `a dire si et seulement si σ P Sn1,...,nk. +□ +D´efinition 2.47. On pose Ą +Ens +k +n :“ tppn1, . . . , nkq, σq|n1 ě 0, . . . , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ n +et σ P Sn1,...,nku. +Lemme 2.48. Il existe une unique bijection +(2.5.4) +Ensk +n Ñ Ą +Ens +k +n +qui envoie ppn1, . . . , nkq, σq vers pv, σq, o`u v est l’image de pn1, . . . , nkq par (2.5.2). +D´emonstration. Cons´equence des lemmes 2.44 et 2.46. +□ +Notons que pour n1 ` ¨ ¨ ¨ ` nk “ n et σ P Sn1,...,nk, l’automorphisme de r0, 1sn dans la +cat´egorie Top donn´e par σ˚ induit un ´el´ement, not´e encore σ˚ de Topp∆n, ∆n1ˆ. . .ˆ∆nkq. Par +ailleurs, ˜γn1 +1 ˆ¨ ¨ ¨ˆ˜γnk +k +P Topp∆n1 ˆ. . .ˆ∆nk, Xnq, donc p˜γn1 +1 ˆ¨ ¨ ¨ˆ˜γnk +k q˝σ˚ P Topp∆n, Xnq. +Lemme 2.49. Les applications Ensk +n Ñ Topp∆n, Xnq et Ą +Ens +k +n Ñ Topp∆n, Xnq donn´ees re- +spectivement par ppn1, . . . , nkq, σq ÞÑ p˜γpn1q +1 +ˆ ¨ ¨ ¨ ˆ ˜γpnkq +k +q ˝ σ˚ et pv, σq ÞÑ ˜γpnq ˝ cpv, σq sont +telles que le diagramme +Ensk +n +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +(2.5.4) +� Ą +Ens +k +n +�rrrrrrrrrrr +Topp∆n, Xnq +est commutatif. + +24 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +D´emonstration. Soit ppn1, . . . , nkq, σq P Ensk +n et pv, σq P Ą +Ens +k +n son image par (2.5.4). Posons +δm :“ p1, . . . , 1q P Rm pour m ě 1. Alors pour px1, . . . , xnq P ∆n, on a +˜γpnq ˝ cpv, σqpx1, . . . , xnq +“˜γpnq +´ +pxσp1q,...,xσpn1qq{k,pδn2 `pxσpn1`1q,...,xσpn1`n2qqq{k,...,ppk´1qδnk`pxσpn1`¨¨¨`nk´1`1q,...,xσpnqqq{k +¯ +“ +´ +˜γn1 +´ +pxσp1q,...,xσpn1qq{k +¯ +,˜γn2 +´ +pδn2`pxσpn1`1q,...,xσpn1`n2qqq{k +¯ +,...,˜γnk +´ +ppk´1qδnk `pxσpn1`¨¨¨`nk´1`1q,...,xσpnqqq{k +¯¯ +“ +´ +˜γn1 +1 pxσp1q, . . . , xσpn1qq, ˜γn2 +2 pxσpn1`1q, . . . , xσpn1`n2qqq, . . . , ˜γnk +k pxσpn1`¨¨¨`nk´1`1q, . . . , xσpnqqq +¯ +“ +´ +˜γpn1q +1 +pxσp1q, . . . , xσpn1qq, ˜γpn2q +2 +pxσpn1`1q, . . . , xσpn1`n2qq, . . . , ˜γpnkq +k +pxσpn1`¨¨¨`nk´1`1q, . . . , xσpnqq +¯ +“ p˜γpn1q +1 +ˆ ¨ ¨ ¨ ˆ ˜γpnkq +k +q ˝ σ˚px1, . . . , xnq, +o`u la premi`ere ´egalit´e suit de ce que cpv, σq est l’application de ∆n dans lui-mˆeme donn´ee par +px1, . . . , xnq ÞÑ +´ +pxσp1q, . . . , xσpn1qq{k, pδn2 ` pxσpn1`1q, . . . , xσpn1`n2qqq{k, . . . , ppk ´ 1qδnk ` pxσpn1`¨¨¨`nk´1`1q, . . . , xσpnqqq{k +¯ +, +la deuxi`eme ´egalit´e suit de la combinaison du fait que ˜γpnq est une restriction de ˜γn et de +l’´egalit´e ˜γn “ ˜γn1 ˆ¨ ¨ ¨ˆ ˜γnk, la troisi`eme ´egalit´e suit de l’identit´e ˜γppi´1`xq{kq “ ˜γipxq pour +x P r0, 1s et i P rr1, kss, la quatri`eme ´egalit´e suit des relations pxσp1q, . . . , xσpn1qq P ∆n1, etc., +pxσpn1`¨¨¨`nk´1`1q, . . . , xσpnqq P ∆nk, elles-mˆemes cons´equences de σ P Sn1,...,nk, la derni`ere +´egalit´e suit de la d´efinition de σ˚ (cf. section 2.5.2). On a donc ˜γpnq ˝ cpv, σq “ p˜γpn1q +1 +ˆ ¨ ¨ ¨ ˆ +˜γpnkq +k +q ˝ σ˚ (´egalit´e dans Topp∆n, Xnq). +□ +Proposition 2.50. Si a1, . . . , ak`1 P X et ˜γi P Chempai, ai`1q pour i P rr1, kss. Alors +pdivk +nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq “ +ÿ +n1,...,nkě0| +n1`¨¨¨`nk“n +ÿ +σPSn1,...,nk +ǫpσqp˜γpn1q +1 +ˆ ¨ ¨ ¨ ˆ ˜γpnkq +k +q ˝ σ˚ +(´egalit´e dans CnpXnq “ ZTopp∆n, Xnq). +D´emonstration. On a +pdivk +nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq “ p˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnq ˝ divk +n “ +ÿ +pσ,vqPEnsk +n +ǫpσqp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnq ˝ cpσ, vq +“ +ÿ +n1,...,nkě0| +n1`¨¨¨`nk“n +ÿ +σPSn1,...,nk +ǫpσqp˜γpn1q +1 +ˆ ¨ ¨ ¨ ˆ ˜γpnkq +k +q ˝ σ˚ +o`u la premi`ere ´egalit´e suit de la d´ef. 2.40, la deuxi`eme ´egalit´e suit de la d´ef. 2.36, et la troisi`eme +suit du lemme 2.49. +□ +2.5.3. Une ´egalit´e dans CnpXnq. On fixe ˜γ P Chempa, bq et α0, . . . , αn P Chempa, aq. On note +pour I Ă rr0, nss, cI :“ p˜γ ˚ ˚iPI ˜αiqpnq P CnpXnq. En appliquant (2.4.5) `a cI (X ´etant remplac´e +par Xn et k par |I| ` 1), on trouve +(2.5.5) +cI ´ pdiv|I|`1 +n +q˚pcIq “ pB˚ +n,n`1 ˝ pL|I|`1 +n`1,nq˚ ` pL|I|`1 +n,n´1q˚ ˝ B˚ +n´1,nqpcIq + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +25 +(relation dans CnpXnq). On a B˚ +n´1,npcIq P Cn´1pY pnq +ab q par le lemme 2.2, le lemme 2.41 implique +alors que pL|I|`1 +n,n´1q˚ ˝ B˚ +n´1,npcIq P CnpY pnq +ab q. D’autre part, pL|I|`1 +n`1,nq˚pcIq P Cn`1pXnq, donc +B˚ +n,n`1 ˝ pL|I|`1 +n`1,nq˚pcIq P B˚ +n,n`1pCn`1pXnqq. Ces deux relations et (2.5.5) impliquent +@I Ă rr0, nss, +cI ´ pdiv|I|`1 +n +q˚pcIq P CnpY pnq +ab q ` B˚ +n,n`1pCn`1pXnqq +(relation dans CnpXnq). Cette relation implique ř +IĂrr0,nssp´1q|I|pcI´pdiv|I|`1 +n +q˚pcIqq P CnpY pnq +ab q` +B˚ +n,n`1pCn`1pXnqq donc +(2.5.6) +ÿ +IĂrr0,nss +p´1q|I|cI ´ +ÿ +IĂrr0,nss +p´1q|I|pdiv|I|`1 +n +q˚pcIq P CnpY pnq +ab q ` B˚ +n,n`1pCn`1pXnqq. +(relation dans CnpXnq). +Lemme 2.51. On a ř +IĂrr0,nssp´1q|I|pdiv|I|`1 +n +q˚pcIq “ 0 (´egalit´e dans CnpXnq). +D´emonstration. Pour ν0, . . . , νn`1 ě 0 avec ν0 ` ¨ ¨ ¨ ` νn`1 “ n, posons +(2.5.7) +fpν0, . . . , νn`1q :“ +ÿ +σPSν0,...,νn`1 +ǫpσqp˜αpν0q ˆ ¨ ¨ ¨ ˆ ˜αpνnq ˆ ˜γpνn`1qq ˝ σ˚ P CnpXnq. +Soit I Ă rr0, nss. Soit α ÞÑ iα l’unique bijection croissante rr1, |I|ss Ñ I. On a les ´egalit´es +pdiv|I|`1 +n +q˚pcIq “ +ÿ +ν:I\tn`1uÑZě0, +ř +xPI\tn`1u νpxq“n +ÿ +σPSνpi1q,...,νpi|I|q,νpn`1q +ǫpσqp˜αpνpi1qq +i1 +ˆ ¨ ¨ ¨ ˆ ˜α +pνpi|I|qq +i|I| +ˆ ˜γpνpn`1qqq ˝ σ˚ +(2.5.8) +“ +ÿ +ν0,...,νn`1ě0| +ν0`¨¨¨`νn`1“n, +ν|rr0,nss´I“0 +ÿ +σPSν0,...,νn`1 +ǫpσqp˜αpν0q ˆ ¨ ¨ ¨ ˆ ˜αpνnq ˆ ˜γpνn`1qq ˝ σ˚ “ +ÿ +ν0,...,νn`1ě0| +ν0`¨¨¨`νn`1“n, +ν|rr0,nss´I“0 +fpν0, . . . , νn`1q +dans CnpXnq o`u la premi`ere ´egalit´e suit de la proposition 2.50, et la deuxi`eme ´egalit´e utilise +la bijection entre applications I \ tn ` 1u Ñ Zě0 et applications rr0, n ` 1ss Ñ Zě0 nulles sur +rr0, nss ´ I fournie par l’extension par la fonctions nulle, et la troisi`eme ´egalit´e suit de (2.5.7). +Alors +ÿ +IĂrr0,nss +p´1q|I|pdiv|I|`1 +n +q˚pcIq “ +ÿ +IĂrr0,nss +ÿ +ν0,...,νn`1ě0| +ν0`¨¨¨`νn`1“n, +ν|rr0,nss´I“0 +p´1q|I|fpν0, . . . , νn`1q +“ +ÿ +ν0,...,νn`1ě0| +ν0`¨¨¨`νn`1“n +ÿ +txPrr0,nss|νx‰0uĂIĂrr0,nss +p´1q|I|fpν0, . . . , νn`1q +“ +ÿ +ν0,...,νn`1ě0| +ν0`¨¨¨`νn`1“n +fpν0, . . . , νn`1q +ÿ +txPrr0,nss|νx‰0uĂIĂrr0,nss +p´1q|I| +(2.5.9) +o`u la premi`ere ´egalit´e suit de (2.5.8), la deuxi`eme ´egalit´e suit de l’´equivalence entre les conditions +ν|rr0,nss´I “ 0 et I Ą tx P rr0, nss|νx ‰ 0u et la troisi`eme ´egalit´e est une factorisation. +Pour tout pν0, . . . , νn`1q P Zn`1 +ě0 +tel que ν0 ` ¨ ¨ ¨ ` νn`1 “ n, on a +(2.5.10) +ÿ +txPrr0,nss|νx‰0uĂIĂrr0,nss +p´1q|I| “ p´1q|txPrr0,nss|νx‰0u| +ÿ +JĂtxPrr0,nss|νx“0u +p´1q|J| “ 0 + +26 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +o`u la premi`ere ´egalit´e suit de la bijection entre l’ensemble des I tels que tx P rr0, nss|νx ‰ +0u Ă I Ă rr0, nss et l’ensemble des parties J de tx P rr0, nss|νx “ 0u fournie par J ÞÑ J Y tx P +rr0, nss|νx ‰ 0u, et la seconde ´egalit´e suit de l’identit´e ř +XĂEp´1q|X| “ 0 pour tout ensemble fini +non vide E, ainsi que de tx P rr0, nss|νx “ 0u ‰ H, qui r´esulte de pν0, . . . , νn`1q P Zn`1 +ě0 +et ν0 ` +¨ ¨ ¨ ` νn`1 “ n. En combinant (2.5.10) et (2.5.9), on obtient ř +IĂrr0,nssp´1q|I|pdiv|I|`1 +n +q˚pcIq “ +0. +□ +2.5.4. D´emonstration de (b) du th´eor`eme 2.3. Le lemme 2.51 et (2.5.6) impliquent +ÿ +IĂrr0,nss +p´1q|I|cI P CnpY pnq +ab q ` B˚ +n,n`1pCn`1pXnqq +(relation dans CnpXnq) c’est-`a-dire +(2.5.11) +ÿ +IĂrr0,nss +p´1q|I|p˜γ ˚ ˚iPI ˜αiqpnq P CnpY pnq +ab q ` B˚ +n,n`1pCn`1pXnqq +(relation dans CnpXnq). Chaque cI “ p˜γ˚˚iPI ˜αiqpnq appartient au sous-espace ZnpXn, Y pnq +ab q “ +tc P CnpXnq|B˚ +n´1,npcq P Cn´1pY pnq +ab qu Ă CnpXnq, donc (2.5.11) peut ˆetre vue comme une +relation dans ZnpXn, Y pnq +ab q. Elle implique la relation +(2.5.12) +ÿ +IĂrr0,nss +p´1q|I|rp˜γ ˚ ˚iPI ˜αiqpnqs “ 0 +(´egalit´e dans HnpXn, Y pnq +ab q “ ZnpXn, Y pnq +ab q{BnpXn, Y pnq +ab q, avec BnpXn, Y pnq +ab q “ CnpY pnq +ab q ` +B˚ +n,n`1pCn`1pXnqq). +Pour I Ă rr0, nss, on a +rp˜γ ˚ ˚iPI ˜αiqpnqs “ Fnpr˜γ ˚ ˚iPI ˜αisq “ Fnpγ ¨ +ź +iPI +αiq +(´egalit´e dans HnpXn, Y pnq +ab q) o`u la premi`ere ´egalit´e suit de (2.1.1) et deuxi`eme suit de ((2.5.1)). +En combinant cette ´egalit´e avec (2.5.12), on en d´eduit l’´egalit´e souhait´ee +ÿ +IĂrr0,nss +p´1q|I|Fnpγ ¨ +ź +iPI +αiq “ 0 +(´egalit´e dans HnpXn, Y pnq +ab q). +3. Lien avec l’isomorphisme de Beilinson +Le but de cette section est la d´emonstration de la proposition 3.1, qui relie l’application +F +pnq +xy obtenue dans le th´eor`eme 2.3 avec l’isomorphisme (0.0.1) de Beilinson. On rappelle la +construction de cet isomorphisme en section 3.1, puis on montre la proposition 3.1 en section +3.2. + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +27 +3.1. Rappels sur l’isomorphisme de Beilinson. Soit M une vari´et´e diff´erentiable connexe, +ayant le type d’homotopie d’un CW-complexe fini et n ě 1. Dans [DG], §3.3 (voir aussi [BGFr], +p. 251), on associe `a chaque couple px, yq d’´el´ements de M un complexe de faisceaux de Q- +espaces vectoriels yKxxny sur M n et une application lin´eaire surjective H‚pM n,x Kxxnyq Ñ Q. +On a les isomorphismes de Q-espaces vectoriels +H‚pM n, Y pnq +yx ; Qq » H‚pM n,y Kxxnyq si x ‰ y, +([BGFr], deux lignes avant (3.282)) et +H‚pM n, Y pnq +xx ; Qq » KerpH‚pM n,x Kxxnyq Ñ Qq, +([BGFr], (3.284)), o`u H‚p´, ´; Qq d´esigne l’homologie singuli`ere relative `a coefficients dans Q +des paires d’espaces topologiques et H‚ l’hypercohomologie des complexes de faisceaux. +Le th´eor`eme de Beilinson (Proposition 3.4 de [DG], ou Theorem 3.298 de [BGFr]) dit qu’il y +a un isomorphisme de Q-espaces vectoriels +βpnq +yx : H‚pM n,y Kxxnyq Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚ +s’ins´erant pour y “ x dans le diagramme commutatif +H‚pM n,x Kxxnyq +� +�▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +pQπ1pxq{pQπ1pxqqn`1 +` +q˚ +�♥♥♥♥♥♥♥♥♥♥♥♥♥♥ +Q +l’application pQπ1pxq{pQπ1pxqqn`1 +` +q˚ Ñ Q ´etant duale de l’application Q Ñ Qπ1pxq{pQπ1pxqqn`1 +` +induite par 1 ÞÑ 1. +On en d´eduit pour tout px, yq une application lin´eaire +(3.1.1) +βpnq +yx : H‚pM n, Y pnq +yx ; Qq Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚, +qui est un isomorphisme si y ‰ x, et qui induit un isomorphisme +H‚pM n, Y pnq +xx ; Qq „ +Ñ KerppQπ1pxq{pQπ1pxqqn`1 +` +q˚ Ñ Qq +si y “ x. +3.2. Relation du th´eor`eme 2.3 avec l’isomorphisme de Beilinson. Notons θ l’involution +de M n donn´ee par px1, . . . , xnq ÞÑ pxn, . . . , x1q. L’image de Y pnq +yx +par cette involution est Y pnq +xy , +donc elle induit un isomorphisme θ˚ : HnpM n, Y pnq +yx ; Qq Ñ HnpM n, Y pnq +xy ; Qq. +Rappelons le +couplage entre homologie et cohomologie relatives, qui induit une application lin´eaire can : +HnpM n, Y pnq +xy ; Qq Ñ HnpM n, Y pnq +xy q˚ +Q, o`u pour A un Z-module, on note A˚ +Q :“ HomZpA, Qq. +Pour f : A Ñ B morphisme de Z-modules, on note aussi f ˚ +Q : B˚ +Q Ñ A˚ +Q le morphisme induit. + +28 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +Proposition 3.1. L’application lin´eaire βpnq +yx (cf. (3.1.1)) est au signe pr`es, la compos´ee du +dual pF +pnq +xy q˚ +Q de l’application lin´eaire F +pnq +xy (cf. th´eor`eme 2.3), de can et de θ˚. Pr´ecis´ement, +on a +βpnq +yx “ p´1qn`1pF +pnq +xy q˚ +Q ˝ can ˝ θ˚. +D´emonstration. Dans [BGFr], on construit un morphisme de complexes de faisceaux nat :y +˜Kxxny Ñy Kxxny ((3.282) et cinq lignes avant cette ´equation) ; si y ‰ x, ce morphisme est +l’identit´e de yKxxny (cf. loc. cit., 6 lignes avant (3.282)). Dans loc. cit., on construit un iso- +morphisme d’espaces vectoriels isoyx +BGF : HnpM n,y ˜Kxxnyq Ñ HnpM n, Y pnq +yx ; Qq (Lemma 3.281). +L’application (3.1.1) est alors donn´ee par la composition +HnpM n, Y pnq +yx ; Qq +pisoyx +BGFq´1 +Ñ +HnpM n,y ˜Kxxnyq Ñ HnpM n,y Kxxnyq +βpnq +yx +Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚ +L’´enonc´e est alors ´equivalent `a la commutativit´e du diagramme suivant +HnpM n,y ˜Kxxnyq +nat˚ � +isoyx +BGF � +HnpM n,y Kxxnyq +βpnq +yx � pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚ +HnpM n, Y pnq +yx ; Qq +θ˚ � HnpM n, Y pnq +xy ; Qq +p´1qn`1can +� HnpM n, Y pnq +xy q˚ +Q +pF pnq +xy q˚ +� +laquelle est ´equivalente `a l’´enonc´e suivant : +@c P HnpM n, Y pnq +xy ; Qq, +@a P Qπ1px, yq{pQπ1px, yqpQπ1pxqqn`1 +` +q, +(3.2.1) +xβpnq +yx ˝ nat˚ ˝ pisoyx +BGFq´1 ˝ pθ˚q´1pcq, ay “ p´1qn`1xc, F +pnq +xy paqyhom, +o`u x´, ´y est le couplage V ˚ ˆ V Ñ Q associ´e `a un Q-espace vectoriel V et x´, ´yhom est +le couplage naturel HnpX, Y ; Qq ˆ HnpX, Y q Ñ Q, que par lin´earit´e il suffit de v´erifier pour +a “ rγs, o`u γ P π1px, yq. +Rappelons quelques constructions de [BGFr]. Soit ˜C :“ pp ˜Cp,qqp,qě0, d1, d2q le bicomplexe tel +que ˜Cp,q :“ ‘IĂrr1,nss||I|“n´pCqpXIq, o`u CqpXq :“ HomQpCqpXq, Qq, o`u d2 est la somme sur +I, q des op´erateurs de cobord CqpXIq Ñ Cq`1pXIq et o`u d2 est la somme sur les couples pI, Jq +avec I Ą J et |I ´ J| “ 1 des applications ǫpI, Jqδ˚ +I,J : CqpXIq Ñ CqpXJq, o`u δI,J : XJ Ñ XI +est le morphisme donn´e par [BGFr], formule apr`es (3.287) et ǫpI, Jq P t˘1u est donn´e par +[BGFr], formule avant (3.278). +Soit C :“ ppCp,qqp,qě0, d1, d2q le bicomplexe quotient de ˜C +donn´e par Cp,q “ ˜Cp,q si q ă n, Cp,n “ 0 pour tout p ě 0. +D’apr`es [BGFr], Lemma 3.289, on a un isomorphisme H‚pTotpCqq » HnpM n,y Kxxnyq. On +montre que cet isomorphisme s’ins`ere dans carr´e commutatif dont les morphismes verticaux + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +29 +sont des isomorphismes +(3.2.2) +H‚pTot ˜Cq +nat +� +„ +˜iC � +H‚pTotCq +„ +iC +� +HnpM n,y ˜Kxxnyq +nat˚ � HnpM n,y Kxxnyq +Alors l’application compos´ee +HnpTotCq iC +Ñ HnpM n,y Kxxnyq +βpnq +yx +Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚ +est telle que pour ω “ pωIqH‰IĂrr1,nss P ZnpTotCq avec ωI P Cn´|I|pXIq, et γ P π1px, yq, on a +(3.2.3) +xβpnq +yx ˝ iCprωsq, rγsy “ +ÿ +H‰IĂrr1,nss +p´1qp1{2qp|I|´1qp|I|´2q`n|I|ǫpIqωIp˜γpIqq, +(cf. [BGFr], (3.292)), o`u ǫpIq est donn´e par [BGFr], (3.278) et γpIq P Topp∆|I|, M Iq Ă C|I|pM Iq +donn´e par ∆|I| Q pt1, . . . , t|I|q ÞÑ pI Q i ÞÑ ˜γptκpiqqq P M I, avec κ l’unique bijection croissante +I Ñ rr1, |I|ss. +Pour tout i P rr1, nss et p ě 0, l’application δ˚ +rr1,nss,rr1,nss´tiu : CppXnq Ñ CppXrr1,nss´iq est +une composition CppXnq Ñ CppY pnq +xy q Ñ CppXrr1,nss´tiuq donc KerpCppXnq Ñ CppY pnq +xy qq Ă +Kerp‘iPrr1,nss : δ˚ +rr1,nss,rr1,nss´tiu : CppXnq Ñ ‘iPrr1,nssCppXrr1,nss´tiuqq. +On a donc pour chaque p ě 0 une application lin´eaire ˜µp : KerpCppXnq Ñ CppY pnq +xy qq Ñ +Totpp ˜Cq donn´ee par KerpCppXnq Ñ CppY pnq +xy qq Q c ÞÑ ppp, 0q ÞÑ c, pp, 0q ‰ pp,1 q1q ÞÑ 0q, qui +d´efinit un morphisme de complexes +(3.2.4) +˜µ‚ : KerpC‚pXnq Ñ C‚pY pnq +xy qq Ñ Tot‚p ˜Cq. +La cohomologie du complexe source de (3.2.4) est la cohomologie singuli`ere relative H‚pXn, Y pnq +xy ; Qq. +On d´eduit du morphisme de complexes (3.2.4) un morphisme en cohomologie +H‚p˜µ‚q : H‚pXn, Y pnq +xy ; Qq Ñ H‚pTotp ˜Cqq +dont on v´erifie qu’il satisfait +(3.2.5) +H‚p˜µ‚q “ p˜iCq´1 ˝ pisoBGFq´1 ˝ pθ˚q´1. +Montrons alors (3.2.1). Soit γ P π1px, yq, c P HnpM n, Y pnq +xy ; Qq. Soit ˜c P ZnpKerpC‚pXnq Ñ +C‚pY pnq +xy qqq un repr´esentant de c. Alors +xβpnq +yx ˝ nat˚ ˝ pisoyx +BGFq´1 ˝ pθ˚q´1pcq, rγsy “ xβpnq +yx ˝ iC ˝ nat ˝ ˜i´1 +C ˝ pisoyx +BGFq´1 ˝ pθ˚q´1pcq, rγsy +“ xβpnq +yx ˝ iC ˝ nat ˝ Hnp˜µ‚qpcq, rγsy “ xβpnq +yx ˝ iC ˝ nat ˝ r˜µnp˜cqs, rγsy “ xβpnq +yx ˝ iC ˝ rµnp˜cqs, rγsy +“ p´1qpn´1qpn´2q{2`n2`npn`1q{2˜cp˜γpnqq “ p´1qn`1xc, F +pnq +xy prγsqyhom +o`u µn est la compos´ee de ˜µn et de la projection ˜Cn Ñ Cn ; la premi`ere ´egalit´e suit de la +commutativit´e de (3.2.2), la deuxi`eme ´egalit´e suite de (3.2.5), la troisi`eme ´egalit´e suit de la +d´efinition de Hnp˜µ‚q, la quatri`eme ´egalit´e suit de ce que nat est la version cohomologiquee + +30 +BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE +de la projection ˜Cn Ñ Cn, la cinqui`eme ´egalit´e suit de l’´egalit´e (3.2.3), dans laquelle seule la +contribution de I “ rr1, nss est non-triviale, la derni`ere ´egalit´e suit de (2.1.1). +□ +La d´emonstration de la proposition 3.1 est illustr´ee par le diagramme suivant +H‚pTot ˜Cq +nat +� +„ +˜iC � +H‚pTotCq +„ +iC +� +βpnq +yx ˝iC +�❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +HnpM n,y ˜Kxxnyq +nat˚ � +isoyx +BGF � +HnpM n,y Kxxnyq +βpnq +yx� pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 +` +q˚ +HnpM n, Y pnq +yx ; Qq +θ˚ � HnpM n, Y pnq +xy ; Qq +can +� HnpM n, Y pnq +xy q˚ +Q +pF pnq +xy q˚ +� +4. Construction de transformations naturelles +On note Top2 la cat´egorie des espaces topologiques munis d’un couple de points marqu´es +(i.e. d’un couple de morphismes de source l’objet initial ˚). Pour pX, a, bq un objet de Top2, +le couple pπ1pa, bq, π1paqq est un torseur `a droite, `a savoir un couple pT, Gq avec T un ensemble +et G un groupe, munis d’une action `a droite libre et transtive de G sur T . La correspondance +pX, a, bq ÞÑ pπ1pa, bq, π1paqq d´efinit un foncteur Top2 Ñ TorDt avec TorDt la cat´egorie des +torseurs `a droite. +D´efinition 4.1. Pour pX, x, yq un objet de Top2, on note Zπ1pX, x, yq le Z-module libre sur +π1pX, x, yq. +Lemme 4.2. L’application pX, a, bq ÞÑ FnpX, a, bq d´efinit un foncteur covariant de Top2 vers +la cat´egorie Ab des groupes ab´eliens. +D´emonstration. Il s’agit de la composition du foncteur Top2 Ñ TorDt envoyant pX, a, bq vers +pπ1pa, bq, π1paqq et du foncteur TorDt Ñ Ab envoyant pT, Gq vers ZT {pZT qpZGqn`1 +` +. +□ +D´efinition 4.3. Pour X “ pX, a, bq un objet de Top2, on note YXpnq :“ Y pnq +ab +(cf. (0.0.2)). +Lemme 4.4. L’application pX, a, bq ÞÑ pXn, YXpnqq est un foncteur covariant Top2 Ñ Paires. +D´emonstration. On v´erifie que si X “ pX, a, bq et X1 “ pX1, a1, b1q sont des objets de Top2 et +si f : X Ñ X1 est un morphisme dans Top tel que fpaq “ a1 et fpbq “ b1, alors fpYXpnqq Ă +YX1pnq. +□ +Lemme-D´efinition 4.5. Pour X P Top2, on d´efinit HnpXq :“ HnpXn, YXpnqq. L’application +X ÞÑ HnpXq d´efinit un foncteur covariant Hn : Top2 Ñ Paires. +D´emonstration. Provient de l’identification de Hn avec la composition du foncteur pX, a, bq ÞÑ +pXn, YXpnqq avec le foncteur homologie relative. +□ + +TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE +31 +Un corollaire du th´eor`eme 2.3 est: +Th´eor`eme 4.6. Pour n ě 0 et X un objet de Top2, le morphisme de groupes Zπ1pa, bq Ñ +HnpXq, γ ÞÑ r˜γns induit un morphisme de groupes νpnq +X +: FnpXq Ñ HnpXq. La correspondance +X ÞÑ νpnq +X +est une transformation naturelle de Fn vers Hn. +Remerciements. Le travail de B.E. a b´en´efici´e du soutien du projet ANR “Project HighAGT +ANR20-CE40-0016”. +Bibliographie +[BGFr] J. +Burgos +Gil, +J. +Fresan, +Multiple +zeta +values: +from +numbers +to +motives, +preprint +http://javier.fresan.perso.math.cnrs.fr/mzv.pdf, `a paraˆıtre dans Clay Mathematics Proceedings. +[DG] +P. Deligne, A. Goncharov, Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. ´Ecole Norm. +Sup. (4) 38 (2005), no. 1, 1–56. +[GrH] +M. Greenberg, J. Harper, Algebraic topology. A first course, Mathematics Lecture Note Series, 58. +Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1981. +[Ha] +A. Hatcher, Algebraic topology. New York, Cambridge University Press, 2001. +Institut de Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e de Strasbourg et CNRS, 7 +rue Ren´e Descartes, 67000 Strasbourg, France +Email address: enriquez@math.unistra.fr +Email address: lecomte@math.unistra.fr + diff --git a/mdAzT4oBgHgl3EQfN_tE/content/tmp_files/load_file.txt b/mdAzT4oBgHgl3EQfN_tE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..35f6b97910c1652e9cb5ab267d65a58fec28a7cc --- /dev/null +++ b/mdAzT4oBgHgl3EQfN_tE/content/tmp_files/load_file.txt @@ -0,0 +1,1831 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf,len=1830 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='01157v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='AT] 3 Jan 2023 TRANSFORMATIONS NATURELLES RELIANT FONCTEURS D’HOMOTOPIE ET D’HOMOLOGIE SINGULI`ERE BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE R´esum´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La cat´egorie des espaces topologiques avec deux points marqu´es est munie de deux familles Fn et Hn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' index´ees par un entier n ě 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' de foncteurs vers la cat´egorie des groupes ab´eliens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' la premi`ere associant `a l’objet pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' yq le quotient de Zπ1pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' yq par un sous- groupe ab´elien associ´e `a la n ` 1-i`eme puissance de d’id´eal d’augmentation de l’alg`ebre de groupe Zπ1pX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' xq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' la seconde associant au mˆeme objet le n-i`eme groupe d’homologie sin- guli`ere relative de Xn par rapport `a un sous-espace d´efini en termes de diagonales partielles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Nous construisons une famille de transformations naturelles νn : Fn Ñ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Nous identifions la transformation naturelle obtenue par restriction de νn `a la sous-cat´egorie des vari´et´es alg´ebriques et tensorisation avec Q avec l’´equivalence naturelle due `a Beilinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Table des mati`eres Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappels 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Espaces topologiques et homologie singuli`ere 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Paires d’espaces topologiques et (co)homologie singuli`ere relative 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Une identit´e en homologie relative 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Mat´eriel de base et r´esultat principal 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (a) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Constructions combinatoires 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant une involution de Zn ˆ Sn ˆ rr0, nss 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Constructions et r´esultats relatifs aux permutations 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant les applications pf, sgnq, p ˜f, Ą sgnq et bij 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Battages et transpositions 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La bijection bij 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant Affp∆n´1, ∆nq ˆ t˘1u 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Construction d’endomorphismes de groupes de chaˆınes 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Morphismes dans une cat´egorie C 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de divk n ˝ Bn´1,n “ Bn´1,n ˝ divk n´1 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Relation dans C entre divk ‚ et id‚ 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Endomorphismes de groupes de chaˆınes singuli`eres 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (b) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Composition de chemins 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Calcul de pdivk nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Une ´egalit´e dans CnpXnq 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (b) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lien avec l’isomorphisme de Beilinson 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappels sur l’isomorphisme de Beilinson 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Relation du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 avec l’isomorphisme de Beilinson 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Construction de transformations naturelles 30 Bibliographie 31 Date: 3 janvier 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 1 2 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE Introduction Pour X un espace topologique connexe et a, b P X, on note π1pa, bq l’ensemble des classes de chemins reliant a `a b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le Z-module Zπ1pa, bq est alors un module `a droite sous l’action de l’alg`ebre du groupe π1paq :“ π1pa, aq, et on d´efinit, pour n ě 0, le Z-module FnpX, a, bq comme son quotient par le sous-module Zπ1pa, bq ¨ pZπ1paqqn`1 ` engendr´e par l’action de la n ` 1-`eme puissance de l’id´eal d’augmentation de l’alg`ebre de groupe de π1paq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si X est une vari´et´e diff´erentiable connexe, ayant le type d’homotopie d’un CW-complexe fini, on dispose d’une interpr´etation cohomologique de HomZpFnpX, a, bq, Qq, sous la forme d’un isomorphisme de Q-espaces vectoriels (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) HnpXn, Y pnq ba ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq „ Ñ HomZpFnpX, a, bq, Qq, o`u Y pnq ba est la partie de Xn d´efinie par (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) Y pnq ba :“ Yn i“0Y pnq ba,i avec Y pnq ba,i “ tpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq P Xn|xi “ xi`1u avec x0 “ b et xn`1 “ a, et H‚p´, ´;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq d´esigne la cohomologie singuli`ere relative `a coefficients dans Q (travail de Beilinson, r´edig´e dans [DG, BGFr]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La construction de cet isomorphisme repose sur des techniques faisceau- tiques : pr´ecis´ement, on construit un morphisme b ˜Ka Ñb Ka de complexes de faisceaux sur X et un isomorphisme isoba BGF : HnpXn,b ˜Kaxnyq Ñ HnpXn, Y pnq ba ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq, o`u Hp´, ´q d´esigne l’hypercohomologie des complexes de faisceaux ([BGFr], lemme 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='281) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) est alors con- struit comme une composition HnpXn, Y pnq ba ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq pisoba BGFq´1 Ñ HnpXn,b ˜Kaxnyq Ñ HnpXn,b Kaxnyq Ñ HomZpFnpX, a, bq, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La nature topologique de la source et du but de l’application (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) sugg`ere la possibilit´e d’une construction topologique de cette application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le but de ce travail est de fournir une telle construction, dans le cadre plus g´en´eral o`u X est un espace topologique, et en travaillant sur Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Plus pr´ecis´ement, nous contruisons un morphisme de Z-modules FnpX, a, bq Ñ HnpXn, Y pnq ab q, o`u Hnp´, ´q est l’homologie singuli`ere relative (`a coefficients dans Z), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Nous montrons, dans le cas o`u X est une vari´et´e diff´erentiable comme ci-dessus, la compatibilit´e de ce morphisme avec (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) et l’isomorphisme HnpXn, Y pnq ab q Ñ HnpXn, Y pnq ba q induit par l’automorphisme de Xn donn´e par px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq ÞÑ pxn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , x1q (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ce travail est organis´e comme suit : la section 1 contient des rappels sur l’homologie sin- guli`ere ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' la section principale est la section 2, qui a pour objectif la d´emonstration du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' la section 3 ´etablit le lien de ce r´esultat avec l’isomorphisme (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) de Beilinson (proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' en section 4, on ´etudie l’aspect fonctoriel de l’application construite dans le th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappels En section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1, on rappelle la construction de l’homologie singuli`ere, et en section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2, celles de l’homologie et la cohomologie relatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Espaces topologiques et homologie singuli`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note Top la cat´egorie des espaces topologiques ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' si X, Y sont deux espaces topologiques, on note ainsi ToppX, Y q l’ensemble des applications continues X Ñ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n ě 0, on note ∆n le simplexe donn´e par ∆n :“ tpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tnq P Rn|0 ď t1 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ď tn ď 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si X est un espace topologique, on pose CnpXq :“ ZTopp∆n, Xq pour n ě 0, ainsi que C´1pXq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Un sous-ensemble Y de X est naturellement muni de la topologie induite, on note YX (ou simplement Y s’il n’y a pas de risque de confusion) l’espace topologique correspondant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors Topp∆n, YXq “ tf P Topp∆n, Xq|fp∆nq Ă Y u et CnpYXq :“ ZTopp∆n, YXq Ă CnpXq pour n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour X un espace topologique et n ě 0, on note B˚ n,n´1 : CnpXq Ñ Cn´1pXq la diff´erentielle singuli`ere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’homologie du complexe pC‚pXq, B˚q est alors l’homologie singuli`ere H‚pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note AbGr celle des groupes ab´eliens Z-gradu´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’homologie singuli`ere d´efinit un foncteur H‚ : Top Ñ AbGr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Paires d’espaces topologiques et (co)homologie singuli`ere relative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si X est un espace topologique et Y est un sous-ensemble de X, alors pC‚pYXq, B˚q est un sous-complexe de pC‚pXq, B˚q, et l’homologie relative H‚pX, Y q est l’homologie du complexe quotient pC‚pXq{C‚pYXq, B˚q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' c’est un groupe ab´elien gradu´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La cohomologie relative H‚pX, Y q est celle du sous-complexe C‚pYXqK du complexe HomZpC‚pXq, Zq, muni de la diff´erentielle duale de B˚ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' on dispose d’un couplage H‚pX, Y q b H‚pX, Y q Ñ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors H‚pX, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq “ H‚pX, Y q b Q est la cohomologie du sous-complexe C‚pYXqK b Q de HomZpC‚pXq, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit Paires la cat´egorie des paires d’espaces topologiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les objets de Paires sont les cou- ples pX, Y q, avec X espace topologique et Y sous-ensemble de X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a PairesppA, Bq, pA1, B1qq :“ tf P EnspA, A1q|f est continue et fpBq Ă B1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors pX, Y q ÞÑ H‚pX, Y q d´efinit un foncteur H‚ : Paires Ñ AbGr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note f˚ le morphisme H‚pA, Bq Ñ H‚pA1, B1q dans AbGr associ´e `a un morphisme f : pA, Bq Ñ pA1, B1q dans Paires (not´e H‚pfq dans [Ha], pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 108, 124).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application identit´e id∆n d´efinit un ´el´ement de Cnp∆nq, dont l’image dans Cnp∆nq{CnpB∆nq est un cycle pour le bord relatif, et d´efinit donc un ´el´ement du groupe d’homologie relative rid∆ns P Hnp∆n, B∆nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Une identit´e en homologie relative Le but de cette section est la d´emonstration du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ce r´esultat est formul´e en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1, et sa premi`ere partie (th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3(a)) est d´emontr´ee en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le reste 4 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE de la section est consacr´e `a la d´emonstration de sa deuxi`eme partie (th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 contient des r´esultats combinatoires, et la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4 l’application de ses r´esultats `a la construction d’endomorphismes divk ‚ des complexes de chaˆınes singuli`eres satisfaisant la relation d’homotopie (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Cette relation est appliqu´ee en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5 `a la d´emonstration du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Mat´eriel de base et r´esultat principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1, on fixe un espace topologique X, des ´el´ements a, b P X, et n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On d´efinit la partie Y pnq ab Ă Xn par (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note Chempa, bq l’ensemble des applications continues ˜γ : r0, 1s Ñ X telles que ˜γp0q “ a et ˜γp1q “ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour ˜γ P Chempa, bq, on note ˜γpnq P Topp∆n, Xnq l’application compos´ee ∆n cann Ñ r0, 1sn ˜γn Ñ Xn o`u cann : ∆n Ñ r0, 1sn est l’injection canonique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' B˚ n,n´1p˜γpnqq P Cn´1pY pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a B˚ n,n´1p˜γpnqq “ řn i“0p´1qi˜γpnq ˝ Bn i , o`u pour i P rr0, nss l’application Bn i : ∆n´1 Ñ ∆n est donn´ee par pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ti, ti, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q (avec par convention t0 “ 0, tn “ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On v´erifie que ˜γpnq ˝ Bn i “ Bn,X i ˝ ˜γpn´1q, avec Bn,X i : Xn´1 Ñ Xn l’application donn´ee par px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xn´1q ÞÑ px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xi, xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xn´1q (avec par convention x0 “ a, xn “ b), donc B˚ n,n´1p˜γpnqq “ řn i“0p´1qiBn,X i ˝ ˜γpn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a pour tout i P rr0, nss les relations Bn,X i pXn´1q “ Y pnq ab,i Ă Y pnq ab qui impliquent la relation annonc´ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ On rappelle que C‚pY pnq ab q est un sous-complexe de C‚pXnq, et que l’homologie du complexe quotient C‚pXnq{C‚pY pnq ab q est l’homologie relative H‚pXn, Y pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il suit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2 que la classe de ˜γpnq dans CnpXnq{CnpY pnq ab q est un cycle du complexe quotient, et d´efinit donc une classe r˜γpnqs P HnpXn, Y pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note π1pa, bq le quotient de Chempa, bq par la relation d’´equivalence donn´ee par l’homotopie entre deux chemins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Il existe une unique application π1pa, bq Ñ HnpXn, Y pnq ab q, γ ÞÑ Fnpγq telle que l’application Chempa, bq Ñ HnpXn, Y pnq ab q, ˜γ ÞÑ r˜γpnqs admette une factorisation Chempa, bq Ñ π1pa, bq Ñ HnpXn, Y pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc pour ˜γ P Chempa, bq et on notant ˜γ Ñ r˜γs l’application canonique Chempa, bq Ñ π1pa, bq, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) r˜γpnqs “ Fnpr˜γsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Pour α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , αn P π1paq, et I Ă rr0, nss, on pose ś iPI αi le produit αip1q ¨ ¨ ¨ αip|I|q, o`u i est l’unique bijection croissante rr1, |I|ss Ñ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors ÿ IĂrr0,nss p´1q|I|Fnpγ ¨ ź iPI αiq “ 0 (´egalit´e dans HnpXn, Y pnq ab q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc factorisation de Fn en une application lin´eaire F pnq ab : Zπ1pa, bq{pZπ1pa, bqpZπ1paqqn`1 ` q Ñ HnpXn, Y pnq ab q, o`u pZπ1paqq` est l’id´eal d’augmentation de Zπ1paq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (a) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Pour ˜γ P Chempa, bq, ˜γpnq induit un morphisme ˜γpnq,Paires : p∆n, B∆nq Ñ pXn, Y pnq ab q dans Paires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Pour ˜γ P Chempa, bq, on a r˜γpnqs “ p˜γpnq,Pairesq˚prid∆nsq P HnpXn, Y pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) suit de la d´emonstration du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) suit de ce que l’´el´ement ˜γpnq P CnpXnq est l’image par le morphisme ∆n Ñ Xn dans Top induit par ˜γpnq de id∆n P Cnp∆nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Le th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3,(a) suit alors du lemme suivant : Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application ChempXq Ñ HnpXn, Y pnq ab q, ˜γ ÞÑ r˜γpnqs est invariante par homo- topie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit ˜γ, ˜γ1 P ChempXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Une homotopie entre ˜γ et ˜γ1 produit une homotopie entre les morphismes de paires p∆n, B∆nq Ñ pXn, Y pnq ab q donn´es par ˜γpnq,Paires et p˜γ1qpnq,Paires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Par l’invariance homotopique de l’homologie relative (Proposition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='14 dans [GrH]), les mor- phismes associ´es Hnp∆n, B∆nq Ñ HnpXn, Y pnq ab q induits en homologie, `a savoir Hnp˜γpnq,Pairesq et Hnpp˜γ1qpnq,Pairesq sont ´egaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les images qu’ils donnent `a rid∆ns sont donc ´egales, et le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4(b) implique alors r˜γpnqs “ r˜γ1pnqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Constructions combinatoires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le but de cette sous-section est la construction, pour tout couple pn, kq avec n, k ě 1 : (a) d’un ensemble AffpRn´1, Rnq, d’une application pf, sgnq : Zn ˆ Sn ˆ rr0, nss Ñ AffpRn´1, Rn´1q ˆ t˘1u et d’une involution invol de Zn ˆ Sn ˆ rr0, nss, telle que le diagramme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) commute ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) d’applications p ˜f, ˜ sgnq : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ AffpRn´1, Rn´1q ˆ t˘1u et bij : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ Zn ˆ Sn ˆ rr0, nss telles que le diagramme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17) commute (c) d’un sous-ensemble Ensk n Ă Zn ˆ Sn et d’une bijection bij : Ensk n´1 ˆ rr0, nss Ñ tx P Ensk n ˆ rr0, nss|involpxq R Ensk n ˆ rr0, nssu, telle que le diagramme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='19) commute (d) d’un sous-ensemble Affp∆n´1, ∆nq de AffpRn´1, Rnq et la construction d’applications f : Ensk n ˆrr0, nss Ñ Affp∆n´1, ∆nq et ˜f : Ensk n´1 ˆrr0, nss Ñ Affp∆n´1, ∆nq telle que le diagramme (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='22) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant une involution de Zn ˆ Sn ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note Aff la cat´egorie des espaces affines, dont les morphismes sont les applications affines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n ‰ 0, on note pen 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , en nq la base canonique de Rn et pour i P rr0, nss, on pose En i :“ en n ` en n´1 ` ¨ ¨ ¨ ` en n´i`1 P Rn (on a en particulier En 0 “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit n, m ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour P0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Pn P Rm, on note rP0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Pns P AffpRn, Rmq l’unique application affine Rn Ñ Rm telle que En i ÞÑ Pi pour i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 6 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour i P rr0, nss, on pose Bn i :“ rEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En n´i´1, En n´i`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En ns P AffpRn´1, Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit Sn le groupe des permutations de rr1, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour σ P Sn, on note σ˚ la permutation de Rn donn´ee par σ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tnq :“ ptσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tσpnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors σ˚pen i q “ en σ´1piq, et pστq˚ “ τ ˚ ˝ σ˚ pour σ, τ P Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour pv, σq P Zn ˆ Sn, on d´efinit ckpv, σq P AffpRn, Rnq comme l’application de Rn dans lui-mˆeme donn´ee par x ÞÑ p1{kqpv ` σ˚pxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors ckpv, σq “ rp1{kqpv ` σ˚En 0 q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , p1{kqpv ` σ˚En nqs pour pv, σq P Zn ˆ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note f : Zn ˆ Sn ˆ rr0, nss Ñ AffpRn´1, Rnq et sgn : Zn ˆ Sn ˆ rr0, nss Ñ t˘1u les applications donn´ees par fpv, σ, iq :“ ckpv, σq ˝ Bn i , sgnpv, σ, iq :“ p´1qiǫpσq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note invol l’application de Zn ˆ Sn ˆ rr0, nss dans lui-mˆeme donn´ee par involpv, σ, iq :“ pv, si,i`1 ˝ σ, iq pour pv, σ, iq P Zn ˆ Sn ˆ rr1, n ´ 1ss, o`u si,i`1 P Sn est la permutation de i et i ` 1, involpv, σ, nq :“ pv ` σ˚pen nq, c ˝ σ, 0q pour pv, σq P Zn ˆ Sn, o`u c P Sn est le n-cycle donn´e par cpiq :“ i ` 1 pour i ‰ n, cpnq “ 1, et involpv, σ, 0q :“ pv ´ σ˚pen 1q, c´1 ˝ σ, nq pour pv, σq P Zn ˆ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) invol est une involution de Zn ˆ Sn ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Le diagramme suivant commute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) Zn ˆ Sn ˆ rr0, nss invol � pf,sgnq �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ Zn ˆ Sn ˆ rr0, nss pf,´sgnq �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ AffpRn´1, Rn´1q ˆ t˘1u D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Soit pv, σ, iq P Zn ˆ Sn ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i ‰ 0, n, alors s2 i,i`1 “ id implique invol ˝ involpv, σ, iq “ pv, σ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ n, alors invol ˝ involpv, σ, nq “ involpv ` σ˚pen nq, c ˝ σ, 0q “ pv ` σ˚pen nq ´ pc ˝ σq˚pen 1q, c´1 ˝ c ˝ σ, nq “ pv, σ, nq car pc ˝ σq˚pen 1q “ σ˚ ˝ c˚pen 1q “ σ˚pen nq et si i “ 0, on a invol ˝ involpv, σ, 0q “ involpv ´ σ˚pen 1q, c´1 ˝ σ, nq “ pv ´ σ˚pen 1q ` pc´1 ˝ σq˚pen nq, c ˝ c´1 ˝ σ, 0q “ pv, σ, 0q car pc´1 ˝ σq˚pen nq “ σ˚ ˝ pc´1q˚pen nq “ σ˚pen 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc dans tous les cas invol ˝ involpv, σ, iq “ pv, σ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Soit pv, σ, iq P Zn ˆ Sn ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i ‰ 0, n, alors sgn ˝ involpv, σ, iq “ ´sgnpv, σ, iq du fait de ǫpsi,i`1q “ ´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ n, alors sgn ˝ involpv, σ, nq “ sgnpv ` σ˚pen 1q, c´1 ˝ σ, 0q “ p´1qnǫpc´1 ˝ σq “ ´p´1q0ǫpσq “ TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 7 ´sgnpv, σ, nq du fait de ǫpcq “ p´1qn´1 et sgn ˝ involpv, σ, 0q “ sgnpv ´ σ˚pen nq, c ˝ σ, nq “ p´1q0ǫpc ˝ σq “ ´p´1qnǫpσq “ ´sgnpv, σ, 0q pour la mˆeme raison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc dans tous les cas sgn ˝ involpv, σ, iq “ ´sgnpv, σ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i ‰ 0, n, alors f ˝ involpv, σ, iq “ fpv, si,i`1 ˝ σ, iq “ rpv ` psi,i`1 ˝ σq˚pEn 0 q{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` psi,i`1 ˝ σq˚pEn nq{ks ˝ rEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En n´i´1, En n´i`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En ns “ rpv ` psi,i`1 ˝ σq˚pEn 0 q{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` psi,i`1 ˝ σq˚pEn n´i´1q{k, pv ` psi,i`1 ˝ σq˚pEn n´i`1q{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` psi,i`1 ˝ σq˚pEn nq{ks “ rpv ` σ˚pEn 0 q{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn n´i´1q{k, pv ` σ˚pEn n´i`1q{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn nq{ks “ fpv, σ, iq en utilisant pσ ¨ τq˚ “ τ ˚ ˝ σ˚ et s˚ i,i`1pEn j q “ En j pour j P rr0, nss et j ‰ n ´ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ n, alors f ˝ involpv, σ, nq “ fpv ` σ˚pen nq, c ˝ σ, 0q “ rpv ` σ˚pen nq ` pc ˝ σq˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pen nq ` pc ˝ σq˚pEn nqq{ks ˝ rEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En n´1s “ rpv ` σ˚pen nq ` pc ˝ σq˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pen nq ` pc ˝ σq˚pEn n´1qq{ks “ rpv ` σ˚pen n ` c˚pEn 0 qqq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pen n ` c˚pEn n´1qqq{ks “ rpv ` σ˚pEn 1 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn nqq{ks “ rpv ` σ˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn nqq{ks ˝ rEn 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En ns “ fpv, σ, nq du fait de c˚pEn i q ` en n “ En i`1 pour i P rr0, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ 0, alors f ˝ involpv, σ, 0q “ fpv ´ σ˚pen 1q, c´1 ˝ σ, nq “ rpv ´ σ˚pen 1q ` pc´1 ˝ σq˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ´ σ˚pen 1q ` pc´1 ˝ σq˚pEn nqq{ks ˝ rEn 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En ns “ rpv ´ σ˚pen 1q ` pc´1 ˝ σq˚pEn 1 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ´ σ˚pen 1q ` pc´1 ˝ σq˚pEn nqq{ks “ rpv ` σ˚p´en 1 ` pc´1q˚pEn 1 qqq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚p´en 1 ` pc´1q˚pEn nqqq{ks “ rpv ` σ˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn n´1qq{ks “ rpv ` σ˚pEn 0 qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pv ` σ˚pEn nqq{ks ˝ rEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En n´1s “ fpv, σ, 0q du fait de pc´1q˚pEn i q´en 1 “ En i´1 pour i P rr1, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc dans tous les cas f ˝involpv, σ, iq “ fpv, σ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Remarque 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On v´erifie directement que invol est sans point fixe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' cela r´esulte aussi de sgn ˝ invol “ ´sgn et du fait que t˘1u n’a pas de point fixe sous le changement de signe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Constructions et r´esultats relatifs aux permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit τ P Sn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour i P rr1, n´1ss, soit invpτ, iq :“ tj P rr1, n´1ss|pj ´iqpτpjq´ τpiqq ă 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors |invpτ, iq| ” τpiq ´ i (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 8 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons l’´enonc´e par r´ecurrence sur i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ 1, invpτ, 1q “ tj P rr1, n ´ 1ss|τpjq ă τp1qu “ τ ´1prr1, τp1q ´ 1ssq, donc |invpτ, 1q| “ |τ ´1prr1, τp1q ´ 1ssq| “ |rr1, τp1q ´ 1ss| “ τp1q ´ 1 o`u la deuxi`eme ´egalit´e suit de la bijectivit´e de τ, ce qui implique l’´enonc´e pour i “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit 1 ď i ă n ´ 1 : supposons l’´enonc´e vrai pour i et montrons-le pour i ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour cela, on note A0 :“ tj|j ă i et τpjq ą τpiqu, A1 :“ tj|j ă i et τpjq ą τpi ` 1qu, B0 :“ tj|j ą i ` 1 et τpjq ă τpiqu, B1 :“ tj|j ą i ` 1 et τpjq ă τpi ` 1qu ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' on observe que Aα X Bβ “ H pour tous α, β P t0, 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Deux cas se pr´esentent : ‚ on a τpiq ă τpi ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors i ` 1 R invpτ, iq et i R invpτ, i ` 1q, ce qui implique invpτ, iq “ A0 Y B0 et invpτ, i ` 1q “ A1 Y B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' En notant △ l’op´eration de diff´erence sym´etrique, on obtient (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) invpτ, iq△invpτ, i ` 1q “ pA0△A1q Y pB0△B1q, compte tenu de pX0YY0q△pX1YY1q “ pX0△X1qYpY0△Y1q pour tous ensembles X0, X1, Y0, Y1 tels que Xα X Yβ “ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Du fait que τpiq ă τpi ` 1q, on a A1 Ă A0 et B1 Ą B0, ce qui implique (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3) pA0△A1q Y pB0△B1q “ pA0 ´ A1q Y pB1 ´ B0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors A0´A1 “ τ ´1prrτpiq`1, τpi`1qssqXtj|j ă iu, B1´B0 “ τ ´1prrτpiq, τpi`1q´1ssqXtj|j ą i`1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme τpi ` 1q R τptj|j ă iuq et τpiq R τptj|j ą i ` 1uq on en d´eduit A0´A1 “ τ ´1prrτpiq`1, τpi`1q´1ssqXtj|j ă iu, B1´B0 “ τ´1prrτpiq`1, τpi`1q´1ssqXtj|j ą i`1u, donc pA0 ´ A1q Y pB1 ´ B0q “ τ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq X tj|j ‰ i, i ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Compte tenu de ti, i ` 1u X τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq “ H, on en d´eduit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) pA0 ´ A1q Y pB1 ´ B0q “ τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors |invpτ, i ` 1q| ´ |invpτ, iq| ” |invpτ, iq△invpτ, i ` 1q| “ |pA0 ´ A1q Y pB1 ´ B0q| “ |τ ´1prrτpiq ` 1, τpi ` 1q ´ 1ssq| “ |rrτpiq ` 1, τpi ` 1q ´ 1ss| “ τpi ` 1q ´ τpiq ´ 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) mod 2, o`u la premi`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6) |A△B| ” |B| ´ |A| mod 2 pour A, B ensembles finis, la deuxi`eme ´egalit´e suit de la combinaison de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3), la troisi`eme ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4), la quatri`eme ´egalit´e suit de la bijectivit´e de τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ‚ on a τpi ` 1q ă τpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors A1 Ą A0 et B1 Ă B0, ce qui implique (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='7) pA0△A1q Y pB0△B1q “ pA1 ´ A0q Y pB0 ´ B1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors A1´A0 “ τ ´1prrτpi`1q`1, τpiqssqXtj|j ă iu, B0´B1 “ τ ´1prrτpi`1q, τpiq´1ssqXtj|j ą i`1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 9 Or τpiq R τptj|j ă iuq, τpi ` 1q R τptj|j ą i ` 1uq, donc A1´A0 “ τ ´1prrτpi`1q`1, τpiq´1ssqXtj|j ă iu, B0´B1 “ τ ´1prrτpi`1q`1, τpiq´1ssqXtj|j ą i`1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc pA1 ´ A0q Y pB0 ´ B1q “ τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq X tj|j ‰ i, i ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Compte tenu de ti, i ` 1u X τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq “ H, on en d´eduit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='8) pA1 ´ A0q Y pB0 ´ B1q “ τ ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, comme τpi`1q ă τpiq, on a invpτ, iq “ A0YB0Yti`1u et invpτ, i`1q “ A1YB1Ytiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a i R invpτ, iq et i ` 1 R invpτ, i ` 1q, donc (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9) invpτ, iq△invpτ, i ` 1q “ ppA0 Y B0q△pA1 Y B1qq Y ti, i ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors |invpτ, iq| ´ |invpτ, i ` 1q| ” |invpτ, iq△invpτ, i ` 1q| “ |pA0 Y B0q△pA1 Y B1q| ´ 2 ” |pA0 Y B0q△pA1 Y B1q| “ |τ´1prrτpi ` 1q ` 1, τpiq ´ 1ssq| “ |rrτpi ` 1q ` 1, τpiq ´ 1ss| “ τpiq ´ τpi ` 1q ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='10) o`u la premi`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6) et x ” ´x mod 2, la deuxi`eme ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9) combin´e `a ti, i ` 1u X ppA0 Y B0q△pA1 Y B1qq Ă ti, i ` 1u X tj|j ‰ i, i ` 1u “ H, la quatri`eme ´egalit´e suit de la combinaison de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='7), la cinqui`eme ´egalit´e suit de la bijectivit´e de τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On d´eduit des ´egalit´es (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) dans le premier cas et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='10) dans le second l’´egalit´e |invpi` 1q| ´ τpi ` 1q ` i ` 1 ” |invpiq| ´ τpiq ` i mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a |invpτ, iq| ´ τpiq ` i ” 0 mod 2 d’apr`es l’hypoth`ese de r´ecurrence, ce qui implique |invpτ, i ` 1q| ´ τpi ` 1q ` i ` 1 ” 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit i P rr1, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) On note pi : rr1, nss Ñ rr1, n´1ss l’application donn´ee par pipxq “ x si x ď i et pipxq “ x´1 si x ě i ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) On note sti : rr1, n ´ 1ss Ñ rr1, nss l’application donn´ee par x ÞÑ x si x ď i et x ÞÑ x ` 1 si x ě i ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme-D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit τ P Sn´1 et i P rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il existe un unique ´el´ement τ piq P Sn satisfaisant les conditions suivantes : (a) pτpiq ˝ τ piq “ τ ˝ pi et τ piqpiq “ τpiq, τ piqpi ` 1q “ τpiq ` 1 si i ‰ 0, n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) τ p0qp1q “ 1 et τ p0qpxq “ τpx ´ 1q ` 1 pour tout x P rr2, nss si i “ 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (c) τ pnqpnq “ n et τ pnqpxq “ τpxq pour tout x P rr1, n ´ 1ss si i “ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Pour j P rr0, nss, stj ˝ pj est l’application de rr1, nss dans lui-mˆeme telle que x ÞÑ x pour x ‰ j ` 1 et j ` 1 ÞÑ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si τpiq est une application de rr1, nss dans lui-mˆeme satisfaisant les conditions dites, on a alors stτpiq ˝ ˝pτpiq ˝ τ piq “ stτpiq ˝ τ ˝ pi ce qui implique τ piqpxq “ stτpiq ˝τ ˝pipxq pour tout x ‰ i, i`1 ainsi que τpiqpiq “ τpiq, τ piqpi`1q “ τpiq`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les conditions dites d´eterminent donc uniquement τpiq comme application de rr1, nss dans lui-mˆeme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 10 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE On v´erifie alors que τ piq P Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b,c) Pour i “ 0, n, l’application τ piq est la juxtaposition de deux permutations, donc est une permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si τ P Sn´1 et i P rr1, n ´ 1ss, alors ǫpτ piqq “ ǫpτqp´1qτpiq´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si p ě 1 et σ P Sp, notons invpσq :“ tpa, bq P rr1, pss2|a ă b et σpaq ą σpbqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11) ǫpσq “ p´1q|invpσq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si τ P Sn´1 et i P rr1, n ´ 1ss, on a une partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='12) invpτq “ A \\ B, avec A :“ tpa, bq P invpτq|a ‰ i et b ‰ iu et B :“ tpa, bq P invpτq|a “ i ou b “ iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a de mˆeme une partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='13) invpτ piqq “ A1 \\ B1 \\ B2, avec A1 :“ tpa, bq P invpτ piqq|a R ti, i ` 1u et b R ti, i ` 1uu, B1 :“ tpa, bq P invpτ piqq|pa “ i et b R ti, i ` 1uq ou pb “ i et a R ti, i ` 1uqu B2 :“ tpa, bq P invpτ piqq|pa “ i ` 1 et b R ti, i ` 1uq ou pb “ i ` 1 et a R ti, i ` 1uqu ceci du fait que pi, i ` 1q R invpτ piqq car τ piqpi ` 1q “ 1 ` τ piqpiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a encore des bijections (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='14) A „ Ñ A1, B „ Ñ B1, B „ Ñ B2 induites respectivement par pa, bq ÞÑ pstipaq, stipbqq (application A Ñ A1), pa, iq ÞÑ pa, iq et pi, bq ÞÑ pi, b ` 1q (application B Ñ B1), pa, iq ÞÑ pa, i ` 1q et pi, bq ÞÑ pi ` 1, b ` 1q (application B Ñ B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Enfin on a une bijection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='15) invpτ, iq „ Ñ B donn´ee par a ÞÑ pa, iq si a ă i et a ÞÑ pi, aq si i ă a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='16) |invpτ piqq| “ |A1| \\ |B1| \\ |B2| “ p|A| \\ |B|q \\ |B| “ |invpτq| ` |invpτ, iq| ” |invpτq| ` τpiq ´ i mod 2, o`u la premi`ere (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' deuxi`eme, troisi`eme, quatri`eme) ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='13) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='15), lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors ǫpτ piqq “ p´1q|invpτ piqq| “ p´1q|invpτq|`τpiq´i “ p´1q|invpτq|p´1qτpiq´i “ ǫpτqp´1qτpiq´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' o`u la premi`ere (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' deuxi`eme, derni`ere) ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='16), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant les applications pf, sgnq, p ˜f, Ą sgnq et bij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note ˜f : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ AffpRn´1, Rnq et Ą sgn : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ t˘1u les applications donn´ees par ˜fp˜v, ˜σ, iq :“ Bn i ˝ ckp˜v, ˜σq, Ą sgnp˜v, ˜σ, iq :“ p´1qiǫp˜σq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note w ÞÑ wp0q et w ÞÑ wpnq les applications Zn´1 Ñ Zn donn´ees par wp0q :“ p0, wq et wpnq :“ pw, k ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note bij : Zn´1 ˆ Sn´1 ˆ rr0, nss Ñ Zn ˆ Sn ˆ rr0, nss l’application donn´ee par pw, τ, iq ÞÑ pw ˝ pi, τ piq, τpiqq si i ‰ 0, n, par pw, τ, 0q ÞÑ pwp0q, τ p0q, 0q, et par pw, τ, nq ÞÑ pwpnq, τ pnq, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le diagramme suivant commute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17) Zn´1 ˆ Sn´1 ˆ rr0, nss bij � p ˜ f,Ą sgnq �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Zn ˆ Sn ˆ rr0, nss pf,sgnq �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ AffpRn´1, Rn´1q ˆ t˘1u D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit pw, τ, iq P Zn´1 ˆ Sn´1 ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Supposons i ‰ 0, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors sgn ˝ bijpw, τ, iq “ sgnpw ˝ pi, τ piq, τpiqq “ ǫpτ piqqp´1qτpiq “ ǫpτqp´1qi “ Ą sgnpw, τ, iq d’apr`es le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus f ˝ bijpw, τ, iq “ fpw ˝ pi, τ piq, τpiqq “ ckpw ˝ pi, τ piqq ˝ Bn τpiq “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ˝ pi ` pτ piqq˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tτpiq, tτpiq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ˝ pi ` pτ piqq˚ptpτpiqp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tpτpiqpnqqq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ˝ pi ` ptpτpiq˝τ piqp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tpτpiq˝τ piqpnqqq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ˝ pi ` ptτ˝pip1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tτ˝pipnqqq{ks “ Bn i ˝ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ` ptτp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tτpn´1qqq{ks “ Bn i ˝ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ` τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{ks “ Bn i ˝ ckpw, τq “ ˜fpw, τ, iq o`u la sixi`eme ´egalit´e provient de τ ˝ pi “ pτpiq ˝ τ piq (´egalit´e d’applications rr1, nss Ñ rr1, n ´ 1ss) et la septi`eme ´egalit´e suit de Bn i pxq “ x ˝ pi pour x P Rn´1 “ Applprr1, n ´ 1ss, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i “ 0, alors f ˝ bijpw, τ, 0q “ fpwp0q, τ p0q, 0q “ ckpwp0q, τ p0qq ˝ Bn 0 “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pwp0q ` pτ p0qq˚p0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pp0, wq ` p0, τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qqq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ p0, w ` τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{ks “ Bn 0 ˝ ckpw, τq “ ˜fpw, τ, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 12 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE Si i “ n, alors f ˝ bijpw, τ, nq “ fpwpnq, τ pnq, nq “ ckpwpnq, τ pnqq ˝ Bn n “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pwpnq ` pτ pnqq˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1, 1qq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ ppw, k ´ 1q ` pτ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q, 1qq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ` τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q, kq{ks “ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ ppw ` τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{k, 1qs “ Bn n ˝ rpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1q ÞÑ pw ` τ ˚pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , tn´1qq{ks “ Bn n ˝ ckpw, τq “ ˜fpw, τ, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Battages et transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si n ě 1 et n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nk sont des entiers positifs ou nuls avec n1 `¨ ¨ ¨`nk “ n, on note Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk l’ensemble des ´el´ements σ P Sn tel que pour tout i “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , k, la restriction de σ `a n1 ` ¨ ¨ ¨ ` ni´1 ` rr1, niss est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si ni “ 0, l’ensemble rr1, niss est vide, la condition relative `a i est alors automatiquement satisfaite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nk des entiers ě 1 et σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk et i P rr1, n ´ 1ss avec n :“ n1`¨ ¨ ¨`nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors si,i`1˝σ R Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk si et seulement si σ´1piq R tn1, n1`n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1`¨ ¨ ¨`nku et σ´1pi ` 1q “ σ´1piq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk et i P rr1, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit j :“ σ´1piq, j1 :“ σ´1pi ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour α P rr1, kss, notons Iα :“ n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nαss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors on a une partition rr1, nss “ \\k α“1Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit α, α1 P rr1, kss les indices tels que j P Iα, j1 P Iα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrer l’´equivalence annonc´ee, on montre d’abord l’´equivalence pα “ α1q ðñ psi,i`1 ˝ σ R Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nkq, puis l’´equivalence pα “ α1q ðñ pσ´1piq R tn1, n1 ` n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nku et σ´1pi ` 1q “ σ´1piq ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Premi`ere ´etape : ´equivalence pα “ α1q ðñ psi,i`1 ˝ σ R Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Supposons α “ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors on a j, j1 P Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction σ|Iα est strictement croissante, et σpjq “ i, σpj1q “ i ` 1, donc j ă j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, si on avait j1 ą j ` 1, alors j ` 1 P Iα et σpjq ă σpj ` 1q ă σpj1q donc i ă σpj ` 1q ă i ` 1 ce qui est impossible, σpj ` 1q ´etant entier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc j1 “ j ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme j ` 1 P Iα, on a n´ecessairement j ‰ n1 ` ¨ ¨ ¨ ` nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, la restriction de si,i`1 ˝ σ `a Iα est telle que j ÞÑ i ` 1 et j ` 1 ÞÑ i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' cette restriction n’est donc pas strictement croissante, donc si,i`1 ˝ σ R Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Supposons α ‰ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme σpIαq S i ` 1, la restriction de si,i`1 ˝ σ `a Iα est ´egale `a a ˝ σ|Iα, o`u a : σpIαq Ñ rr1, nss est donn´ee par x ÞÑ x si x ‰ i et i ÞÑ i`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application a est croissante, donc il est de mˆeme de psi,i`1 ˝ σq|Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De mˆeme, σpIα1q S i, donc la restriction de si,i`1 ˝ σ `a Iα1 est ´egale `a a1 ˝ σ|Iα, o`u a1 : σpIα1q Ñ rr1, nss est donn´ee par x ÞÑ x si x ‰ i ` 1 et i ` 1 ÞÑ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 13 L’application a1 est croissante, donc il est de mˆeme de psi,i`1 ˝σq|Iα1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Enfin pour tout β ‰ α, α1, on a psi,i`1 ˝ σq|Iβ “ σ|Iβ donc psi,i`1 ˝ σq|Iβ est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc si,i`1 ˝ σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc ´equivalence entre si,i`1 ˝ σ R Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk et α “ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Seconde ´etape : pα “ α1q ðñ pσ´1piq R tn1, n1 ` n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nku et σ´1pi ` 1q “ σ´1piq ` 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a vu que si α “ α1, alors j1 “ j ` 1 et j ‰ n1 ` ¨ ¨ ¨ ` nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme j P Iα, on a aussi j ‰ n1 ` ¨ ¨ ¨ ` nβ pour tout β ‰ α, donc j R tn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Inversement, si j1 “ j ` 1 et j R tn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nku, on a j ‰ n1 ` ¨ ¨ ¨ ` nα donc j P n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nα ´ 1ss donc j1 “ j ` 1 P n1 ` ¨ ¨ ¨ ` nα´1 ` rr1, nαss “ Iα, donc α “ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La bijection bij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) On note rr0, k ´ 1ssn ď :“ tpv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , vnq P Zn|0 ď v1 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ď vn ď k ´ 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) On note Ensk n l’ensemble des couples pv, σq P rr0, k´1ssn ďˆSn, tels que σ P S|v´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|v´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si pv, σq P Ensk n, on a pour tout i P rr1, n ´ 1ss l’implication (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='18) pvi “ vi`1q ùñ pσpiq ă σpi ` 1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit pv, σ, iq P Ensk n ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Si i ‰ 0, n, la condition involpv, σ, iq R Ensk n ˆ rr0, nss est ´equivalente `a la conjonction de σ´1piq R t|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u et σ´1pi ` 1q “ σ´1piq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Si i “ n, la condition involpv, σ, iq R Ensk n ˆ rr0, nss est ´equivalente `a la conjonction σpnq “ n et vpnq “ k ´ 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (c) Si i “ 0, la condition involpv, σ, iq R Ensk n ˆ rr0, nss est ´equivalente `a la conjonction σp1q “ 1 et vp1q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) On a σ P S|v´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|v´1p0q|`¨¨¨`|v´1pk´1q| et involpv, σ, iq “ pv, si,i`1 ˝ σ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a v P rr0, k ´ 1ssn ď donc on a ´equivalence entre involpv, σ, iq R Ensk n ˆ rr0, nss et si,i`1 ˝ σ R S|v´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|v´1p0q|`¨¨¨`|v´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le r´esultat est alors cons´equence du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) Rappelons que involpv, σ, nq “ pv ` en σ´1pnq, c ˝ σ, 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons nα :“ |v´1pα´1q| pour α “ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors on a une partition rr1, nss “ \\k α“1Iα, avec Iα “ n1`¨ ¨ ¨`nα´1`rr1, nαss (avec la convention que Iα “ H si nα “ 0), et v prend la valeur α´1 sur Iα, pour tout α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons l’identification de rr0, k´1ssn ď `a l’ensemble Applďprr1, nss, rr0, k´ 1ssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a σ´1pnq P tn1 ` ¨ ¨ ¨ ` nβ|nβ ‰ 0u donc v ` en σ´1pnq P Applďprr1, nss, rr0, kssq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, v`en σ´1pnq atteint la valeur k si et seulement si nk ‰ 0 et σ´1pnq “ n1`¨ ¨ ¨`nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La conjonction de ces conditions est ´equivalente `a celle de vpnq “ k ´ 1 et σpnq “ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ceci montre l’´equivalence entre v ` en σ´1pnq R rr0, k ´ 1ssn ď et la conjonction des conditions vpnq “ k ´ 1 et σpnq “ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors si vpnq “ k ´ 1 et σpnq “ n, on a v ` en σ´1pnq R rr0, k ´ 1ssn ď donc involpv, σ, 0q R Ensk n ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 14 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE Si vpnq ă k ´ 1 ou σpnq ‰ n, montrons que involpv, σ, 0q P Ensk n ˆ rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D’une part, on a w :“ v ` en σ´1pnq P rr0, k ´ 1ssn ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons qu’on a d’autre part c ˝ σ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|, ceci dans chacun des cas σpnq ‰ n et (σpnq “ n et vpnq ă k ´ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Premier cas : σpnq ‰ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit α l’indice tel que σ´1pnq P Iα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a n´ecessairement α ă k et nα ą 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' de plus σ´1pnq “ maxpIαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons que p|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1pk ´ 1q|q “ pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, alors p|w´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |w´1pk ´ 1q|q “ pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nα ´ 1, nα`1 ` 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, et la partition de rr1, nss associ´ee `a w est pJ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Jkq avec Jβ “ Iβ pour β P rr1, kss ´ tα, α ` 1u, Jα “ Iα ´ tmaxpIαqu, Jα`1 “ tmaxpIαqu Y Iα`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons que c ˝ σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nα´1,nα`1`1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si β P rr1, kss ´ tαu, alors σpIβq S n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de c `a σpIβq est donc x ÞÑ x ` 1, qui est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc la restriction de c˝σ `a Iβ est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' En particulier, si β P rr1, kss´tα, α`1u, la restriction de c ˝ σ `a Jβ “ Iβ est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons que la restriction de c˝σ `a Iα`1 est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a Jα “ tmaxpIαquY Iα`1 avec maxpIαq ď Iα`1 et c ˝ σpmaxpIαqq “ 0, donc la restriction de c ˝ σ `a Jα`1 est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme Jα “ Iα ´ tmaxpIαqu et que maxpIαq “ σ´1pnq, on a σpJαq S n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de c `a σpJβq est donc x ÞÑ x ` 1, qui est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc la restriction de c ˝ σ `a Jα est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc on a c ˝ σ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Deuxi`eme cas : σpnq “ n et vpnq ă k ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit p :“ 1 ` maxti P rr0, k ´ 1ss||v´1piq| ‰ 0u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' alors p ă k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors p|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1pk ´ 1q|q “ pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , np, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme σpnq “ n, on a σ´1pnq “ n “ n1 ` ¨ ¨ ¨ ` np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc p|w´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |w´1pk ´ 1q|q “ pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , np ´ 1, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', 0q, la partition correspondant `a w ´etant donn´ee par pJ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Jkq avec Jα “ Iα pour α ‰ p, p ` 1, Jp :“ Ip ´ tnu, et Jp`1 “ tnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si α ‰ p, p ` 1, σpIαq S n, donc la restriction de c `a σpIαq est x ÞÑ x ` 1 qui est strictement croissante, donc la restriction de c ˝ σ `a Jα “ Iα est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a Jp “ Ip ´ tnu et σpnq “ n, donc σpJpq S n, donc la restriction de c `a σpJpq est x ÞÑ x ` 1 qui est strictement croissante, donc la restriction de c ˝ σ `a Jp est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Enfin, Jp`1 est un singleton, donc la restriction de c˝σ `a cet ensemble est strictement croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc c ˝ σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',np´1,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',0 “ S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (c) D´emonstration semblable `a celle de (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' bij induit une bijection bij entre Ensk n´1ˆrr0, nss et tx P Ensk nˆrr0, nss|involpxq R Ensk n ˆ rr0, nssu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors le diagramme commutatif (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='19) Ensk n´1 ˆ rr0, nss � � � bij „ � tx P Ensk n ˆ rr0, nss|involpxq R Ensk n ˆ rr0, nssuu � � � Zn´1 ˆ Sn´1 ˆ rr0, nss bij � Zn ˆ Sn ˆ rr0, nss TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 15 D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D’apr`es le lemme 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='29, il suffit de montrer que bij induit une bijection entre Ensk n´1 ˆ rr0, nss et E :“ En \\ E0 \\ Err1,n´1ss o`u En :“ tpv, σ, nq|pv, σq P Ensk n et σpnq “ n et vpnq “ k ´ 1u, E0 :“ tpv, σ, 0q|pv, σq P Ensk n et σp1q “ 1 et vp1q “ 0u, Err1,n´1ss :“tpv, σ, iq P Ensk n ˆ rr1, n ´ 1ss|σ´1piq R t|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u et σ´1pi ` 1q “ σ´1piq ` 1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour cela, on montre s´epar´ement que bij induit une bijection entre (a) Ensk n´1 ˆ tnu et En, (b) Ensk n´1 ˆ t0u et E0, et (c) Ensk n´1 ˆ rr1, n ´ 1ss et Err1,n´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Montrons que bij envoie Ensk n´1 ˆ tnu dans En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit pw, τq P Ensk n´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a w P rr0, k ´ 1ssn´1 ď , ce qui implique wpnq “ pw, k ´ 1q P rr0, k ´ 1ssn ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Posons v :“ wpnq, alors pour tout i P rr0, k ´ 1ss, on a v´1piq “ w´1piq si i ‰ k ´ 1 et v´1pk ´ 1q “ w´1pk ´ 1q \\ tnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si i ‰ k ´ 1, la restriction de τ pnq `a v´1piq co¨ıncide avec la restriction de τ `a w´1piq, qui est croissante car τ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de τ pnq `a v´1pk ´ 1q est l’union disjointe de la restriction de τ `a w´1pk ´ 1q, qui est croissante et `a valeurs dans rr1, n ´ 1ss et de l’application n ÞÑ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme on a w´1pk ´ 1q ă n, cette union disjointe est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc τ pnq P S|v´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|v´1pk´1q|, ce qui implique pwpnq, τ pnqq P Ensk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, on a wpnqpnq “ k ´ 1 et τ pnqpnq “ n, donc bijpw, τ, nq “ pwpnq, τ pnq, nq P En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc bijpEnsk n´1 ˆ tnuq Ă En, notons bijn : Ensk n´1 ˆ tnu Ñ En l’application induite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit invbijn : En Ñ Zn´1ˆSn´1ˆtnu l’application donn´ee par pv, σ, nq ÞÑ pv|rr1,n´1ss, σ|rr1,n´1ss, nq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' comme σ P Sn et σpnq “ n, on a bien σ|rr1,n´1ss P Sn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons que invbijn envoie En dans Ensk n´1 ˆ tnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit pv, σ, nq P En, et posons w :“ v|rr1,n´1ss, τ :“ σ|rr1,n´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors v P rr0, k ´ 1ssn ď ce qui implique w P rr0, k ´ 1ssn ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, pour chaque i P r0, k ´ 1s, w´1piq est contenu dans w´1piq (on a mˆeme ´egalit´e si i ‰ k ´ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de τ `a w´1piq co¨ıncide avec la restriction de σ au mˆeme ensemble, qui est croissante par la croissance de σ en restriction `a v´1piq, qui contient w´1piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc pw, τq P Ensk n´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc invbijn envoie En dans Ensk n´1 ˆ tnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons invbijn : En Ñ Ensk n´1 ˆ tnu l’application induite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On v´erifie que les compositions invbijn˝bijn et bijn˝invbijn sont l’identit´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit que bijn est une bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) D´emonstration semblable `a celle de (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (c) Montrons que bij envoie Ensk n´1 ˆ rr1, n ´ 1ss dans Err1,n´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit pw, τ, iq P Ensk n´1 ˆ rr1, n ´ 1ss et posons v :“ w ˝ pi, σ :“ τ piq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme pi : rr1, nss Ñ rr1, n ´ 1ss est croissante et 16 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE w P rr0, k ´ 1ssn´1 ď , on a v P rr0, k ´ 1ssn ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors pour j P rr0, k ´ 1ss, v´1pjq “ w´1pjq si j ă wpiq, l’´egalit´e v´1pjq “ w´1pjq`1 si j ą wpiq, et v´1pwpiqq “ w´1pwpiqqYpw´1pwpiqq`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si j ă wpiq, la restriction de σ `a v´1pjq “ w´1pjq est ´egale `a la composition stτpiq ˝ τ|w´1pjq, qui est croissante par croissance de chacun des termes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si j ą wpiq, la restriction de σ `a v´1pjq “ w´1pjq ` 1 est ´egale `a la composition stτpiq´1 ˝ τ|w´1pjq ˝ px ÞÑ x ´ 1q, qui est croissante par croissance de chacun des termes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons α, β les ´el´ements minimaux et maximaux de w´1pwpiqq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' comme cet ensemble est un intervalle, on a w´1pwpiqq “ rrα, βss Q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors v´1pwpiqq “ rrα, β ` 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de σ `a v´1pwpiqq “ rrα, β ` 1ss est donn´ee par x ÞÑ τpxq si x P rrα, iss et x ÞÑ τpx ´ 1q ` 1 si x P rri ` 1, β ` 1ss ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' les restrictions de τ `a rrα, iss et rri, βss sont croissantes, ce qui implique que les restrictions de σ `a rrα, iss et rri ` 1, β ` 1ss le sont aussi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a σpi ` 1q “ τpiq ` 1 “ σpiq ` 1 ce qui, combin´e `a la croissance de σ sur rrα, iss et rri ` 1, β ` 1ss implique la croissance de σ sur rrα, β ` 1ss “ v´1pwpiqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc σ P S|v´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|v´1pk´1q| donc pv, σq “ pw ˝ pi, τ piqq P Ensk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On sait que τ piqpiq “ τpiq et τ piqpi ` 1q “ τpiq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors pτ piqq´1pτpiqq “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’ensemble t|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u est celui des maxima des intervalles de la partition rr1, nss “ v´1p0q \\ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' \\ v´1pk ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Celui de ces intervalles auquel appartient i est v´1pvpiqq “ rrα, β ` 1ss, on a donc pour j ‰ vpiq, i ‰ maxpvpjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme i ď β, on a i ‰ β ` 1 “ maxpv´1pvpiqqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc pour tout j P rr0, k ´ 1ss, i ‰ maxpv´1pjqq, donc pτ piqq´1pτpiqq “ i R t|v´1p0q|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , |v´1p0q| ` ¨ ¨ ¨ ` |v´1pk ´ 1q|u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, on a pτ piqq´1pτpiq ` 1q “ i ` 1 “ pτ piqq´1pτpiqq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit pw ˝ pi, τ piq, τpiqq P Err1,n´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc bijpEnsk n´1 ˆ rr1, n ´ 1ssq Ă Err1,n´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons bijrr1,n´1ss : Ensk n´1 ˆ rr1, n ´ 1ss Ñ Err1,n´1ss l’application induite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour pv, σ, jq P Err1,n´1ss posons (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='20) invbijrr1,n´1sspv, σ, jq :“ pv ˝ stσ´1pjq, pj ˝ σ ˝ stσ´1pjq, σ´1pjqq “ pw, τ, iq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors stσ´1pjq est une application rr1, n ´ 1ss Ñ rr1, nss, donc w P Zn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a aussi i P rr1, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Enfin τ “ pj ˝ σ ˝ stσ´1pjq est une application de rr1, n ´ 1ss dans lui-mˆeme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons son injectivit´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Supposons x ‰ x1 P rr1, n ´ 1ss et τpxq “ τ 1pxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Par injectivit´e de σ et stσ´1pjq on a σ ˝ stσ´1pjqpxq ‰ σ ˝ stσ´1pjqpx1q donc on a (quitte `a ´echanger x et x1) σ ˝ stσ´1pjqpxq “ j et σ ˝ stσ´1pjqpx1q “ j ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc stσ´1pjqpxq “ σ´1pjq et stσ´1pjqpxq “ σ´1pjq ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Or σ´1pjq ` 1 n’est pas dans l’image de stσ´1pjq, contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc l’injectivit´e de τ, donc τ P Sn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ceci montre que (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='20) d´efinit une application invbijrr1,n´1ss : Err1,n´1ss Ñ Zn ˆ Sn ˆ rr1, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons que invbijrr1,n´1ss envoie Err1,n´1ss dans Ensk n ˆ rr1, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='20), on a w “ v ˝stσ´1pjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a v P rr0, k ´1ssn´1 ď et stσ´1pjq est une application croissante rr1, n´1ss Ñ rr1, nss, ce qui implique w P rr1, k ´ 1ssn ď.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 17 Montrons que τ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ceci revient `a montrer que la restriction de τ `a w´1pℓq est croissante pour tout ℓ P rr0, k ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons que v´1pvpσ´1piqqq est un intervalle de rr1, nss dont σ´1piq n’est pas le plus grand ´el´ement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons α, β les minimum et maximum de cet intervalle, alors v´1pvpσ´1piqqq “ rrα, βss avec α ď σ´1piq ă β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si ℓ ă vpσ´1piqq, alors w´1pℓq “ v´1pℓq, et la restriction de τ `a w´1pℓq co¨ıncide avec la restriction de pσ´1pjq ˝ σ au mˆeme intervalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme σ|v´1pℓq et pσ´1pjq sont croissantes, τ|w´1pℓq est donc croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si ℓ ą vpσ´1piqq, alors w´1pℓq “ v´1pℓq ´ 1, et la restriction de τ `a w´1pℓq co¨ıncide avec la restriction de pσ´1pjq ˝ σ ˝ px ÞÑ x ` 1q au mˆeme intervalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme pσ ˝ px ÞÑ x ` 1qq|v´1pℓq´1 et pσ´1pjq sont croissantes, τ|w´1pℓq est donc croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a w´1pvpσ´1piqqq “ rrα, β ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de τ `a rrα, σ´1piqss co¨ıncide avec celle de σ `a cet intervalle qui est croissante, cet intervalle ´etant contenu dans v´1pvpσ´1piqqq, donc τ|rrα,σ´1piqss est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La restriction de τ `a rrσ´1piq, β ´ 1ss co¨ıncide avec celle de σ ˝ px ÞÑ x ` 1q `a cet intervalle qui est croissante par croissance de σ sur v´1pvpσ´1piqqq, donc τ|rrσ´1piq,β´1ss est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il suit que la restriction de τ `a rrα, β ´ 1ss “ w´1pvpσ´1piqqq est croissante.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Tout ceci implique τ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme τ P S|w´1p0q|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',|w´1pk´1q| et w P rr1, k ´ 1ssn ď, on a pw, τq P Ensk n´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc invbijrr1,n´1sspErr1,n´1ssq Ă Ensk n ˆ rr1, n ´ 1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons invbijrr1,n´1ss : Err1,n´1ss Ñ Ensk n ˆ rr1, n ´ 1ss l’application induite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On v´erifie que les compositions invbijn˝bijn et bijn˝invbijn sont l’identit´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit que bijrr1,n´1ss est une bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Diagramme commutatif impliquant Affp∆n´1, ∆nq ˆ t˘1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n, m ě 0, on note Affp∆n, ∆mq l’ensemble des applications φ : ∆n Ñ ∆m telles qu’il existe une application affine φ : Rn Ñ Rm telle que φ ˝ cann “ canm ˝ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (avec cann, canm comme en d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit n, m ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) Si φ P Affp∆n, ∆mq, une application affine φ : Rn Ñ Rm comme en d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='26 est unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) L’application Affp∆n, ∆mq Ñ AffpRn, Rmq, φ ÞÑ φ est injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (a) est une cons´equence de ce que ∆n contient une base affine de Rn, `a savoir pEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (b) provient de ce qu’une application affine est uniquement d´etermin´ee par l’image d’une base affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ On d´eduit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='27 une famille d’inclusions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='21) Affp∆n, ∆mq Ă AffpRn, Rmq pour n, m ě 0, compatible avec les compositions d’applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 18 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE Lemme-D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans la situation de la d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6, si P0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Pn P ∆m, l’application rP0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Pns envoie ∆n dans ∆m, donc appartient `a l’image de l’injection Affp∆n, ∆mq ãÑ AffpRn, Rmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note rP0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , Pns l’´el´ement de Affp∆n, ∆mq correspondant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Suit de la convexit´e de ∆n et ∆m, et de ce que pEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En nq forme une base du convexe ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Lemme-D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si pv, σq P Ensk n, alors ckpv, σq appartient `a l’image de l’injection Affp∆n, ∆nq ãÑ AffpRn, Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note ckpv, σq P Affp∆n, ∆nq la pr´eimage de ckpv, σq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application associ´ee `a pv, σq P Ensk n est donn´ee par px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq ÞÑ ppv1 ` xσp1qq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , pvn ` xσpnqq{kq “: py1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ynq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' c’est une endo-application continue de Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Supposons px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq P ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Comme v1 ě 0 et xσp1q ě 0, on a y1 ě 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' de mˆeme, vn ď k ´ 1 et xσpnq ď 1 implique yn ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Enfin pour i P rr1, n ´ 1ss, la d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='23 implique que vi “ vi`1 ou vi ă vi`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans le premier cas (vi “ vi`1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='18) implique σpiq ă σpi`1q, ce qui implique par croissance de i ÞÑ xi l’in´egalit´e xσpiq ď xσpi`1q, ce qui combin´e avec vi “ vi`1 implique l’in´egalit´e dans yi “ pvi ` xσpiqq{k ď pvi`1 ` xσpi`1qq{k “ yi`1, o`u les ´egalit´es extrˆemes proviennent des d´efinitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans le second cas (vi ă vi`1), on a yi “ pvi ` xσpiqq{k ď pvi ` 1q{k ď vi`1{k ď pvi`1 ` xσpi`1qq{k “ yi`1, o`u la premi`ere et derni`ere ´egalit´e proviennent des d´efinitions, o`u la premi`ere et derni`ere in´egalit´e proviennent respectivement de xσpiq ě 0 et xσpi`1q ď 1, et o`u l’in´egalit´e centrale provient de vi ă vi`1 et du caract`ere entier des composantes de v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc dans tous les cas yi ď yi`1, ce qui ach`eve de montrer que py1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ynq P ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ceci montre l’´enonc´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ On a ckpv, σq “ rp1{kqpv ` σ˚En 0 q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , p1{kqpv ` σ˚En nqs pour pv, σq P Ensk n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note pour i P rr0, nss, Bn i :“ rEn 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En n´i´1, En n´i`1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , En ns P Affp∆n´1, ∆nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Bn i est l’image de Bn i par l’application Affp∆n´1, ∆nq Ñ AffpRn´1, Rnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Imm´ediat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note f : Ensk n ˆ rr0, nss Ñ Affp∆n´1, ∆nq, ˜f : Ensk n´1 ˆ rr0, nss Ñ Affp∆n´1, ∆nq les applications donn´ees par fpv, σ, iq :“ ckpv, σq ˝ Bn i , ˜fpw, τ, iq :“ Bn i ˝ ckpw, τq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 19 Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les diagrammes suivants commutent Ensk n ˆ rr0, nss f � � � � Affp∆n´1, ∆nq � � � Zn ˆ Sn ˆ ˆrr0, nss f � AffpRn´1, Rnq Ensk n´1 ˆ rr0, nss ˜ f � � � � Affp∆n´1, ∆nq � � � Zn´1 ˆ Sn´1 ˆ ˆrr0, nss ˜ f � AffpRn´1, Rnq D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Cela provient de la compatibilit´e des applications (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='21) avec la composition, du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='31, et ce que pour pv, σq P Ensk n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' pw, τq P Ensk n´1), l’image de ckpv, σq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ckpw, τq) sous Affp∆n, ∆nq Ñ AffpRn, Rnq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Affp∆n´1, ∆n´1q Ñ AffpRn´1, Rn´1q) est ckpv, σq (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ckpw, τq) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le diagramme suivant commute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='22) Ensk n´1 ˆ rr0, nss bij „ � p ˜ f,Ą sgnq �❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ tx P Ensk n ˆ rr0, nss|involpxq R Ensk n ˆ rr0, nssuu pf,sgnq �❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ Affp∆n´1, ∆nq ˆ t˘1u D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ceci provient de la commutativit´e des diagrammes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17) et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='19) ainsi que du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Construction d’endomorphismes de groupes de chaˆınes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les r´esultats de cette section seront utilis´es en section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5 afin de montrer le (b) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Morphismes dans une cat´egorie C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit C la petite cat´egorie dont l’ensemble d’objets est Zě0, avec Cpn, mq :“ ZTopp∆n, ∆mq, et dont la composition est donn´ee par la lin´earisation de la composition dans Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Toute application affine ´etant continue, on a une famille de diagrammes Affp∆n, ∆mq Ă Cpn, mq pour n, m ě 0, compatible avec les compositions d’applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On pose divk n :“ ÿ pv,σqPEnsk n ǫpσqckpv, σq P ZAffp∆n, ∆nq Ă Cpn, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note Bn´1,n :“ nÿ i“0 p´1qiBn i P ZAffp∆n´1, ∆nq Ă Cpn ´ 1, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 20 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE On sait que Bn,n`1 ˝ Bn´1,n “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' De plus, pour tout p ě 0, le complexe ¨ ¨ ¨ Ñ Cpk, pq ´˝Bk´1,k Ñ Cpk ´ 1, pq Ñ ¨ ¨ ¨ Ñ Cp0, pq Ñ Z Ñ 0 est acyclique (l’homologie de ∆p ´etant donn´ee par Hkp∆pq “ 0 si k ą 0 et “ Z si k “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de divk n ˝Bn´1,n “ Bn´1,n ˝divk n´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2, on fixe n, k ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) divk n ˝ Bn´1,n “ ÿ xPEnsk nˆrr0,nss sgnpxqfpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' en combinant les d´efs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='32, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='36, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='37, et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) Bn´1,n ˝ divk n´1 “ ÿ ˜xPEnsk n´1ˆrr0,nss Ą sgnp˜xq ˜fp˜xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' en combinant les d´efs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='17, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='32, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='36, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='37 (´egalit´es dans ZAffp∆n´1, ∆nq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a pour tous n, k ě 1 divk n ˝ Bn´1,n “ Bn´1,n ˝ divk n´1 (´egalit´e dans Cpn ´ 1, nq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a ÿ xPEnsk nˆrr0,nss| involpxqPEnsk nˆrr0,nss sgnpxqfpxq “ ÿ xPEnsk nˆrr0,nss| involpxqPEnsk nˆrr0,nss sgnpinvolpxqqfpinvolpxqq “ ´ ÿ xPEnsk nˆrr0,nss| involpxqPEnsk nˆrr0,nss sgnpxqfpxq o`u la premi`ere ´egalit´e suit de ce que invol est une involution de tx P Ensk n ˆ rr0, nss|involpxq P Ensk nˆrr0, nssu (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11(a)) et la deuxi`eme ´egalit´e suit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11(b) et de la premi`ere ´egalit´e du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3) ÿ xPEnsk nˆrr0,nss| involpxqPEnsk nˆrr0,nss sgnpxqfpxq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 21 Alors divk n ˝ Bn´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='n “ ÿ xPEnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss sgnpxqfpxq “ ÿ xPEnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss| involpxqPEnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss sgnpxqfpxq ` ÿ xPEnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss| involpxqREnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss sgnpxqfpxq “ ÿ xPEnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss| involpxqREnsk nˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss sgnpxqfpxq “ ÿ yPEnsk n´1ˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss sgnpbijpyqqfpbijpyqq “ ÿ yPEnsk n´1ˆrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss Ą sgnpyq ˜fpyq “ Bn´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='n ˝ divk n´1 (´egalit´e dans ZAffp∆n´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ∆nq) o`u la premi`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1), la troisi`eme ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3), la quatri`eme ´egalit´e suit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='25, la cinqui`eme ´egalit´e suit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='20 et des deux ´egalit´es du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='33, la sixi`eme ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit l’´egalit´e annonc´ee, les applications Affp∆n, ∆mq Ñ Cpn, mq ´etant compatibles aux compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Relation dans C entre divk ‚ et id‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour tout k ě 0, il existe une famille pLk n`1,nqně0 avec Lk n`1,n P Cpn ` 1, nq, telle que pour tout n ě 0, on a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) idn ´ divk n “ Lk n`1,n ˝ Bn,n`1 ` Bn´1,n ˝ Lk n,n´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons par r´ecurrence sur n ě 0 l’existence d’une famille pLk m`1,mqmďn telle que pour tout m ď n, on a idn ´ divk n “ Lk n`1,n ˝ Bn,n`1 ` Bn´1,n ˝ Lk n,n´1 (´enonc´e Epnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Posons Lk 1,0 :“ 0, alors on a id0 ´ divk 0 “ Lk 1,0 ˝ B0,1 d’o`u l’´enonc´e Ep0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit n ě 1, et supposons Epn ´ 1q v´erifi´e avec une famille pLk m`1,mqmďn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors pidn ´ divk n ´ Bn´1,n ˝ Lk n,n´1q ˝ Bn´1,n “ Bn´1,n ´ divk n ˝ Bn´1,n ´ Bn´1,n ˝ Lk n,n´1 ˝ Bn´1,n “ Bn´1,n ´ Bn´1,n ˝ divk n´1 ´ Bn´1,n ˝ Lk n,n´1 ˝ Bn´1,n “ Bn´1,n ˝ pidn´1 ´ divk n´1 ´ Lk n,n´1 ˝ Bn´1,nq “ Bn´1,n ˝ pidn´1 ´ divk n´1 ´ Lk n,n´1 ˝ Bn´1,n ´ Bn´2,n´1 ˝ Lk n´1,n´2q “ Bn´1,n ˝ 0 “ 0 o`u la seconde ´egalit´e suit de la proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='38, la quatri`eme ´egalit´e suit de Bn´1,n ˝Bn´2,n´1 “ 0, la cinqui`eme ´egalit´e suit de Epn ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc idn ´divk n ´Bn´1,n ˝Lk n,n´1 appartient au noyau de l’application ´˝Bn´1,n : Cpn, nq Ñ Cpn´1, nq, qui par acyclicit´e est ´egal `a l’image de l’application ´˝Bn,n`1 : Cpn`1, nq Ñ Cpn, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il existe donc Lk n`1,n P Cpn ` 1, nq tel que idn ´ divk n ´ Bn´1,n ˝ Lk n,n´1 “ Lk n`1,n ˝ Bn,n`1, ce qui implique Epnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Endomorphismes de groupes de chaˆınes singuli`eres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit X un espace topologique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n ě 0, on note CnpXq :“ ZTopp∆n, Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n, m ě 0, on a une application CnpXq ˆ Cpm, nq Ñ CmpXq induite par la composition pc, xq ÞÑ c ˝ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 22 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour x P Cpm, nq, on note x˚ : CnpXq Ñ CmpXq l’application c ÞÑ c ˝ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors px ˝ yq˚ “ y˚ ˝ x˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si Y est un sous-ensemble de X, on a x˚pCnpY qq Ă CmpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Provient de ce que x˚ est une composition `a la source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ De plus pour n ě 1, B˚ n´1,n : CnpXq Ñ Cn´1pXq co¨ıncide avec la diff´erentielle singuli`ere Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) implique (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) idCnpXq ´ pdivk nq˚ “ B˚ n,n`1 ˝ pLk n`1,nq˚ ` pLk n,n´1q˚ ˝ B˚ n´1,n (´egalit´e d’endomorphismes de CnpXq) pour tout n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (b) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On se place dans le cadre de la section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1: X est un espace topologique et a, b P X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Composition de chemins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si s, t, u, v P R avec s ‰ t, on note au,v s,t l’unique application affine de R dans lui-mˆeme telle que s ÞÑ u et t ÞÑ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit X un espace topologique, soit a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , am P X et ˜γi P Chempai, ai`1q pour i P rr0, m´1ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ˜γm´1 ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='˚ ˜γ0 P Chempa0, amq est le chemin tel que pour pour i P rr0, m´1ss, la restriction p˜γm´1 ˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ˚ ˜γ0q|ri{m,pi`1q{ms `a ri{m, pi ` 1q{ms co¨ıncide avec ˜γi ˝ a0,1 i{m,pi`1q{m (conditions coh´erentes car ˜γip1q “ ˜γi`1p0q pour i P rr0, m ´ 2ss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a alors, en notant r´s l’application canonique Chempa, bq Ñ π1pX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' a, bq pour a, b P X quelconques, l’´egalit´e (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) r˜γn´1 ˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ˚ ˜γ0s “ r˜γn´1s ¨ ¨ ¨ r˜γ0s (´egalit´e dans π1pX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' a0, amq, le produit dans le membre de droite ´etant celui dans le groupo¨ıde π1pXq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Calcul de pdivk nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il existe une unique application (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) tpn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq|n1 ě 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ nu Ñ rr0, k ´ 1ssn qui envoie pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq vers l’´el´ement v P rr0, k ´ 1ssn tel que pour i “ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , k ´ 1 on a v|n1`¨¨¨`ni`rr1,ni`1ss “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Cette application est bijective, et la bijection r´eciproque envoie v P rr0, k ´ 1ssn vers pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq donn´e par ni “ |v´1pi ´ 1q| pour i P rr1, kss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Imm´ediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour v P rr0, k ´ 1ssn, on pose Spvq :“ tσ P Sn|pv, σq P Ensk nu (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 23 Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq P tpn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq|n1 ě 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ nu et si v est l’image de pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq par la bijection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2), alors on a Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk “ Spvq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application v : rr1, nss Ñ rr0, k ´ 1ss ´etant constante sur chaque sous-ensemble n1`¨ ¨ ¨`ni`rr1, ni`1ss et les valeurs prises sur des sous-ensembles cons´ecutifs ´etant strictement croissantes, on a pour i P rr1, nss l’´equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3) pvi ă vi`1q ðñ pi P tn1, n1 ` n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nk´1uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit alors σ P Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a σ P Spvq si et seulement si @i P rr1, nss, pσpiq ą σpi ` 1qq ùñ pvi ă vi`1q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' d’apr`es (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3) une condition ´equivalente est @i P rr1, nss, pσpiq ą σpi ` 1qq ùñ pi P tn1, n1 ` n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , n1 ` ¨ ¨ ¨ ` nk´1uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Donc σ P Spvq si et seulement si σ est croissante sur les sous-ensembles rr1, n1ss, n1 ` rr1, n2ss, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', n1 ` ¨ ¨ ¨ ` nk´1 ` rr1, nkss, c’est `a dire si et seulement si σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ D´efinition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On pose Ą Ens k n :“ tppn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, σq|n1 ě 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nk ě 0 et n1 ` ¨ ¨ ¨ ` nk “ n et σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il existe une unique bijection (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) Ensk n Ñ Ą Ens k n qui envoie ppn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, σq vers pv, σq, o`u v est l’image de pn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq par (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Cons´equence des lemmes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='44 et 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Notons que pour n1 ` ¨ ¨ ¨ ` nk “ n et σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk, l’automorphisme de r0, 1sn dans la cat´egorie Top donn´e par σ˚ induit un ´el´ement, not´e encore σ˚ de Topp∆n, ∆n1ˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='ˆ∆nkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Par ailleurs, ˜γn1 1 ˆ¨ ¨ ¨ˆ˜γnk k P Topp∆n1 ˆ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='ˆ∆nk, Xnq, donc p˜γn1 1 ˆ¨ ¨ ¨ˆ˜γnk k q˝σ˚ P Topp∆n, Xnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Les applications Ensk n Ñ Topp∆n, Xnq et Ą Ens k n Ñ Topp∆n, Xnq donn´ees re- spectivement par ppn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, σq ÞÑ p˜γpn1q 1 ˆ ¨ ¨ ¨ ˆ ˜γpnkq k q ˝ σ˚ et pv, σq ÞÑ ˜γpnq ˝ cpv, σq sont telles que le diagramme Ensk n �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) � Ą Ens k n �rrrrrrrrrrr Topp∆n, Xnq est commutatif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 24 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit ppn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , nkq, σq P Ensk n et pv, σq P Ą Ens k n son image par (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Posons δm :“ p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , 1q P Rm pour m ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors pour px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq P ∆n, on a ˜γpnq ˝ cpv, σqpx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq “˜γpnq ´ pxσp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpn1qq{k,pδn2 `pxσpn1`1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpn1`n2qqq{k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',ppk´1qδnk`pxσpn1`¨¨¨`nk´1`1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpnqqq{k ¯ “ ´ ˜γn1 ´ pxσp1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpn1qq{k ¯ ,˜γn2 ´ pδn2`pxσpn1`1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpn1`n2qqq{k ¯ ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',˜γnk ´ ppk´1qδnk `pxσpn1`¨¨¨`nk´1`1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',xσpnqqq{k ¯¯ “ ´ ˜γn1 1 pxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1qq, ˜γn2 2 pxσpn1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1`n2qqq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ˜γnk k pxσpn1`¨¨¨`nk´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpnqqq ¯ “ ´ ˜γpn1q 1 pxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1qq, ˜γpn2q 2 pxσpn1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1`n2qq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ˜γpnkq k pxσpn1`¨¨¨`nk´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpnqq ¯ “ p˜γpn1q 1 ˆ ¨ ¨ ¨ ˆ ˜γpnkq k q ˝ σ˚px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq, o`u la premi`ere ´egalit´e suit de ce que cpv, σq est l’application de ∆n dans lui-mˆeme donn´ee par px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq ÞÑ ´ pxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1qq{k, pδn2 ` pxσpn1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1`n2qqq{k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ppk ´ 1qδnk ` pxσpn1`¨¨¨`nk´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpnqqq{k ¯ , la deuxi`eme ´egalit´e suit de la combinaison du fait que ˜γpnq est une restriction de ˜γn et de l’´egalit´e ˜γn “ ˜γn1 ˆ¨ ¨ ¨ˆ ˜γnk, la troisi`eme ´egalit´e suit de l’identit´e ˜γppi´1`xq{kq “ ˜γipxq pour x P r0, 1s et i P rr1, kss, la quatri`eme ´egalit´e suit des relations pxσp1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpn1qq P ∆n1, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', pxσpn1`¨¨¨`nk´1`1q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xσpnqq P ∆nk, elles-mˆemes cons´equences de σ P Sn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk, la derni`ere ´egalit´e suit de la d´efinition de σ˚ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc ˜γpnq ˝ cpv, σq “ p˜γpn1q 1 ˆ ¨ ¨ ¨ ˆ ˜γpnkq k q ˝ σ˚ (´egalit´e dans Topp∆n, Xnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Si a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , ak`1 P X et ˜γi P Chempai, ai`1q pour i P rr1, kss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors pdivk nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq “ ÿ n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nkě0| n1`¨¨¨`nk“n ÿ σPSn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk ǫpσqp˜γpn1q 1 ˆ ¨ ¨ ¨ ˆ ˜γpnkq k q ˝ σ˚ (´egalit´e dans CnpXnq “ ZTopp∆n, Xnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a pdivk nq˚pp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnqq “ p˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnq ˝ divk n “ ÿ pσ,vqPEnsk n ǫpσqp˜γk ˚ ¨ ¨ ¨ ˚ ˜γ1qpnq ˝ cpσ, vq “ ÿ n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nkě0| n1`¨¨¨`nk“n ÿ σPSn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',nk ǫpσqp˜γpn1q 1 ˆ ¨ ¨ ¨ ˆ ˜γpnkq k q ˝ σ˚ o`u la premi`ere ´egalit´e suit de la d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='40, la deuxi`eme ´egalit´e suit de la d´ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='36, et la troisi`eme suit du lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Une ´egalit´e dans CnpXnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On fixe ˜γ P Chempa, bq et α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , αn P Chempa, aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On note pour I Ă rr0, nss, cI :“ p˜γ ˚ ˚iPI ˜αiqpnq P CnpXnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' En appliquant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) `a cI (X ´etant remplac´e par Xn et k par |I| ` 1), on trouve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) cI ´ pdiv|I|`1 n q˚pcIq “ pB˚ n,n`1 ˝ pL|I|`1 n`1,nq˚ ` pL|I|`1 n,n´1q˚ ˝ B˚ n´1,nqpcIq TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 25 (relation dans CnpXnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a B˚ n´1,npcIq P Cn´1pY pnq ab q par le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2, le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='41 implique alors que pL|I|`1 n,n´1q˚ ˝ B˚ n´1,npcIq P CnpY pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D’autre part, pL|I|`1 n`1,nq˚pcIq P Cn`1pXnq, donc B˚ n,n`1 ˝ pL|I|`1 n`1,nq˚pcIq P B˚ n,n`1pCn`1pXnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ces deux relations et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) impliquent @I Ă rr0, nss, cI ´ pdiv|I|`1 n q˚pcIq P CnpY pnq ab q ` B˚ n,n`1pCn`1pXnqq (relation dans CnpXnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Cette relation implique ř IĂrr0,nssp´1q|I|pcI´pdiv|I|`1 n q˚pcIqq P CnpY pnq ab q` B˚ n,n`1pCn`1pXnqq donc (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6) ÿ IĂrr0,nss p´1q|I|cI ´ ÿ IĂrr0,nss p´1q|I|pdiv|I|`1 n q˚pcIq P CnpY pnq ab q ` B˚ n,n`1pCn`1pXnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (relation dans CnpXnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a ř IĂrr0,nssp´1q|I|pdiv|I|`1 n q˚pcIq “ 0 (´egalit´e dans CnpXnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour ν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1 ě 0 avec ν0 ` ¨ ¨ ¨ ` νn`1 “ n, posons (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='7) fpν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q :“ ÿ σPSν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1 ǫpσqp˜αpν0q ˆ ¨ ¨ ¨ ˆ ˜αpνnq ˆ ˜γpνn`1qq ˝ σ˚ P CnpXnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit I Ă rr0, nss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit α ÞÑ iα l’unique bijection croissante rr1, |I|ss Ñ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a les ´egalit´es pdiv|I|`1 n q˚pcIq “ ÿ ν:I\\tn`1uÑZě0, ř xPI\\tn`1u νpxq“n ÿ σPSνpi1q,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νpi|I|q,νpn`1q ǫpσqp˜αpνpi1qq i1 ˆ ¨ ¨ ¨ ˆ ˜α pνpi|I|qq i|I| ˆ ˜γpνpn`1qqq ˝ σ˚ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='8) “ ÿ ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1ě0| ν0`¨¨¨`νn`1“n, ν|rr0,nss´I“0 ÿ σPSν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1 ǫpσqp˜αpν0q ˆ ¨ ¨ ¨ ˆ ˜αpνnq ˆ ˜γpνn`1qq ˝ σ˚ “ ÿ ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1ě0| ν0`¨¨¨`νn`1“n, ν|rr0,nss´I“0 fpν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q dans CnpXnq o`u la premi`ere ´egalit´e suit de la proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='50, et la deuxi`eme ´egalit´e utilise la bijection entre applications I \\ tn ` 1u Ñ Zě0 et applications rr0, n ` 1ss Ñ Zě0 nulles sur rr0, nss ´ I fournie par l’extension par la fonctions nulle, et la troisi`eme ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors ÿ IĂrr0,nss p´1q|I|pdiv|I|`1 n q˚pcIq “ ÿ IĂrr0,nss ÿ ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1ě0| ν0`¨¨¨`νn`1“n, ν|rr0,nss´I“0 p´1q|I|fpν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q “ ÿ ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1ě0| ν0`¨¨¨`νn`1“n ÿ txPrr0,nss|νx‰0uĂIĂrr0,nss p´1q|I|fpν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q “ ÿ ν0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=',νn`1ě0| ν0`¨¨¨`νn`1“n fpν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q ÿ txPrr0,nss|νx‰0uĂIĂrr0,nss p´1q|I| (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9) o`u la premi`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='8), la deuxi`eme ´egalit´e suit de l’´equivalence entre les conditions ν|rr0,nss´I “ 0 et I Ą tx P rr0, nss|νx ‰ 0u et la troisi`eme ´egalit´e est une factorisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour tout pν0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q P Zn`1 ě0 tel que ν0 ` ¨ ¨ ¨ ` νn`1 “ n, on a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='10) ÿ txPrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss|νx‰0uĂIĂrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss p´1q|I| “ p´1q|txPrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss|νx‰0u| ÿ JĂtxPrr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='nss|νx“0u p´1q|J| “ 0 26 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE o`u la premi`ere ´egalit´e suit de la bijection entre l’ensemble des I tels que tx P rr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' nss|νx ‰ 0u Ă I Ă rr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' nss et l’ensemble des parties J de tx P rr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' nss|νx “ 0u fournie par J ÞÑ J Y tx P rr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' nss|νx ‰ 0u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' et la seconde ´egalit´e suit de l’identit´e ř XĂEp´1q|X| “ 0 pour tout ensemble fini non vide E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' ainsi que de tx P rr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' nss|νx “ 0u ‰ H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' qui r´esulte de pν0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , νn`1q P Zn`1 ě0 et ν0 ` ¨ ¨ ¨ ` νn`1 “ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' En combinant (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='10) et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='9), on obtient ř IĂrr0,nssp´1q|I|pdiv|I|`1 n q˚pcIq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration de (b) du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le lemme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='51 et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6) impliquent ÿ IĂrr0,nss p´1q|I|cI P CnpY pnq ab q ` B˚ n,n`1pCn`1pXnqq (relation dans CnpXnq) c’est-`a-dire (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11) ÿ IĂrr0,nss p´1q|I|p˜γ ˚ ˚iPI ˜αiqpnq P CnpY pnq ab q ` B˚ n,n`1pCn`1pXnqq (relation dans CnpXnq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Chaque cI “ p˜γ˚˚iPI ˜αiqpnq appartient au sous-espace ZnpXn, Y pnq ab q “ tc P CnpXnq|B˚ n´1,npcq P Cn´1pY pnq ab qu Ă CnpXnq, donc (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='11) peut ˆetre vue comme une relation dans ZnpXn, Y pnq ab q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Elle implique la relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='12) ÿ IĂrr0,nss p´1q|I|rp˜γ ˚ ˚iPI ˜αiqpnqs “ 0 (´egalit´e dans HnpXn, Y pnq ab q “ ZnpXn, Y pnq ab q{BnpXn, Y pnq ab q, avec BnpXn, Y pnq ab q “ CnpY pnq ab q ` B˚ n,n`1pCn`1pXnqq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour I Ă rr0, nss, on a rp˜γ ˚ ˚iPI ˜αiqpnqs “ Fnpr˜γ ˚ ˚iPI ˜αisq “ Fnpγ ¨ ź iPI αiq (´egalit´e dans HnpXn, Y pnq ab q) o`u la premi`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) et deuxi`eme suit de ((2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' En combinant cette ´egalit´e avec (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='12), on en d´eduit l’´egalit´e souhait´ee ÿ IĂrr0,nss p´1q|I|Fnpγ ¨ ź iPI αiq “ 0 (´egalit´e dans HnpXn, Y pnq ab q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lien avec l’isomorphisme de Beilinson Le but de cette section est la d´emonstration de la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1, qui relie l’application F pnq xy obtenue dans le th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 avec l’isomorphisme (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) de Beilinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On rappelle la construction de cet isomorphisme en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1, puis on montre la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1 en section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappels sur l’isomorphisme de Beilinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit M une vari´et´e diff´erentiable connexe, ayant le type d’homotopie d’un CW-complexe fini et n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans [DG], §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 (voir aussi [BGFr], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 251), on associe `a chaque couple px, yq d’´el´ements de M un complexe de faisceaux de Q- espaces vectoriels yKxxny sur M n et une application lin´eaire surjective H‚pM n,x Kxxnyq Ñ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a les isomorphismes de Q-espaces vectoriels H‚pM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq » H‚pM n,y Kxxnyq si x ‰ y, ([BGFr], deux lignes avant (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='282)) et H‚pM n, Y pnq xx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq » KerpH‚pM n,x Kxxnyq Ñ Qq, ([BGFr], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='284)), o`u H‚p´, ´;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq d´esigne l’homologie singuli`ere relative `a coefficients dans Q des paires d’espaces topologiques et H‚ l’hypercohomologie des complexes de faisceaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le th´eor`eme de Beilinson (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4 de [DG], ou Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='298 de [BGFr]) dit qu’il y a un isomorphisme de Q-espaces vectoriels βpnq yx : H‚pM n,y Kxxnyq Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚ s’ins´erant pour y “ x dans le diagramme commutatif H‚pM n,x Kxxnyq � �▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ pQπ1pxq{pQπ1pxqqn`1 ` q˚ �♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Q l’application pQπ1pxq{pQπ1pxqqn`1 ` q˚ Ñ Q ´etant duale de l’application Q Ñ Qπ1pxq{pQπ1pxqqn`1 ` induite par 1 ÞÑ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On en d´eduit pour tout px, yq une application lin´eaire (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) βpnq yx : H‚pM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚, qui est un isomorphisme si y ‰ x, et qui induit un isomorphisme H‚pM n, Y pnq xx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq „ Ñ KerppQπ1pxq{pQπ1pxqqn`1 ` q˚ Ñ Qq si y “ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Relation du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 avec l’isomorphisme de Beilinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Notons θ l’involution de M n donn´ee par px1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , xnq ÞÑ pxn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , x1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’image de Y pnq yx par cette involution est Y pnq xy , donc elle induit un isomorphisme θ˚ : HnpM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq Ñ HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons le couplage entre homologie et cohomologie relatives, qui induit une application lin´eaire can : HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq Ñ HnpM n, Y pnq xy q˚ Q, o`u pour A un Z-module, on note A˚ Q :“ HomZpA, Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour f : A Ñ B morphisme de Z-modules, on note aussi f ˚ Q : B˚ Q Ñ A˚ Q le morphisme induit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' 28 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application lin´eaire βpnq yx (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1)) est au signe pr`es, la compos´ee du dual pF pnq xy q˚ Q de l’application lin´eaire F pnq xy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3), de can et de θ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pr´ecis´ement, on a βpnq yx “ p´1qn`1pF pnq xy q˚ Q ˝ can ˝ θ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans [BGFr], on construit un morphisme de complexes de faisceaux nat :y ˜Kxxny Ñy Kxxny ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='282) et cinq lignes avant cette ´equation) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' si y ‰ x, ce morphisme est l’identit´e de yKxxny (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', 6 lignes avant (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='282)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Dans loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', on construit un iso- morphisme d’espaces vectoriels isoyx BGF : HnpM n,y ˜Kxxnyq Ñ HnpM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='281).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) est alors donn´ee par la composition HnpM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq pisoyx BGFq´1 Ñ HnpM n,y ˜Kxxnyq Ñ HnpM n,y Kxxnyq βpnq yx Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚ L’´enonc´e est alors ´equivalent `a la commutativit´e du diagramme suivant HnpM n,y ˜Kxxnyq nat˚ � isoyx BGF � HnpM n,y Kxxnyq βpnq yx � pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚ HnpM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq θ˚ � HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq p´1qn`1can � HnpM n, Y pnq xy q˚ Q pF pnq xy q˚ � laquelle est ´equivalente `a l’´enonc´e suivant : @c P HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq, @a P Qπ1px, yq{pQπ1px, yqpQπ1pxqqn`1 ` q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1) xβpnq yx ˝ nat˚ ˝ pisoyx BGFq´1 ˝ pθ˚q´1pcq, ay “ p´1qn`1xc, F pnq xy paqyhom, o`u x´, ´y est le couplage V ˚ ˆ V Ñ Q associ´e `a un Q-espace vectoriel V et x´, ´yhom est le couplage naturel HnpX, Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq ˆ HnpX, Y q Ñ Q, que par lin´earit´e il suffit de v´erifier pour a “ rγs, o`u γ P π1px, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Rappelons quelques constructions de [BGFr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit ˜C :“ pp ˜Cp,qqp,qě0, d1, d2q le bicomplexe tel que ˜Cp,q :“ ‘IĂrr1,nss||I|“n´pCqpXIq, o`u CqpXq :“ HomQpCqpXq, Qq, o`u d2 est la somme sur I, q des op´erateurs de cobord CqpXIq Ñ Cq`1pXIq et o`u d2 est la somme sur les couples pI, Jq avec I Ą J et |I ´ J| “ 1 des applications ǫpI, Jqδ˚ I,J : CqpXIq Ñ CqpXJq, o`u δI,J : XJ Ñ XI est le morphisme donn´e par [BGFr], formule apr`es (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='287) et ǫpI, Jq P t˘1u est donn´e par [BGFr], formule avant (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='278).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit C :“ ppCp,qqp,qě0, d1, d2q le bicomplexe quotient de ˜C donn´e par Cp,q “ ˜Cp,q si q ă n, Cp,n “ 0 pour tout p ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D’apr`es [BGFr], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='289, on a un isomorphisme H‚pTotpCqq » HnpM n,y Kxxnyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On montre que cet isomorphisme s’ins`ere dans carr´e commutatif dont les morphismes verticaux TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 29 sont des isomorphismes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2) H‚pTot ˜Cq nat � „ ˜iC � H‚pTotCq „ iC � HnpM n,y ˜Kxxnyq nat˚ � HnpM n,y Kxxnyq Alors l’application compos´ee HnpTotCq iC Ñ HnpM n,y Kxxnyq βpnq yx Ñ pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚ est telle que pour ω “ pωIqH‰IĂrr1,nss P ZnpTotCq avec ωI P Cn´|I|pXIq, et γ P π1px, yq, on a (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3) xβpnq yx ˝ iCprωsq, rγsy “ ÿ H‰IĂrr1,nss p´1qp1{2qp|I|´1qp|I|´2q`n|I|ǫpIqωIp˜γpIqq, (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' [BGFr], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='292)), o`u ǫpIq est donn´e par [BGFr], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='278) et γpIq P Topp∆|I|, M Iq Ă C|I|pM Iq donn´e par ∆|I| Q pt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' , t|I|q ÞÑ pI Q i ÞÑ ˜γptκpiqqq P M I, avec κ l’unique bijection croissante I Ñ rr1, |I|ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour tout i P rr1, nss et p ě 0, l’application δ˚ rr1,nss,rr1,nss´tiu : CppXnq Ñ CppXrr1,nss´iq est une composition CppXnq Ñ CppY pnq xy q Ñ CppXrr1,nss´tiuq donc KerpCppXnq Ñ CppY pnq xy qq Ă Kerp‘iPrr1,nss : δ˚ rr1,nss,rr1,nss´tiu : CppXnq Ñ ‘iPrr1,nssCppXrr1,nss´tiuqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On a donc pour chaque p ě 0 une application lin´eaire ˜µp : KerpCppXnq Ñ CppY pnq xy qq Ñ Totpp ˜Cq donn´ee par KerpCppXnq Ñ CppY pnq xy qq Q c ÞÑ ppp, 0q ÞÑ c, pp, 0q ‰ pp,1 q1q ÞÑ 0q, qui d´efinit un morphisme de complexes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) ˜µ‚ : KerpC‚pXnq Ñ C‚pY pnq xy qq Ñ Tot‚p ˜Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La cohomologie du complexe source de (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) est la cohomologie singuli`ere relative H‚pXn, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On d´eduit du morphisme de complexes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4) un morphisme en cohomologie H‚p˜µ‚q : H‚pXn, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq Ñ H‚pTotp ˜Cqq dont on v´erifie qu’il satisfait (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5) H‚p˜µ‚q “ p˜iCq´1 ˝ pisoBGFq´1 ˝ pθ˚q´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Montrons alors (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit γ P π1px, yq, c P HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Soit ˜c P ZnpKerpC‚pXnq Ñ C‚pY pnq xy qqq un repr´esentant de c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Alors xβpnq yx ˝ nat˚ ˝ pisoyx BGFq´1 ˝ pθ˚q´1pcq, rγsy “ xβpnq yx ˝ iC ˝ nat ˝ ˜i´1 C ˝ pisoyx BGFq´1 ˝ pθ˚q´1pcq, rγsy “ xβpnq yx ˝ iC ˝ nat ˝ Hnp˜µ‚qpcq, rγsy “ xβpnq yx ˝ iC ˝ nat ˝ r˜µnp˜cqs, rγsy “ xβpnq yx ˝ iC ˝ rµnp˜cqs, rγsy “ p´1qpn´1qpn´2q{2`n2`npn`1q{2˜cp˜γpnqq “ p´1qn`1xc, F pnq xy prγsqyhom o`u µn est la compos´ee de ˜µn et de la projection ˜Cn Ñ Cn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' la premi`ere ´egalit´e suit de la commutativit´e de (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2), la deuxi`eme ´egalit´e suite de (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5), la troisi`eme ´egalit´e suit de la d´efinition de Hnp˜µ‚q, la quatri`eme ´egalit´e suit de ce que nat est la version cohomologiquee 30 BENJAMIN ENRIQUEZ ET FLORENCE LECOMTE de la projection ˜Cn Ñ Cn, la cinqui`eme ´egalit´e suit de l’´egalit´e (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3), dans laquelle seule la contribution de I “ rr1, nss est non-triviale, la derni`ere ´egalit´e suit de (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ La d´emonstration de la proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1 est illustr´ee par le diagramme suivant H‚pTot ˜Cq nat � „ ˜iC � H‚pTotCq „ iC � βpnq yx ˝iC �❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ HnpM n,y ˜Kxxnyq nat˚ � isoyx BGF � HnpM n,y Kxxnyq βpnq yx� pQπ1px, yq{Qπ1px, yqpQπ1pxqqn`1 ` q˚ HnpM n, Y pnq yx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq θ˚ � HnpM n, Y pnq xy ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Qq can � HnpM n, Y pnq xy q˚ Q pF pnq xy q˚ � 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Construction de transformations naturelles On note Top2 la cat´egorie des espaces topologiques munis d’un couple de points marqu´es (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' d’un couple de morphismes de source l’objet initial ˚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour pX, a, bq un objet de Top2, le couple pπ1pa, bq, π1paqq est un torseur `a droite, `a savoir un couple pT, Gq avec T un ensemble et G un groupe, munis d’une action `a droite libre et transtive de G sur T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La correspondance pX, a, bq ÞÑ pπ1pa, bq, π1paqq d´efinit un foncteur Top2 Ñ TorDt avec TorDt la cat´egorie des torseurs `a droite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´efinition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour pX, x, yq un objet de Top2, on note Zπ1pX, x, yq le Z-module libre sur π1pX, x, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application pX, a, bq ÞÑ FnpX, a, bq d´efinit un foncteur covariant de Top2 vers la cat´egorie Ab des groupes ab´eliens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Il s’agit de la composition du foncteur Top2 Ñ TorDt envoyant pX, a, bq vers pπ1pa, bq, π1paqq et du foncteur TorDt Ñ Ab envoyant pT, Gq vers ZT {pZT qpZGqn`1 ` .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ D´efinition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour X “ pX, a, bq un objet de Top2, on note YXpnq :“ Y pnq ab (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Lemme 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application pX, a, bq ÞÑ pXn, YXpnqq est un foncteur covariant Top2 Ñ Paires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' On v´erifie que si X “ pX, a, bq et X1 “ pX1, a1, b1q sont des objets de Top2 et si f : X Ñ X1 est un morphisme dans Top tel que fpaq “ a1 et fpbq “ b1, alors fpYXpnqq Ă YX1pnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ Lemme-D´efinition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour X P Top2, on d´efinit HnpXq :“ HnpXn, YXpnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' L’application X ÞÑ HnpXq d´efinit un foncteur covariant Hn : Top2 Ñ Paires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' D´emonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Provient de l’identification de Hn avec la composition du foncteur pX, a, bq ÞÑ pXn, YXpnqq avec le foncteur homologie relative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' □ TRANSFORMATIONS NATURELLES RELIANT FONCTEURS HOMOTOPIQUE ET HOMOLOGIQUE 31 Un corollaire du th´eor`eme 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='3 est: Th´eor`eme 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Pour n ě 0 et X un objet de Top2, le morphisme de groupes Zπ1pa, bq Ñ HnpXq, γ ÞÑ r˜γns induit un morphisme de groupes νpnq X : FnpXq Ñ HnpXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' La correspondance X ÞÑ νpnq X est une transformation naturelle de Fn vers Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Remerciements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Le travail de B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' a b´en´efici´e du soutien du projet ANR “Project HighAGT ANR20-CE40-0016”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Bibliographie [BGFr] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Burgos Gil, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Fresan, Multiple zeta values: from numbers to motives, preprint http://javier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='fresan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='perso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='cnrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='fr/mzv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='pdf, `a paraˆıtre dans Clay Mathematics Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' [DG] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Deligne, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Goncharov, Groupes fondamentaux motiviques de Tate mixte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Ann.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Harper, Algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' A first course, Mathematics Lecture Note Series, 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Benjamin/Cummings Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', Advanced Book Program, Reading, Mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=', 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' [Ha] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Hatcher, Algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' New York, Cambridge University Press, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content=' Institut de Recherche Math´ematique Avanc´ee, UMR 7501, Universit´e de Strasbourg et CNRS, 7 rue Ren´e Descartes, 67000 Strasbourg, France Email address: enriquez@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdAzT4oBgHgl3EQfN_tE/content/2301.01157v1.pdf'} +page_content='unistra.' metadata={'source': 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We construct simple geometric operations on faces of the Cayley sum +of two polytopes. These operations can be thought of as convex geometric coun- +terparts of divided difference operators in Schubert calculus. We show that these +operations give a uniform construction of Knutson–Miller mitosis (in type A) and +(simplified) Fujita mitosis (in type C) on Kogan faces of Gelfand–Zetlin polytopes. +1. Introduction +Mitosis as a mathematical notion was introduced by Knutson and Miller to gen- +erate combinatorial objects (reduced pipe dreams or rc-graphs) from a single object +[KnM, M]. Originally, pipe dreams were used in Fomin–Kirillov theorem to enu- +merate the coefficients of Schubert polynomials in type A. Recall that Schubert +polynomials are defined inductively by applying divided difference operators to a +single Schubert polynomial. Hence, mitosis operations can be thought of as com- +binatorial counterparts of divided difference operators from Schubert calculus. A +convex geometric version of mitosis was defined in [K16]: pipe dreams were replaced +by faces of polytopes from representation theory (such as Nakashima–Zelevinsky +polytopes) and divided difference operators were replaced by Demazure operators. +In the present paper, we introduce a different version of geometric mitosis (simple +geometric mitosis). It is more general and can be applied to classical Gelfand–Zetlin +polytopes in all types. The starting point for this paper was the desire to understand +the results of [F] from a geometric rather than representation theoretic viewpoint. +Simple geometric mitosis is defined on a convex polytope ∆ such that ∆ can be +represented as the Cayley sum of two of its facets. Let v be a vertex of ∆. Faces of +∆ that contain v play the role of pipe dreams. Mitosis operation acts only on faces +that contain v and assigns to a face a collection of faces of dimension one greater. +In applications to Schubert calculus, there are several different mitosis operations +associated with ∆. This is because Gelfand–Zetlin polytopes (and other polytopes +from representation theory) admit several decompositions into Cayley sums of facets. +In this case, mitosis operations generate faces (that contain v) from the single vertex +v. The faces generated this way can be used to represent Schubert cycles as sums +of faces. +Key words and phrases. Schubert calculus, mitosis, push-pull operator. +The study has been partially funded within the framework of the HSE University Basic Research +Program. +1 + +2 +VALENTINA KIRITCHENKO +The paper is organized as follows. In Section 2, we give a definition of simple +geometric mitosis. In Section 3, we recall the combinatorial definition of Knutson– +Miller mitosis on pipe dreams in type A and Fujita mitosis on skew pipe dreams in +type C. In Section 4, we relate the constructions of the previous two sections using +Gelfand–Zetlin polytopes in types A and C. In Section 4, we outline applications +to Schubert calculus. +2. Main construction +In this section, we give an elementary definition of simple geometric mitosis. In +Section 5, we explain the meaning of this definition in the setting of intersection +theory. +Let P, Q ⊂ Rd be two convex polytopes of full dimension. +Denote by ∆ := +∆(P, Q) ⊂ Rd × R their Cayley sum P ∗ Q, that is, the convex hull of +(P × 0) ∪ (Q × 1). +In what follows, we identify P and Q with facets P ×0 and Q×1 of ∆. The mitosis +operation acts only on those faces of ∆ that are contained in P (“horizontal faces”), +and produces faces of ∆ that are not contained in P. +Definition 1. The face F ⊂ P is called admissible, if there exists a unique face +exp(F) ̸= F with the property +F = exp(F) ∩ P. +It is easy to check that if F is admissible then dim exp(F) = dim F + 1. The face +exp(F) can be thought of as an expansion of F from P to ∆. +Example 2.1. Let d = 2, and let P, Q ⊂ R2 be two triangles obtained from a +unit square by cutting along a diagonal. +More precisely, take P = {(x1, x2) ∈ +R2 | x1, x2 ≤ 1, x1 + x2 ≥ 1} and Q = {(x1, x2) ∈ R2 | x1, x2 ≥ 0, x1 + x2 ≤ 1}. +Then ∆ = {(x1, x2, x3) ∈ R3 | x1, x2, x3 ≤ 1, 1 ≤ x1 +x2 +x3 ≤ 2}. The face P ⊂ ∆ +and edges of P are admissible faces, while vertices of P are not. +Let v ∈ P be a vertex. In the definition below, we consider only those faces of +P and ∆ that contain v. We call them v-faces. In particular, the mitosis operation +mitosisv(·) will depend on the choice of v. Let F ⊂ P be an admissible v-face of +dimension ℓ. We will define mitosisv(F) geometrically as a set of v-faces {E1, . . . , Ek} +of dimension ℓ + 1. +Definition 2. A v-face Ei ⊂ ∆ belongs to mitosisv(F) if Ei satisfies the following +two conditions: +(1) Ei ̸⊂ P, Q; +(2) Ei ∩ Q ⊂ exp(F) ∩ Q. +Faces in mitosisv(F) will be called offsprings of F. + +SIMPLE GEOMETRIC MITOSIS +3 +Informally, the first condition means that Ei is not a “horizontal” face, that is, Ei +intersects both P and Q. In particular, it is not possible to apply the same mitosis +operation twice because none of the faces Ei ∈ mitosisv(F) lies in P. The second +condition tells us that the face Ei is not in general position with respect to Q (unless +Ei = exp(F)). +Example 2.2. Let ∆ ⊂ R3 be a Feigin–Fourier–Littelmann–Vinberg (FFLV) poly- +tope given by inequalities 0 ≤ x1, x3 ≤ 1, 0 ≤ x2, x1 + x2 + x3 ≤ 2. Consider +faces: +P1 = {x3 = 0}, +Q1 = {x3 = 1}, +P2 = {x1 = 1}, +Q2 = {x1 = 0}. +Let v be the vertex (1, 1, 0). Clearly, there are two ways to decompose ∆ as the +Cayley sum of two polygons: ∆ = ∆(P1, Q1) and ∆ = ∆(P2, Q2). Hence, there are +two different mitosis operations mitosisv +1 and mitosisv +2 associated with these decom- +positions. +For instance, mitosisv +1(F) for the edge F = {x3 = 0, x1 + x2 + x3 = 2} consists of +two offsprings, namely, exp(F) = {x1 + x2 + x3 = 2} and P2. However, mitosisv +2(F) +consists of a single offspring for all admissible v-faces F ⊂ P2. +Remark 2.3. In what follows, we sometimes apply the mitosis operation to a set S +of faces of the same dimension. By mitosisv(S) we mean � +F ∈S mitosisv(F). +3. Combinatorial mitosis +In this section, we recall two combinatorial rules: Knutson–Miller mitosis on pipe +dreams (type A) and Fujita mitosis on skew pipe dreams (type C). Both rules can +be defined uniformly using the same combinatorial operation called two-row mitosis. +A similar operation is implicitly used in the original definition of Knutson–Miller +mitosis [KnM, M]. We first define two-row mitosis explicitly, and then define mitosis +operations in types A and C by reducing them to suitable two-row mitosis. Note +that the term row here does not necessarily mean a horizontal collection of items as +in the original definition of Knutson–Miller mitosis. +3.1. Two-row mitosis. Let A and B be two finite collections of squares such that +the number of squares in A is greater by one than the number of squares in B. +Denote the number of squares in B by ℓ. We will label the squares in A by a1, a2,. . . , +a(ℓ+1), and the squares in B by b1, b2,. . . , bℓ (like in chess notation). Symbolically, +we may represent A and B by (ℓ + 1)-row and ℓ-row of squares, respectively. In real +life examples, we might need to arrange squares of A and B in more intricate ways. +Namely, we will identify squares in A and B with specific cells in various tables. +To get a basic pipe dream we fill some squares in A and B with +. The other +squares remain empty. By size of a basic pipe dream D we mean the total number +of + in the squares of D. Two-row mitosis M is an operation on basic pipe dreams +that sends a basic pipe dream D of size s to a (possibly empty) set M(D) of basic +pipe dreams of size (s − 1). + +4 +VALENTINA KIRITCHENKO +To construct the set M(D) we use the following rules: +(1) If square a1 is empty, then M(D) = ∅. +(2) If square a1 contains +, then find the maximal index rD such that the squares +a1,. . . , arD are all filled with +. +(3) If square a1 contains +, define the set J (D) of indices: +J (D) := {j ≤ rD | square aj contains +, square bj is empty or j = ℓ + 1}. +(4) For every p ∈ J (D), construct the offspring Dp as follows. First, erase + in +square ap. Then move + from square aj down to square bj for all j ∈ J (D) +such that j < p. +Example 3.1. (cf. [M, Example 7]) Let ℓ = 5. Figure 1 shows a pipe dream D +of size 6 and three pipe dreams of size 5 that form the set M(D). In this case, +J (D) = {1, 2, 4}, and rD = 4. ++ + + + ++ +↓ ++ + + ++ ++ + ++ ++ ++ ++ + + +Figure 1. +3.2. Mitosis in type A. Recall that a pipe dream in type An can be defined as +an n × n table whose cells are either filled with + or empty. Furthermore, + is not +allowed in cell (i, j) if the cell is below the main antidiagonal of the table (that is, +i + j > n + 1). An example of pipe dream for n = 6 is given on Figure 2 (left). Pipe +dreams have an elegant interpretation in terms of networks of pipes, and are related +to permutations from Sn+1 (see [M] for details). ++ + + + ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +Figure 2. Pipe dreams in type A6 (left) and C4 (right). +There are n different mitosis operations MA +1 ,. . . , MA +n on pipe dreams in type An. +Informally, mitosis operation MA +i can be defined as the two-row mitosis applied to +rows i and i + 1 of a pipe dream (the other rows are not affected by MA +i ). Note +that mitosis operation MA +n is also well-defined though row n + 1 is an empty set of + +SIMPLE GEOMETRIC MITOSIS +5 +boxes. This is because row n might have at most one +, that is, no + will have to +be moved down according to the mitosis rules. +We now define mitosis operation MA +i more formally. Let D be a pipe dream in type +An. Put ℓ = n−i, and label cells (i, 1),. . . , (i, n−i+1) by a1,. . . , a(ℓ+1), respectively. +Label cells (i+1, 1),. . . , (i+1, n−i) by b1,. . . , bℓ, respectively. Extract the basic pipe +dream Di from D by setting Ai = (a1, . . . , a(ℓ + 1)) and Bi = (b1, . . . , bℓ). Apply +two-row mitosis to Di. Complete every resulting offspring Di +p to a pipe dream Dp +by replacing Di with Di +p in D. Define MA +i (D) as the set {Dp | p ∈ J (Di)}. +Example 3.2. Let D be a pipe dream depicted on Figure 2 (left). For i = 1, the +basic pipe dream D1 coincides with the one on Figure 1 (top) from Example 3.1. +Hence, MA +1 (D) consists of the pipe dreams shown on Figure 3. If i = 2, 3, 4, 6, then ++ + + ++ ++ ++ ++ ++ + ++ ++ ++ ++ ++ ++ ++ + + ++ ++ ++ +Figure 3. +MA +i (D) is empty. Finally, MA +5 (D) consists of a single offspring obtained from D by +erasing the + in the fifth row. +Remark 3.3. It is easy to check that MA +i +coincides with the i-th mitosis operator +mitosisi introduced in [M, Definition 6] (Knutson–Miller mitosis). There is also a +strong relationship between MA +i +and the operator Mi introduced in [F, Section 5] +using representation theoretic considerations. These operations coincide whenever +D does not have any + in cells to the right of rDi. While MA +i and Mi might differ on +the other pipe dreams (the latter operation might produce more offsprings than the +former) it is interesting that both operations lead to the same results in Schubert +calculus (see [F, Corollary 5.13] and the preceding discussion for more details). In +particular, [M, Theorem 15] still holds if mitosisi is replaced by Mi, that is, Mi can +be viewed as an alternative version of the Knutson–Miller mitosis. +3.3. Mitosis in type C. Similarly to type A case, pipe dreams in type Cn (also +called skew pipe dreams) can be defined as n × (2n − 1) tables filled with +. In +type Cn case, + is not allowed in cell (i, j) if i + j > 2n or i > j. An example of +a skew pipe dream for n = 4 is given on Figure 2 (right). Skew pipe dreams were +recently used in [F] to construct a convex geometric model for Schubert calculus +in type C in terms of symplectic GZ polytopes (we say more about this in Section +4), in particular, they are related to signed permutation. It would be interesting to +find an interpretation of skew pipe dreams in terms of networks of pipes. In [ST], +c-signed pipe dreams were defined as networks of pipes with extra features. We do +not know of any direct relation between skew pipe dreams and c-signed pipe dreams. + +6 +VALENTINA KIRITCHENKO +We now define n mitosis operations MC +1 ,. . . , MC +n on a skew pipe dream D. The +case i = 1 is special. In this case, set ℓ = n − 1. Label cells (1, n), (2, n),. . . , (n, n) +by a1, a2,. . . , a(ℓ + 1), respectively (these are cells in the middle column of D). +Label cells (1, n + 1), (2, n + 1),. . . , (n − 1, n + 1) by b1, b2,. . . , bℓ, respectively. If +i = 2,. . . , n, set ℓ = 2(n−i)+1. Label cells (1, n−i+1), (1, n+i−1), (2, n−i+1), +(2, n + i + 1),. . . , (n − i + 1, n − i + 1), (n − i + 1, n + i − 1) by a1, a2, a3, a4,. . . , +aℓ, a(ℓ + 1), respectively (these are all cells in columns n ± (i − 1) of D). Label cells +(1, n − i + 2), (1, n + i), (2, n − i + 2), (2, n + i),. . . , (n − i + 1, n − i + 2) by b1, b2, +b3, b4,. . . , bℓ, respectively (here we alternate first n − i + 1 cells in column n − i + 2 +with cells in column n + i of D). The rest of the definition of MC +i +is completely +analogous to type A case. +Example 3.4. Let D be a skew pipe dream from Figure 2 (right). Using Example +3.1 again we get that MC +1 (D) consists of skew pipe dreams depicted on Figure 4. +For i = 2, 3, the set MC +i (D) is empty. The set MC +4 (D) consists of a single offspring ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ + ++ ++ ++ ++ +Figure 4. +obtained by erasing the + in the first column of D. +Remark 3.5. Mitosis operation MC +i +on skew pipe dreams is different from operator +Mi defined in [F, Section 6]. As in type A (see Remark 3.3), the difference lies in +the restriction j ≤ rD in Definition 2. It would be interesting to check whether both +operations still produce the same results in applications to Schubert calculus. +4. Mitosis on GZ polytopes in types A and C +In this section, we apply simple geometric mitosis to GZ polytopes in types A and +C. We compare the resulting operations with geometric realizations of Knutson– +Miller mitosis and Fujita mitosis. +4.1. Gelfand–Zetlin polytopes in type An. Put d := n(n+1) +2 +. A GZ table of type +An is a collection of cells organized according to the pattern on Figure 5 (left). Let +λ := (λ1 ≥ λ2 ≥ . . . ≥ λn+1) be a decreasing collection of real numbers. Identify a +point (x1 +1, x1 +2, . . . , x1 +n; x2 +1, . . . , x2 +n−1; . . . ; xn−1 +1 +, xn−1 +2 +; xn +1) ∈ Rd with the GZ table whose +i-th row is filled with coordinates (xi +1, xi +2, . . . , xi +n−i+1) for 1 ≤ i ≤ n. Define the +GZ polytope GZA +λ ⊂ Rd by 2d interlacing inequalities xi−1 +j +≥ xi +j ≥ xi−1 +j+1 (we put +x0 +j := λj). In terms of the GZ table, these inequalities tell us that the coordinate in +any given cell lies between coordinates in two upper neighbors of this cell. +Let v be the Kogan vertex of GZA +λ . Recall that the Kogan vertex is a unique +vertex of the polytope GZA +λ that satisfies simultaneously all d equations of type + +SIMPLE GEOMETRIC MITOSIS +7 +λ1 +λ2 +λ3 +λ4 +λ5 +λ6 +λ7 +λ1 +λ2 +λ3 +λ4 +0 +0 +0 +0 +Figure 5. Gelfand–Zetlin patterns in type A6 (left) and type C4 (right). +xi−1 +j += xi +j. A face Γ ⊂ GZA +λ is called a Kogan face if Γ contains v. These faces were +first considered by Mikhail Kogan [Ko]. In particular, v-faces used in the definition +of mitosis in Section 2 are exactly Kogan faces for this choice of v. +Note that the polytope GZA +λ for λ = (n, n − 1, . . . , 1, 0) can be represented as +the Cayley sum of two polytopes in n differents ways. Namely, let Pi and Qi be +the facets of GZA +λ given by equations x1 +i = λi and x1 +i = λi+1, respectively. Then +∆(Pi, Qi) coincides with GZA +λ up to a parallel translation. Clearly, v ∈ P1,. . . , Pn. +Hence, we can define n different mitosis operations mitosisv +1,. . . , mitosisv +n on Kogan +faces of GZA +λ as in Section 2. In particular, Example 2.2 describes these operations +for n = 2 because the GZ polytope in this case is unimodularly equivalent to the +FFLV polytope. +By definition, any Kogan face is given only by equations of type xi−1 +j += xi +j. Hence, +every Kogan face F can be encoded by a GZ table filled with +. Namely, if F lies +in the hyperplane {xi−1 +j += xi +j} then put + in cell (i, j) (that is, in the j-th cell of +the i-th row). Otherwise, leave cell (i, j) empty. The resulting GZ table filled with ++ will be called the diagram of the face F and denoted by D(F). +Define a bijective correspondence between pipe dreams in type An and diagrams +of Kogan faces of GZA +λ by identifying cell (i, j) in a GZ table with cell (j, i) in a pipe +dream. The correspondence is illustrated by Figure 7 (left). Namely, inscribe the +words “GELFAND ZETLIN POLYTOPE” (without spaces) inside a pipe dream of +type A6 in usual way, that is, start from the top row and write from left to right. +After switching to GZ table, the words will transform to the sequence of letters on +Figure 7 (left). +Example 4.1. The pipe dream on Figure 2 (left) corresponds to the Kogan face F +given by eight equations λ1 = x1 +1 = x2 +1 = x3 +1 = x4 +1, x2 +2 = x3 +2, λ3 = x1 +3, λ4 = x1 +4, +λ5 = x1 +5. In particular, dim F = 13. It is easy to check that mitosisv +1(F) consists of +three Kogan faces of dimension 14 whose diagrams correspond to pipe dreams on +Figure 3. +Using the bijection between pipe dreams and GZ tables we can extend mitosis +operations MA +1 ,. . . , MA +n from pipe dreams to GZ tables. Informally speaking, we +replace rows with columns when applying two-row mitosis. We will use the same + +8 +VALENTINA KIRITCHENKO +G +E +L +F +A +N +D +Z +E +T +L +I +N +P +O +L +Y +T +O +P +E +S H C U +B E R +T C A +L C +U L +U +S +Figure 6. Correspondence between pipe dreams and GZ patterns in +type A6 (left) and type C4 (right). +notation for mitosis operations on GZ tables as it will be clear from the context +whether we talk about pipe dreams or about GZ tables. +It turns out that geometric operation mitosisv +i on certain Kogan faces is combina- +torially equivalent to mitosis operation MA +i on the corresponding pipe dreams. The +following theorem describes the precise relation between the geometric mitosis on +Kogan faces of a GZ polytope and the combinatorial mitosis on GZ tables. +Theorem 4.2. Let F ⊂ GZA +λ be a Kogan face such that its diagram D(F) has + in +cell (1, i) and no + in cells (j, i + 1) for all j. Then mitosisv +i (F) consists of Kogan +faces whose diagrams are obtained from D(F) by the mitosis operation MA +i +on GZ +tables. +Proof. In order to compute mitosisv +i (F) we have to consider not only Kogan faces. +In general, a face of GZA +λ is given by equations of two types: either xi−1 +j += xi +j (type +A) or xi +j = xi−1 +j+1 (type B). +Definition 3. Following [Ko] define equations Ai,j and Bi,j, respectively, as xi−1 +j += xi +j +and xi +j = xi−1 +j+1. +We now apply Definition 2 to P := Pi and Q := Qi. Note that any Kogan face +is admissible because it contains a simple vertex, namely, the Kogan vertex v. The +face E1 := exp(F) is obtained from F by removing a single equation A1,i. Hence, the +face exp(F) ∩ Q is given by the same equations as F with one exception: equation +A1,i is replaced by equation B1,i. +Let E be a Kogan face of dimension ℓ + 1 (where ℓ := dim F) such that E ∩ Q ⊂ +exp(F)∩Q. Let (X1,. . . , Xd−ℓ−1) be the collection of equations of type A that define +E. If E ̸= exp(F) then there exists an equation Y (of type A) such that exp(F) +satisfies Y but Y does not follow from (X1,. . . , Xd−ℓ−1). However, Y should follow +from equations B1,i, X1,. . . , Xd−ℓ−1. Hence, the collection of equations (B1,i, Y , +X1,. . . , Xd−ℓ−1) is redundant, while the collection (Y , X1,. . . , Xd−ℓ−1) is not. This +is only possible if A1,i+1 is contained among the equations X1,. . . , Xd−ℓ−1 as the +following lemma shows: +Lemma 4.3. Let (X1, . . . , Xs) be a collection of equations of type A that does not +contain equation Ap,q. If Ap,q follows from (Bj,i, X1, . . . , Xs) then three conditions +hold: + +SIMPLE GEOMETRIC MITOSIS +9 +(1) q = i; +(2) p > j; +(3) (X1, . . . , Xs) contains Ak,i+1 for all k such that j ≤ k < p. +Proof. The statement follows directly from Definition 3. +□ +If (X1, X2,. . . , Xd−ℓ−1) contains A1,i+1, A2,i+1,. . . , Ak,i+1 but does not contain +Ak+1,i+1 for some k ≥ 1, then A2,i,. . . , Ak+1,i are the only equations that follow from +(B1,i, X1,. . . , Xd−ℓ−1). Hence, if F satisfies Ap,q, and Ap,q is not contained among +(X1, X2,. . . , Xd−ℓ−1)) then Ap,q must coincide with one of the equations A2,i,. . . , +Ak+1,i. For dimension reasons, all k equations A2,i,. . . , Ak+1,i must be absent in (X1, +X2,. . . , Xd−ℓ−1) (to compensate for the presence of k equations A1,i+1, A2,i+1,. . . , +Ak,i+1). In particular, if F does not satisfy A2,i then mitosisv +i (F) consists of a single +offspring exp(F). +We now proceed by induction on r where r is the maximal number such that F +satisfies the equations A1,i,. . . , Ar,i. Let Γr,i be the facet given by equation Ar,i, and +F ′ the face given by the same equations as F except for Ar,i, that is, F = F ′ ∩ Γr,i. +It is easy to check that there is a bijection between mitosisv +i (F ′) and the subset +S ⊂ mitosisv +i (F) that consists of all faces Ei ∈ mitosisv +i (F) such that Ei ⊂ Γr,i. +Namely, a face E′ ∈ mitosisv +i (F ′) corresponds to the face E = E′∩Γr,i ∈ mitosisv +i (F). +Hence, the faces in S can be described by the induction hypothesis applied to F ′. +It remains to describe the faces Ei ∈ mitosisv +i (F)\S. By condition (2) of Definition +2 and part (3) of Lemma 4.3 applied to Ar,i, we have that if Ei ̸⊂ Γr,i, then Ei +must satisfy equations A1,i+1, A2,i+1. . . , Ar−1,i+1. Hence, there is a single offspring +Er ∈ mitosisv +i (F) \ S obtained from F by removing equation Ar,i and by replacing +equations Aj,i by equations Aj,i+1 for all j < r. +□ +We now adapt geometric mitosis operations so that Theorem 4.2 holds for all +reduced Kogan faces. The definition of reduced pipe dreams can be found in [M, Def- +inition 1] (these pipe dreams are mostly used in applications to Schubert calculus). +A Kogan face F is reduced if its diagram D(F) is reduced. Below we use notation +of Definition 3 from the proof of Theorem 4.2. +Corollary 4.4. Let F ⊂ GZA +λ be a reduced Kogan face. Define an adapted geometric +mitosis operation mitosisv +i as follows. +(1) Consider the face env(F) ⊂ GZA +λ given by equations Ak,i and Ak,i+1 for all +k such that F satisfies both Ak,i and Ak,i+1. +(2) Find the minimal index s such that env(F) does not satisfy equation As,i. +Consider the facets P F +i , QF +i +⊂ env(F) given by equations As,i and Bs,i, +respectively. +(3) Define mitosisv +i (F) by applying Definition 2 to ∆ = env(F), P = P F +i +and +Q = QF +i . +Then mitosisv +i (F) consists of exactly those Kogan faces whose diagrams lie in the set +MA +i (D(F)). + +10 +VALENTINA KIRITCHENKO +The proof of Corollary 4.4 is completely analogous to the proof of Theorem 4.2. +Remark 4.5. Note that the set mitosisv +i (F) might, in general, contain more faces +than the set mitosisv +i (F). By applying mitosis inside the face env(F) instead of the +whole GZA +λ we get rid of these extra faces. The definition of env(F) in Corollary +4.4 is ad hoc, it relies heavily on combinatorics of GZ polytopes in type A. It would +be interesting to find a more geometric definition of env(F). +The choice of P F +i +and QF +i stems from the classical inductive construction of GZ +bases and polytopes based on the chain of subgroups GL2(C) ⊂ GL3(C) ⊂ . . . ⊂ +GLn+1(C). +4.2. Gelfand–Zetlin polytopes in type C. We now describe analogous construc- +tions in type C. We omit details that are the same as in type A case and focus on +unique features of type C case. Put d = n2, and put λ := (λ1 ≥ λ2 ≥ . . . ≥ λn ≥ +λn+1 = 0). A GZ table of type Cn is defined according to the pattern on Figure 5 +(right). Roughly speaking, a GZ table of type Cn is a half of a GZ table of type +A2n. As in type A case, the GZ polytope GZC +λ ⊂ Rd is defined by 2d interlacing +inequalities that come from a GZ pattern in type C. Again, the polytope GZC +λ +for λ = (n, n − 1, . . . , 0) can be represented as Cayley sum of two polytopes in n +different ways. +Let v be the symplectic Kogan vertex of GZC +λ as defined in [F, Section 6]. Using +notation from Definition 3 (see the proof of Theorem 4.2) and regarding a GZ pattern +of type Cn as part of a GZ pattern of type A2n−1 we may define v by equations Ai,j +for all odd i ≤ 2n − 1 and equations Bi,j for all even i < 2n − 1. Similarly to type +A case, a face Γ ⊂ GZC +λ is called a symplectic Kogan face if Γ contains v. Again, +there are n mitosis operations mitosisv +1,. . . , mitosisv +n. +As in type A case, every symplectic Kogan face F can be encoded by a GZ table +D(F) of type C filled with +. Define bijective correspondence between skew pipe +dreams and diagrams of Kogan faces as in [F, Section 6]. For instance, if words +“SCHUBERT CALCULUS” (without spaces) are inscribed into a skew pipe dream +of type C4 in usual way then they get transformed into boustrophedon1 writing in +a GZ pattern of type C4 on Figure 7 (right). Again, we extend mitosis operations +MC +1 ,. . . , MC +n from skew pipe dreams to GZ tables in type Cn. +Note that MC +1 is combinatorially different from MC +2 ,. . . , MC +n . In applications to +Schubert calculus, mitosis operations correspond to generators of the group of signed +permutations Bn on n elements. There is a special generator s1 : (1, 2, . . . , n) �→ +(−1, 2, . . . , n) (change of sign), and the elementary transpositions s2,. . . ,sn, namely, +si = (i − 1 i). If we regard Bn as the Weyl group of the symplectic group Sp2n(C), +then s1 is the simple reflection corresponding to the longer root. The mitosis opera- +tion MC +n corresponds to the special generator s1, hence, it is natural to expect that +MC +n will be special. +1I am grateful to Evgeny Smirnov from whom I learnt the proper name of this writing style. + +SIMPLE GEOMETRIC MITOSIS +11 +The following theorem is analogous to Theorem 4.2 and can be proved using +similar arguments. +Theorem 4.6. Let F ⊂ GZC +λ be a symplectic Kogan face. +(1) Let i = 1,. . . , n − 1. If the diagram D(F) has + in cell (1, i), and no + in +cells (2k + 1, i − k + 1) and (2k, i − k) for all k, then mitosisv +i (F) consists of +faces whose diagrams are obtained from D(F) by applying MC +n−i+1. +(2) If the diagram D(F) has + in cell (1, n), and no + in cells (2k, n − k) then +mitosisv +n(F) consists of faces whose diagrams are obtained from D(F) by +applying MC +1 . +5. Applications to Schubert calculus +We now explain how the notion of simple geometric mitosis fits into in the context +of intersection theory. In particular, we relate Corollary 4.4 and Theorem 4.6 with +the analogous results on Schubert calculus in types A and C. Below we use the +notion of polytope ring (aka Khovanskii–Pukhlikov ring). +The definition can be +found, for instance, in [K21, Section 2]. We refer the reader to [F] for more details +on applications of Khovanskii–Pukhlikov rings to Schubert calculus. +We use notation of Section 2. Let R∆ and RP be the polytope rings of ∆ and P. +Assume that R∆ and RP, respectively, are isomorphic to the Chow rings of smooth +varieties Y and X, where Y = P(E) is the projectivization of a rank two vector +bundle E on X (see [K21, Section 4] for motivation behind such an assumption). +Let p be the natural projection from Y to X. Then there is a push-pull operator +p∗p∗ : CH∗(Y ) → CH∗−1(Y ), +which is a homomorphism of CH∗(X)-modules. The simple geometric mitosis is an +attempt to describe explicitly the action of the push-pull operator on R∆ ≃ CH∗(Y ) +using representations of elements of R∆ by linear combinations of faces of ∆. It +would be interesting to formalize the connection between Definition 2 and the action +of p∗p∗ on faces of ∆ in the general setting. Below we will exhibit such a connection +in the special case of GZ polytopes and flag varieties. +Let ∆ be the GZ polytope for a classical group G and a strictly dominant weight +λ. Let B ⊂ G denote a Borel subgroup of G, and let Y := G/B denote the complete +flag variety for G. The ring R∆ is isomorphic to the subring of CH∗(Y ) generated by +the first Chern classes of line bundles on G/B [Ka, Corollary 5.3, Remark 2.4]. For +G = GLn+1(C) and Sp2n(C), this subring coincides with CH∗(Y ) (in general, the +discrepancy between the subring and the whole CH∗(Y ) is measured by the torsion +index of G, see [T] for more details). In particular, if λ = ρ := (n, n − 1, . . . , 1, 0), +then R∆ ≃ CH∗(Y ) in types A and C. +Denote by α1,. . . , αn the simple roots of G. Let Xi be the partial flag variety +G/Pi for the minimal parabolic subgroup Pi corresponding to αi. Note that the +natural projection pi : Y → Xi turns Y into a P1-fibration (it can be realized as the +projectivization of the rank two bundle Ei := pi∗L(ρ) where L(ρ) is the line bundle + +12 +VALENTINA KIRITCHENKO +on Y corresponding to the weight ρ). The classical divided difference (or push-pull) +operator in Schubert calculus is defined as ∂i := p∗ +i pi∗. +Recall that the operators ∂1,. . . , ∂n are used to generate the Schubert classes +[Xw] ∈ CH∗(Y ) starting from the class [Xid] of a point. More precisely, let w ∈ W +be an element of the Weyl group of G. +Choose a reduced decomposition w = +si1 · · · siℓ. Here si denotes the reflection with respect to the root αi. Then [Xw] = +∂iℓ · · ·∂i1[Xid]. +If we choose the Kogan vertex v as a representative of the class +[Xid] in R∆, then the action of ∂iℓ · · · ∂i1 on [Xid] can be computed using mitosis +operations as follows. +Theorem 5.1. Let G = GLn+1(C) and ∆ = GZA +ρ . +Under the isomorphism +CH∗(Y ) ≃ R∆ the Schubert cycle [Xw] can be represented as the class of the sum of +faces F ⊂ ∆ where F runs through the set +Sw = mitosisv +iℓ · · ·mitosisv +i1(v). +In particular, the action of divided difference operator ∂i on the Schubert cycle [Xw] +gets represented by the action of mitosisv +i on faces from Sw: +Swsi = mitosisv +i (Sw). +This theorem follows from [F, Corollary 5.18] together with Corollary 4.4. Note +that the proof of [F, Theorem 5.17] (which implies [F, Corollary 5.18]) uses repre- +sentation theoretic arguments. It would be interesting to find a convex geometric +proof. +Remark 5.2. In [K16], different geometric mitosis operations are defined. +They +mimick Demazure operators rather than push-pull operators ∂i. In type A, they +can also be used to prove theorems analogous to Theorem 5.1 or [F, Corollary 5.18] +due to a special symmetry of GZ diagrams in type A. Namely, the diagrams are +symmetric with respect to the reflection (i, j) �→ (j, i). +Note that the induction step in [K16, Corollary 3.6] goes from siw to w (not from +wsi to w as in Theorem 5.1 and in [F, Theorem 5.17]). In other words, induction goes +along initial subwords of si1si2 . . . siℓ (not along terminal subwords). This difference +is matched by the difference between mitosis as defined in [K16] and its transpose (or +mirror) mitosis as defined in the present paper. In particular, all arguments with +the mitosis operations applied to w = si1si2 . . . siℓ can be immediately translated +into analogous arguments with the transpose mitosis operations applied to w−1 = +siℓsiℓ−1 . . . si1 (see [F, Section 5] for more details). +In type C, Theorem 4.6 together with Remark 3.5 and [F, Theorem 6.8] suggest +that mitosisv +i can be adapted so that under the isomorphism CH∗(Y ) ≃ R∆ the +Schubert class [Xw] can be represented as the class of the sum of faces F ⊂ ∆ where +F runs through the set +Sw = mitosisv +n−iℓ+1 · · ·mitosisv +n−i1+1(v). + +SIMPLE GEOMETRIC MITOSIS +13 +In [F, Corollary 6.13], another presentation of Schubert cycles by faces of GZC +ρ +is obtained using the dual Kogan faces. Similarly to the definition of symplectic +Kogan vertex in Section 4.2, the dual Kogan vertex v∗ can be defined by equations +Bi,j for all odd i ≤ 2n−1 and equations Ai,j for all even i < 2n−1. Again there are +n geometric mitosis operations mitosisv∗ +1 ,. . . , mitosisv∗ +n where mitosisv∗ +i +corresponds +to the decomposition GZC +ρ = {x1 +i = λi+1} ⋆ {x1 +i = λi} (that is, Pi and Qi from +the definition of mitosisv +i switch places when defining mitosisv∗ +i ). +In the case of +Sp4(C), these operations produce part of the presentation of Schubert cycles from +[F, Corollary 6.13] as the following example shows. +Example 5.3. Let n = 2. We encode dual symplectic Kogan faces by their diagrams. +We put A and B instead of + just for clarity (not because it carries any extra infor- +mation). This way it is easier to distinguish diagrams of Kogan faces from diagrams +of dual Kogan faces. In the second row, we use an adapted mitosis once (this adap- +tation is based on the chain Sp2(C) = SL2(C) ⊂ Sp4(C)). These collections of faces +B B +mitosisv∗ +1 +−→ +id +A +B +B +mitosisv∗ +2 +−→ +s2 +A +B +B +& +s2s1 +B +s2s1 +mitosisv∗ +1 +−→ +A +B +B +& +s2s1s2 +s2s1s2 +B +B B +mitosisv∗ +2 +−→ +id +A +B +B B +mitosisv∗ +1 +−→ +s1 +A +B +mitosisv∗ +2 +−→ +s1s2 +A +B +& +s1s2s1 +s1s2s1 +A +Figure 7. Mitosis for the dual Kogan vertex in type C2. +are the same as the presentations of Schubert cycles [Xw] in [F, Corollary 6.13] for +all w ∈ W except for s2s1. In the case w = s2s1, one face is missing. +Recall that according to [F, Corollary 6.13], the dual Kogan faces that represent +[Xw] are in bijective correspondence with those reduced subwords of the longest +word +w0 = (s1)(s2s1s2)(s3s2s1s2s3) . . . (snsn−1 . . . s2s1s2 . . . sn−1sn) +that represent the element w0w. The bijection is obtained by inscribing the word +w0 into the symplectic GZ pattern using reverse boustrophedon (see Figure 8 left). +For instance, the subword +(s1)(✚✚ +s2✚✚ +s1s2)(s3s2s1s2✚✚ +s3)(✚✚ +s4s3✚✚ +s2✚✚ +s1s2s3s4) +in type C4 corresponds to the dual Kogan face on Figure 8 right. +In particular, there are three dual Kogan faces that represent [Xw] for w = s2s1 +in type C2. Indeed, w0w = s1s2, and there are three reduced subwords of s1(s2s1s2) +that represent s1s2. +Remark 5.4. In type A, there is no combinatorial difference between presentation +of Schubert cycles by Kogan faces and presentation by dual Kogan faces (see [F, + +14 +VALENTINA KIRITCHENKO +s4 s3 s2 s1 +s2 +s3 +s4 +s3 s2 s1 +s2 +s3 +s2 s1 +s2 +s1 +B B B +A +B B +A +A +B +B +Figure 8. Correspondence between subwords of w0 and dual Kogan +faces in type C4 +Theorem 5.25]). Both presentations can be related using the automorphism of the +Dynkin diagram of type An (on the level of GZ diagrams this automorphism corre- +sponds to the reflection (i, j) �→ (i, n+2−i−j)). In type C, there is a combinatorial +difference already for n = 2 (see [F, Section 1] for more details). In type A, one +can immediately recover mitosis on dual Kogan faces from mitosis on Kogan faces. +Both operations will be combinatorially equivalent to the Knutson–Miller mitosis. +In type C, the Fujita mitosis on Kogan faces does not yield a combinatorially equiv- +alent mitosis on dual Kogan faces. +Example 5.3 shows that Definition 2 is too simple to capture completely the action +of p∗p∗ on faces of ∆ in the general setting. However, it may be regarded as the first +approximation of this action. It can also be used to make an educated guess about +a possible combinatorial mitosis on dual Kogan faces in type C and mitosis in type +D. +References +[F] +Naoki Fujita, Schubert calculus from polyhedral parametrizations of Demazure crystals, +Adv. in Math., 397 (2022) +[Ka] Kiumars Kaveh, Note on cohomology rings of spherical varieties and volume polynomial, +J. Lie Theory 21 (2011), no. 2, 263–283 +[K16] Valentina Kiritchenko, Geometric mitosis, Math. Res. Lett., 23 (2016), no. 4, 1069– +1096 +[K21] Valentina +Kiritchenko, +Push-pull +operators +on +convex +polytopes, +IMRN, +https://doi.org/10.1093/imrn/rnab331 +[KP] Valentina Kiritchenko and Maria Padalko, Schubert calculus on Newton–Okounkov +polytopes, Chapter 11 in Interactions with lattice polytopes, Springer, 2022 +[KnM] Allen Knutson and Ezra Miller, Gr¨obner geometry of Schubert polynomials, Ann. of +Math. (2), 161 (2005), 1245–1318 +[Ko] Mikhail Kogan, Schubert geometry of flag varieties and Gelfand–Cetlin theory, Ph.D. the- +sis, MIT, 2000 +[M] +Ezra Miller, Mitosis recursion for coefficients of Schubert polynomials, J. Comb. Theory +A, 103 (2003), no. 2, 223–235 +[ST] Evgeny Smirnov and Anna Tutubalina, Pipe dreams for Schubert polynomials of the +classical groups, preprint arXiv:2009.14120 [math.CO] +[T] +Burt Totaro, The torsion index of the spin groups, Duke Math. J., 129 (2005), no.2, +249–290 + +SIMPLE GEOMETRIC MITOSIS +15 +Email address: vkiritch@hse.ru +Laboratory of Algebraic Geometry and Faculty of Mathematics, National Re- +search University Higher School of Economics, Usacheva str. 6, 119048 Moscow, +Russia +Institute for Information Transmission Problems, Moscow, Russia + diff --git a/q9AyT4oBgHgl3EQfZfeI/content/tmp_files/load_file.txt b/q9AyT4oBgHgl3EQfZfeI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2f37650c653f971e7477540de7998974f396c43 --- /dev/null +++ b/q9AyT4oBgHgl3EQfZfeI/content/tmp_files/load_file.txt @@ -0,0 +1,665 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf,len=664 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='00225v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='CO] 31 Dec 2022 SIMPLE GEOMETRIC MITOSIS VALENTINA KIRITCHENKO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We construct simple geometric operations on faces of the Cayley sum of two polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' These operations can be thought of as convex geometric coun- terparts of divided difference operators in Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We show that these operations give a uniform construction of Knutson–Miller mitosis (in type A) and (simplified) Fujita mitosis (in type C) on Kogan faces of Gelfand–Zetlin polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Introduction Mitosis as a mathematical notion was introduced by Knutson and Miller to gen- erate combinatorial objects (reduced pipe dreams or rc-graphs) from a single object [KnM, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Originally, pipe dreams were used in Fomin–Kirillov theorem to enu- merate the coefficients of Schubert polynomials in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Recall that Schubert polynomials are defined inductively by applying divided difference operators to a single Schubert polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, mitosis operations can be thought of as com- binatorial counterparts of divided difference operators from Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A convex geometric version of mitosis was defined in [K16]: pipe dreams were replaced by faces of polytopes from representation theory (such as Nakashima–Zelevinsky polytopes) and divided difference operators were replaced by Demazure operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In the present paper, we introduce a different version of geometric mitosis (simple geometric mitosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It is more general and can be applied to classical Gelfand–Zetlin polytopes in all types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The starting point for this paper was the desire to understand the results of [F] from a geometric rather than representation theoretic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Simple geometric mitosis is defined on a convex polytope ∆ such that ∆ can be represented as the Cayley sum of two of its facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let v be a vertex of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Faces of ∆ that contain v play the role of pipe dreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis operation acts only on faces that contain v and assigns to a face a collection of faces of dimension one greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In applications to Schubert calculus, there are several different mitosis operations associated with ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This is because Gelfand–Zetlin polytopes (and other polytopes from representation theory) admit several decompositions into Cayley sums of facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In this case, mitosis operations generate faces (that contain v) from the single vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The faces generated this way can be used to represent Schubert cycles as sums of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Schubert calculus, mitosis, push-pull operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The study has been partially funded within the framework of the HSE University Basic Research Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 1 2 VALENTINA KIRITCHENKO The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In Section 2, we give a definition of simple geometric mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In Section 3, we recall the combinatorial definition of Knutson– Miller mitosis on pipe dreams in type A and Fujita mitosis on skew pipe dreams in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In Section 4, we relate the constructions of the previous two sections using Gelfand–Zetlin polytopes in types A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In Section 4, we outline applications to Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Main construction In this section, we give an elementary definition of simple geometric mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In Section 5, we explain the meaning of this definition in the setting of intersection theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let P, Q ⊂ Rd be two convex polytopes of full dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Denote by ∆ := ∆(P, Q) ⊂ Rd × R their Cayley sum P ∗ Q, that is, the convex hull of (P × 0) ∪ (Q × 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In what follows, we identify P and Q with facets P ×0 and Q×1 of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The mitosis operation acts only on those faces of ∆ that are contained in P (“horizontal faces”), and produces faces of ∆ that are not contained in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The face F ⊂ P is called admissible, if there exists a unique face exp(F) ̸= F with the property F = exp(F) ∩ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It is easy to check that if F is admissible then dim exp(F) = dim F + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The face exp(F) can be thought of as an expansion of F from P to ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let d = 2, and let P, Q ⊂ R2 be two triangles obtained from a unit square by cutting along a diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' More precisely, take P = {(x1, x2) ∈ R2 | x1, x2 ≤ 1, x1 + x2 ≥ 1} and Q = {(x1, x2) ∈ R2 | x1, x2 ≥ 0, x1 + x2 ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then ∆ = {(x1, x2, x3) ∈ R3 | x1, x2, x3 ≤ 1, 1 ≤ x1 +x2 +x3 ≤ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The face P ⊂ ∆ and edges of P are admissible faces, while vertices of P are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let v ∈ P be a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In the definition below, we consider only those faces of P and ∆ that contain v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We call them v-faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, the mitosis operation mitosisv(·) will depend on the choice of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let F ⊂ P be an admissible v-face of dimension ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We will define mitosisv(F) geometrically as a set of v-faces {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ek} of dimension ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A v-face Ei ⊂ ∆ belongs to mitosisv(F) if Ei satisfies the following two conditions: (1) Ei ̸⊂ P, Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2) Ei ∩ Q ⊂ exp(F) ∩ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Faces in mitosisv(F) will be called offsprings of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' SIMPLE GEOMETRIC MITOSIS 3 Informally, the first condition means that Ei is not a “horizontal” face, that is, Ei intersects both P and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, it is not possible to apply the same mitosis operation twice because none of the faces Ei ∈ mitosisv(F) lies in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The second condition tells us that the face Ei is not in general position with respect to Q (unless Ei = exp(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let ∆ ⊂ R3 be a Feigin–Fourier–Littelmann–Vinberg (FFLV) poly- tope given by inequalities 0 ≤ x1, x3 ≤ 1, 0 ≤ x2, x1 + x2 + x3 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Consider faces: P1 = {x3 = 0}, Q1 = {x3 = 1}, P2 = {x1 = 1}, Q2 = {x1 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let v be the vertex (1, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Clearly, there are two ways to decompose ∆ as the Cayley sum of two polygons: ∆ = ∆(P1, Q1) and ∆ = ∆(P2, Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, there are two different mitosis operations mitosisv 1 and mitosisv 2 associated with these decom- positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For instance, mitosisv 1(F) for the edge F = {x3 = 0, x1 + x2 + x3 = 2} consists of two offsprings, namely, exp(F) = {x1 + x2 + x3 = 2} and P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' However, mitosisv 2(F) consists of a single offspring for all admissible v-faces F ⊂ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In what follows, we sometimes apply the mitosis operation to a set S of faces of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' By mitosisv(S) we mean � F ∈S mitosisv(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Combinatorial mitosis In this section, we recall two combinatorial rules: Knutson–Miller mitosis on pipe dreams (type A) and Fujita mitosis on skew pipe dreams (type C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Both rules can be defined uniformly using the same combinatorial operation called two-row mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A similar operation is implicitly used in the original definition of Knutson–Miller mitosis [KnM, M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We first define two-row mitosis explicitly, and then define mitosis operations in types A and C by reducing them to suitable two-row mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the term row here does not necessarily mean a horizontal collection of items as in the original definition of Knutson–Miller mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Two-row mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let A and B be two finite collections of squares such that the number of squares in A is greater by one than the number of squares in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Denote the number of squares in B by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We will label the squares in A by a1, a2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , a(ℓ+1), and the squares in B by b1, b2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , bℓ (like in chess notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Symbolically, we may represent A and B by (ℓ + 1)-row and ℓ-row of squares, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In real life examples, we might need to arrange squares of A and B in more intricate ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, we will identify squares in A and B with specific cells in various tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' To get a basic pipe dream we fill some squares in A and B with +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The other squares remain empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' By size of a basic pipe dream D we mean the total number of + in the squares of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Two-row mitosis M is an operation on basic pipe dreams that sends a basic pipe dream D of size s to a (possibly empty) set M(D) of basic pipe dreams of size (s − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 4 VALENTINA KIRITCHENKO To construct the set M(D) we use the following rules: (1) If square a1 is empty, then M(D) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2) If square a1 contains +, then find the maximal index rD such that the squares a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , arD are all filled with +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (3) If square a1 contains +, define the set J (D) of indices: J (D) := {j ≤ rD | square aj contains +, square bj is empty or j = ℓ + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (4) For every p ∈ J (D), construct the offspring Dp as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' First, erase + in square ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then move + from square aj down to square bj for all j ∈ J (D) such that j < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' [M, Example 7]) Let ℓ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Figure 1 shows a pipe dream D of size 6 and three pipe dreams of size 5 that form the set M(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In this case, J (D) = {1, 2, 4}, and rD = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' + + + + + ↓ + + + + + + + + + + + + Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Recall that a pipe dream in type An can be defined as an n × n table whose cells are either filled with + or empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Furthermore, + is not allowed in cell (i, j) if the cell is below the main antidiagonal of the table (that is, i + j > n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' An example of pipe dream for n = 6 is given on Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Pipe dreams have an elegant interpretation in terms of networks of pipes, and are related to permutations from Sn+1 (see [M] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' + + + + + + + + + + + + + + + + + Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Pipe dreams in type A6 (left) and C4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' There are n different mitosis operations MA 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , MA n on pipe dreams in type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Informally, mitosis operation MA i can be defined as the two-row mitosis applied to rows i and i + 1 of a pipe dream (the other rows are not affected by MA i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that mitosis operation MA n is also well-defined though row n + 1 is an empty set of SIMPLE GEOMETRIC MITOSIS 5 boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This is because row n might have at most one +, that is, no + will have to be moved down according to the mitosis rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We now define mitosis operation MA i more formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let D be a pipe dream in type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Put ℓ = n−i, and label cells (i, 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (i, n−i+1) by a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , a(ℓ+1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Label cells (i+1, 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (i+1, n−i) by b1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , bℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Extract the basic pipe dream Di from D by setting Ai = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , a(ℓ + 1)) and Bi = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , bℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Apply two-row mitosis to Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Complete every resulting offspring Di p to a pipe dream Dp by replacing Di with Di p in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Define MA i (D) as the set {Dp | p ∈ J (Di)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let D be a pipe dream depicted on Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For i = 1, the basic pipe dream D1 coincides with the one on Figure 1 (top) from Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, MA 1 (D) consists of the pipe dreams shown on Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If i = 2, 3, 4, 6, then + + + + + + + + + + + + + + + + + + + + + Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' MA i (D) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Finally, MA 5 (D) consists of a single offspring obtained from D by erasing the + in the fifth row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It is easy to check that MA i coincides with the i-th mitosis operator mitosisi introduced in [M, Definition 6] (Knutson–Miller mitosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' There is also a strong relationship between MA i and the operator Mi introduced in [F, Section 5] using representation theoretic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' These operations coincide whenever D does not have any + in cells to the right of rDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' While MA i and Mi might differ on the other pipe dreams (the latter operation might produce more offsprings than the former) it is interesting that both operations lead to the same results in Schubert calculus (see [F, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='13] and the preceding discussion for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, [M, Theorem 15] still holds if mitosisi is replaced by Mi, that is, Mi can be viewed as an alternative version of the Knutson–Miller mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Similarly to type A case, pipe dreams in type Cn (also called skew pipe dreams) can be defined as n × (2n − 1) tables filled with +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type Cn case, + is not allowed in cell (i, j) if i + j > 2n or i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' An example of a skew pipe dream for n = 4 is given on Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Skew pipe dreams were recently used in [F] to construct a convex geometric model for Schubert calculus in type C in terms of symplectic GZ polytopes (we say more about this in Section 4), in particular, they are related to signed permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It would be interesting to find an interpretation of skew pipe dreams in terms of networks of pipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In [ST], c-signed pipe dreams were defined as networks of pipes with extra features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We do not know of any direct relation between skew pipe dreams and c-signed pipe dreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 6 VALENTINA KIRITCHENKO We now define n mitosis operations MC 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , MC n on a skew pipe dream D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The case i = 1 is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In this case, set ℓ = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Label cells (1, n), (2, n),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (n, n) by a1, a2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , a(ℓ + 1), respectively (these are cells in the middle column of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Label cells (1, n + 1), (2, n + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (n − 1, n + 1) by b1, b2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , bℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If i = 2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , n, set ℓ = 2(n−i)+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Label cells (1, n−i+1), (1, n+i−1), (2, n−i+1), (2, n + i + 1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (n − i + 1, n − i + 1), (n − i + 1, n + i − 1) by a1, a2, a3, a4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , aℓ, a(ℓ + 1), respectively (these are all cells in columns n ± (i − 1) of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Label cells (1, n − i + 2), (1, n + i), (2, n − i + 2), (2, n + i),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , (n − i + 1, n − i + 2) by b1, b2, b3, b4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , bℓ, respectively (here we alternate first n − i + 1 cells in column n − i + 2 with cells in column n + i of D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The rest of the definition of MC i is completely analogous to type A case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let D be a skew pipe dream from Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Using Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1 again we get that MC 1 (D) consists of skew pipe dreams depicted on Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For i = 2, 3, the set MC i (D) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The set MC 4 (D) consists of a single offspring + + + + + + + + + + + + + + + + + + + + + + + + Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' obtained by erasing the + in the first column of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis operation MC i on skew pipe dreams is different from operator Mi defined in [F, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' As in type A (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3), the difference lies in the restriction j ≤ rD in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It would be interesting to check whether both operations still produce the same results in applications to Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis on GZ polytopes in types A and C In this section, we apply simple geometric mitosis to GZ polytopes in types A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We compare the resulting operations with geometric realizations of Knutson– Miller mitosis and Fujita mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Gelfand–Zetlin polytopes in type An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Put d := n(n+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A GZ table of type An is a collection of cells organized according to the pattern on Figure 5 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let λ := (λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' ≥ λn+1) be a decreasing collection of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Identify a point (x1 1, x1 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , x1 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' x2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , x2 n−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' xn−1 1 , xn−1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' xn 1) ∈ Rd with the GZ table whose i-th row is filled with coordinates (xi 1, xi 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , xi n−i+1) for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Define the GZ polytope GZA λ ⊂ Rd by 2d interlacing inequalities xi−1 j ≥ xi j ≥ xi−1 j+1 (we put x0 j := λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In terms of the GZ table, these inequalities tell us that the coordinate in any given cell lies between coordinates in two upper neighbors of this cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let v be the Kogan vertex of GZA λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Recall that the Kogan vertex is a unique vertex of the polytope GZA λ that satisfies simultaneously all d equations of type SIMPLE GEOMETRIC MITOSIS 7 λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ1 λ2 λ3 λ4 0 0 0 0 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Gelfand–Zetlin patterns in type A6 (left) and type C4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' xi−1 j = xi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A face Γ ⊂ GZA λ is called a Kogan face if Γ contains v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' These faces were first considered by Mikhail Kogan [Ko].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, v-faces used in the definition of mitosis in Section 2 are exactly Kogan faces for this choice of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the polytope GZA λ for λ = (n, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , 1, 0) can be represented as the Cayley sum of two polytopes in n differents ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, let Pi and Qi be the facets of GZA λ given by equations x1 i = λi and x1 i = λi+1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then ∆(Pi, Qi) coincides with GZA λ up to a parallel translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Clearly, v ∈ P1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, we can define n different mitosis operations mitosisv 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , mitosisv n on Kogan faces of GZA λ as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2 describes these operations for n = 2 because the GZ polytope in this case is unimodularly equivalent to the FFLV polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' By definition, any Kogan face is given only by equations of type xi−1 j = xi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, every Kogan face F can be encoded by a GZ table filled with +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, if F lies in the hyperplane {xi−1 j = xi j} then put + in cell (i, j) (that is, in the j-th cell of the i-th row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Otherwise, leave cell (i, j) empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The resulting GZ table filled with + will be called the diagram of the face F and denoted by D(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Define a bijective correspondence between pipe dreams in type An and diagrams of Kogan faces of GZA λ by identifying cell (i, j) in a GZ table with cell (j, i) in a pipe dream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The correspondence is illustrated by Figure 7 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, inscribe the words “GELFAND ZETLIN POLYTOPE” (without spaces) inside a pipe dream of type A6 in usual way, that is, start from the top row and write from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' After switching to GZ table, the words will transform to the sequence of letters on Figure 7 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The pipe dream on Figure 2 (left) corresponds to the Kogan face F given by eight equations λ1 = x1 1 = x2 1 = x3 1 = x4 1, x2 2 = x3 2, λ3 = x1 3, λ4 = x1 4, λ5 = x1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, dim F = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It is easy to check that mitosisv 1(F) consists of three Kogan faces of dimension 14 whose diagrams correspond to pipe dreams on Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Using the bijection between pipe dreams and GZ tables we can extend mitosis operations MA 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , MA n from pipe dreams to GZ tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Informally speaking, we replace rows with columns when applying two-row mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We will use the same 8 VALENTINA KIRITCHENKO G E L F A N D Z E T L I N P O L Y T O P E S H C U B E R T C A L C U L U S Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Correspondence between pipe dreams and GZ patterns in type A6 (left) and type C4 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' notation for mitosis operations on GZ tables as it will be clear from the context whether we talk about pipe dreams or about GZ tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It turns out that geometric operation mitosisv i on certain Kogan faces is combina- torially equivalent to mitosis operation MA i on the corresponding pipe dreams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The following theorem describes the precise relation between the geometric mitosis on Kogan faces of a GZ polytope and the combinatorial mitosis on GZ tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let F ⊂ GZA λ be a Kogan face such that its diagram D(F) has + in cell (1, i) and no + in cells (j, i + 1) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then mitosisv i (F) consists of Kogan faces whose diagrams are obtained from D(F) by the mitosis operation MA i on GZ tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In order to compute mitosisv i (F) we have to consider not only Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In general, a face of GZA λ is given by equations of two types: either xi−1 j = xi j (type A) or xi j = xi−1 j+1 (type B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Following [Ko] define equations Ai,j and Bi,j, respectively, as xi−1 j = xi j and xi j = xi−1 j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We now apply Definition 2 to P := Pi and Q := Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that any Kogan face is admissible because it contains a simple vertex, namely, the Kogan vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The face E1 := exp(F) is obtained from F by removing a single equation A1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, the face exp(F) ∩ Q is given by the same equations as F with one exception: equation A1,i is replaced by equation B1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let E be a Kogan face of dimension ℓ + 1 (where ℓ := dim F) such that E ∩ Q ⊂ exp(F)∩Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let (X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1) be the collection of equations of type A that define E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If E ̸= exp(F) then there exists an equation Y (of type A) such that exp(F) satisfies Y but Y does not follow from (X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' However, Y should follow from equations B1,i, X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, the collection of equations (B1,i, Y , X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1) is redundant, while the collection (Y , X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This is only possible if A1,i+1 is contained among the equations X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1 as the following lemma shows: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xs) be a collection of equations of type A that does not contain equation Ap,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If Ap,q follows from (Bj,i, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xs) then three conditions hold: SIMPLE GEOMETRIC MITOSIS 9 (1) q = i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2) p > j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (3) (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xs) contains Ak,i+1 for all k such that j ≤ k < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The statement follows directly from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' □ If (X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1) contains A1,i+1, A2,i+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ak,i+1 but does not contain Ak+1,i+1 for some k ≥ 1, then A2,i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ak+1,i are the only equations that follow from (B1,i, X1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, if F satisfies Ap,q, and Ap,q is not contained among (X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1)) then Ap,q must coincide with one of the equations A2,i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ak+1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For dimension reasons, all k equations A2,i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ak+1,i must be absent in (X1, X2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Xd−ℓ−1) (to compensate for the presence of k equations A1,i+1, A2,i+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ak,i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, if F does not satisfy A2,i then mitosisv i (F) consists of a single offspring exp(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We now proceed by induction on r where r is the maximal number such that F satisfies the equations A1,i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ar,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let Γr,i be the facet given by equation Ar,i, and F ′ the face given by the same equations as F except for Ar,i, that is, F = F ′ ∩ Γr,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It is easy to check that there is a bijection between mitosisv i (F ′) and the subset S ⊂ mitosisv i (F) that consists of all faces Ei ∈ mitosisv i (F) such that Ei ⊂ Γr,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, a face E′ ∈ mitosisv i (F ′) corresponds to the face E = E′∩Γr,i ∈ mitosisv i (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, the faces in S can be described by the induction hypothesis applied to F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It remains to describe the faces Ei ∈ mitosisv i (F)\\S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' By condition (2) of Definition 2 and part (3) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3 applied to Ar,i, we have that if Ei ̸⊂ Γr,i, then Ei must satisfy equations A1,i+1, A2,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , Ar−1,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Hence, there is a single offspring Er ∈ mitosisv i (F) \\ S obtained from F by removing equation Ar,i and by replacing equations Aj,i by equations Aj,i+1 for all j < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' □ We now adapt geometric mitosis operations so that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2 holds for all reduced Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The definition of reduced pipe dreams can be found in [M, Def- inition 1] (these pipe dreams are mostly used in applications to Schubert calculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A Kogan face F is reduced if its diagram D(F) is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Below we use notation of Definition 3 from the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let F ⊂ GZA λ be a reduced Kogan face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Define an adapted geometric mitosis operation mitosisv i as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (1) Consider the face env(F) ⊂ GZA λ given by equations Ak,i and Ak,i+1 for all k such that F satisfies both Ak,i and Ak,i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2) Find the minimal index s such that env(F) does not satisfy equation As,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Consider the facets P F i , QF i ⊂ env(F) given by equations As,i and Bs,i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (3) Define mitosisv i (F) by applying Definition 2 to ∆ = env(F), P = P F i and Q = QF i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then mitosisv i (F) consists of exactly those Kogan faces whose diagrams lie in the set MA i (D(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 10 VALENTINA KIRITCHENKO The proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4 is completely analogous to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the set mitosisv i (F) might, in general, contain more faces than the set mitosisv i (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' By applying mitosis inside the face env(F) instead of the whole GZA λ we get rid of these extra faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The definition of env(F) in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4 is ad hoc, it relies heavily on combinatorics of GZ polytopes in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It would be interesting to find a more geometric definition of env(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The choice of P F i and QF i stems from the classical inductive construction of GZ bases and polytopes based on the chain of subgroups GL2(C) ⊂ GL3(C) ⊂ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' ⊂ GLn+1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Gelfand–Zetlin polytopes in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We now describe analogous construc- tions in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We omit details that are the same as in type A case and focus on unique features of type C case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Put d = n2, and put λ := (λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' ≥ λn ≥ λn+1 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' A GZ table of type Cn is defined according to the pattern on Figure 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Roughly speaking, a GZ table of type Cn is a half of a GZ table of type A2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' As in type A case, the GZ polytope GZC λ ⊂ Rd is defined by 2d interlacing inequalities that come from a GZ pattern in type C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Again, the polytope GZC λ for λ = (n, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , 0) can be represented as Cayley sum of two polytopes in n different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let v be the symplectic Kogan vertex of GZC λ as defined in [F, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Using notation from Definition 3 (see the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2) and regarding a GZ pattern of type Cn as part of a GZ pattern of type A2n−1 we may define v by equations Ai,j for all odd i ≤ 2n − 1 and equations Bi,j for all even i < 2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Similarly to type A case, a face Γ ⊂ GZC λ is called a symplectic Kogan face if Γ contains v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Again, there are n mitosis operations mitosisv 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , mitosisv n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' As in type A case, every symplectic Kogan face F can be encoded by a GZ table D(F) of type C filled with +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Define bijective correspondence between skew pipe dreams and diagrams of Kogan faces as in [F, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For instance, if words “SCHUBERT CALCULUS” (without spaces) are inscribed into a skew pipe dream of type C4 in usual way then they get transformed into boustrophedon1 writing in a GZ pattern of type C4 on Figure 7 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Again, we extend mitosis operations MC 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , MC n from skew pipe dreams to GZ tables in type Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that MC 1 is combinatorially different from MC 2 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , MC n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In applications to Schubert calculus, mitosis operations correspond to generators of the group of signed permutations Bn on n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' There is a special generator s1 : (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , n) �→ (−1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , n) (change of sign), and the elementary transpositions s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' ,sn, namely, si = (i − 1 i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If we regard Bn as the Weyl group of the symplectic group Sp2n(C), then s1 is the simple reflection corresponding to the longer root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The mitosis opera- tion MC n corresponds to the special generator s1, hence, it is natural to expect that MC n will be special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 1I am grateful to Evgeny Smirnov from whom I learnt the proper name of this writing style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' SIMPLE GEOMETRIC MITOSIS 11 The following theorem is analogous to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2 and can be proved using similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let F ⊂ GZC λ be a symplectic Kogan face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (1) Let i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If the diagram D(F) has + in cell (1, i), and no + in cells (2k + 1, i − k + 1) and (2k, i − k) for all k, then mitosisv i (F) consists of faces whose diagrams are obtained from D(F) by applying MC n−i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2) If the diagram D(F) has + in cell (1, n), and no + in cells (2k, n − k) then mitosisv n(F) consists of faces whose diagrams are obtained from D(F) by applying MC 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Applications to Schubert calculus We now explain how the notion of simple geometric mitosis fits into in the context of intersection theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, we relate Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='6 with the analogous results on Schubert calculus in types A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Below we use the notion of polytope ring (aka Khovanskii–Pukhlikov ring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The definition can be found, for instance, in [K21, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We refer the reader to [F] for more details on applications of Khovanskii–Pukhlikov rings to Schubert calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We use notation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let R∆ and RP be the polytope rings of ∆ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Assume that R∆ and RP, respectively, are isomorphic to the Chow rings of smooth varieties Y and X, where Y = P(E) is the projectivization of a rank two vector bundle E on X (see [K21, Section 4] for motivation behind such an assumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let p be the natural projection from Y to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then there is a push-pull operator p∗p∗ : CH∗(Y ) → CH∗−1(Y ), which is a homomorphism of CH∗(X)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The simple geometric mitosis is an attempt to describe explicitly the action of the push-pull operator on R∆ ≃ CH∗(Y ) using representations of elements of R∆ by linear combinations of faces of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It would be interesting to formalize the connection between Definition 2 and the action of p∗p∗ on faces of ∆ in the general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Below we will exhibit such a connection in the special case of GZ polytopes and flag varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let ∆ be the GZ polytope for a classical group G and a strictly dominant weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let B ⊂ G denote a Borel subgroup of G, and let Y := G/B denote the complete flag variety for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The ring R∆ is isomorphic to the subring of CH∗(Y ) generated by the first Chern classes of line bundles on G/B [Ka, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For G = GLn+1(C) and Sp2n(C), this subring coincides with CH∗(Y ) (in general, the discrepancy between the subring and the whole CH∗(Y ) is measured by the torsion index of G, see [T] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, if λ = ρ := (n, n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , 1, 0), then R∆ ≃ CH∗(Y ) in types A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Denote by α1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , αn the simple roots of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let Xi be the partial flag variety G/Pi for the minimal parabolic subgroup Pi corresponding to αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the natural projection pi : Y → Xi turns Y into a P1-fibration (it can be realized as the projectivization of the rank two bundle Ei := pi∗L(ρ) where L(ρ) is the line bundle 12 VALENTINA KIRITCHENKO on Y corresponding to the weight ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The classical divided difference (or push-pull) operator in Schubert calculus is defined as ∂i := p∗ i pi∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Recall that the operators ∂1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , ∂n are used to generate the Schubert classes [Xw] ∈ CH∗(Y ) starting from the class [Xid] of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' More precisely, let w ∈ W be an element of the Weyl group of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Choose a reduced decomposition w = si1 · · · siℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Here si denotes the reflection with respect to the root αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Then [Xw] = ∂iℓ · · ·∂i1[Xid].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' If we choose the Kogan vertex v as a representative of the class [Xid] in R∆, then the action of ∂iℓ · · · ∂i1 on [Xid] can be computed using mitosis operations as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let G = GLn+1(C) and ∆ = GZA ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Under the isomorphism CH∗(Y ) ≃ R∆ the Schubert cycle [Xw] can be represented as the class of the sum of faces F ⊂ ∆ where F runs through the set Sw = mitosisv iℓ · · ·mitosisv i1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, the action of divided difference operator ∂i on the Schubert cycle [Xw] gets represented by the action of mitosisv i on faces from Sw: Swsi = mitosisv i (Sw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This theorem follows from [F, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='18] together with Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the proof of [F, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='17] (which implies [F, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='18]) uses repre- sentation theoretic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It would be interesting to find a convex geometric proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In [K16], different geometric mitosis operations are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' They mimick Demazure operators rather than push-pull operators ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type A, they can also be used to prove theorems analogous to Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1 or [F, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='18] due to a special symmetry of GZ diagrams in type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Namely, the diagrams are symmetric with respect to the reflection (i, j) �→ (j, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Note that the induction step in [K16, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='6] goes from siw to w (not from wsi to w as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1 and in [F, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In other words, induction goes along initial subwords of si1si2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' siℓ (not along terminal subwords).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This difference is matched by the difference between mitosis as defined in [K16] and its transpose (or mirror) mitosis as defined in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, all arguments with the mitosis operations applied to w = si1si2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' siℓ can be immediately translated into analogous arguments with the transpose mitosis operations applied to w−1 = siℓsiℓ−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' si1 (see [F, Section 5] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type C, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='6 together with Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='5 and [F, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='8] suggest that mitosisv i can be adapted so that under the isomorphism CH∗(Y ) ≃ R∆ the Schubert class [Xw] can be represented as the class of the sum of faces F ⊂ ∆ where F runs through the set Sw = mitosisv n−iℓ+1 · · ·mitosisv n−i1+1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' SIMPLE GEOMETRIC MITOSIS 13 In [F, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='13], another presentation of Schubert cycles by faces of GZC ρ is obtained using the dual Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Similarly to the definition of symplectic Kogan vertex in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2, the dual Kogan vertex v∗ can be defined by equations Bi,j for all odd i ≤ 2n−1 and equations Ai,j for all even i < 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Again there are n geometric mitosis operations mitosisv∗ 1 ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' , mitosisv∗ n where mitosisv∗ i corresponds to the decomposition GZC ρ = {x1 i = λi+1} ⋆ {x1 i = λi} (that is, Pi and Qi from the definition of mitosisv i switch places when defining mitosisv∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In the case of Sp4(C), these operations produce part of the presentation of Schubert cycles from [F, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='13] as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Let n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We encode dual symplectic Kogan faces by their diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' We put A and B instead of + just for clarity (not because it carries any extra infor- mation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' This way it is easier to distinguish diagrams of Kogan faces from diagrams of dual Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In the second row, we use an adapted mitosis once (this adap- tation is based on the chain Sp2(C) = SL2(C) ⊂ Sp4(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' These collections of faces B B mitosisv∗ 1 −→ id A B B mitosisv∗ 2 −→ s2 A B B & s2s1 B s2s1 mitosisv∗ 1 −→ A B B & s2s1s2 s2s1s2 B B B mitosisv∗ 2 −→ id A B B B mitosisv∗ 1 −→ s1 A B mitosisv∗ 2 −→ s1s2 A B & s1s2s1 s1s2s1 A Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Mitosis for the dual Kogan vertex in type C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' are the same as the presentations of Schubert cycles [Xw] in [F, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='13] for all w ∈ W except for s2s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In the case w = s2s1, one face is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Recall that according to [F, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='13], the dual Kogan faces that represent [Xw] are in bijective correspondence with those reduced subwords of the longest word w0 = (s1)(s2s1s2)(s3s2s1s2s3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (snsn−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' s2s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' sn−1sn) that represent the element w0w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' The bijection is obtained by inscribing the word w0 into the symplectic GZ pattern using reverse boustrophedon (see Figure 8 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' For instance, the subword (s1)(✚✚ s2✚✚ s1s2)(s3s2s1s2✚✚ s3)(✚✚ s4s3✚✚ s2✚✚ s1s2s3s4) in type C4 corresponds to the dual Kogan face on Figure 8 right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In particular, there are three dual Kogan faces that represent [Xw] for w = s2s1 in type C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Indeed, w0w = s1s2, and there are three reduced subwords of s1(s2s1s2) that represent s1s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type A, there is no combinatorial difference between presentation of Schubert cycles by Kogan faces and presentation by dual Kogan faces (see [F, 14 VALENTINA KIRITCHENKO s4 s3 s2 s1 s2 s3 s4 s3 s2 s1 s2 s3 s2 s1 s2 s1 B B B A B B A A B B Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Correspondence between subwords of w0 and dual Kogan faces in type C4 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Both presentations can be related using the automorphism of the Dynkin diagram of type An (on the level of GZ diagrams this automorphism corre- sponds to the reflection (i, j) �→ (i, n+2−i−j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type C, there is a combinatorial difference already for n = 2 (see [F, Section 1] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type A, one can immediately recover mitosis on dual Kogan faces from mitosis on Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Both operations will be combinatorially equivalent to the Knutson–Miller mitosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' In type C, the Fujita mitosis on Kogan faces does not yield a combinatorially equiv- alent mitosis on dual Kogan faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='3 shows that Definition 2 is too simple to capture completely the action of p∗p∗ on faces of ∆ in the general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' However, it may be regarded as the first approximation of this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' It can also be used to make an educated guess about a possible combinatorial mitosis on dual Kogan faces in type C and mitosis in type D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' References [F] Naoki Fujita, Schubert calculus from polyhedral parametrizations of Demazure crystals, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=', 397 (2022) [Ka] Kiumars Kaveh, Note on cohomology rings of spherical varieties and volume polynomial, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Lie Theory 21 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 2, 263–283 [K16] Valentina Kiritchenko, Geometric mitosis, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=', 23 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 4, 1069– 1096 [K21] Valentina Kiritchenko, Push-pull operators on convex polytopes, IMRN, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='1093/imrn/rnab331 [KP] Valentina Kiritchenko and Maria Padalko, Schubert calculus on Newton–Okounkov polytopes, Chapter 11 in Interactions with lattice polytopes, Springer, 2022 [KnM] Allen Knutson and Ezra Miller, Gr¨obner geometry of Schubert polynomials, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' (2), 161 (2005), 1245–1318 [Ko] Mikhail Kogan, Schubert geometry of flag varieties and Gelfand–Cetlin theory, Ph.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='14120 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='CO] [T] Burt Totaro, The torsion index of the spin groups, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=', 129 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='2, 249–290 SIMPLE GEOMETRIC MITOSIS 15 Email address: vkiritch@hse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content='ru Laboratory of Algebraic Geometry and Faculty of Mathematics, National Re- search University Higher School of Economics, Usacheva str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} +page_content=' 6, 119048 Moscow, Russia Institute for Information Transmission Problems, Moscow, Russia' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AyT4oBgHgl3EQfZfeI/content/2301.00225v1.pdf'} diff --git a/rNFJT4oBgHgl3EQfbCz-/vector_store/index.pkl b/rNFJT4oBgHgl3EQfbCz-/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..12b5a6267945307adcbecd9ea3de055d80e9455b --- /dev/null +++ b/rNFJT4oBgHgl3EQfbCz-/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8c90339e97e37efc9b381f83e171e4661cc0c4639faf2c765eb60eff89927623 +size 88735 diff --git a/rtE2T4oBgHgl3EQfLAbq/content/2301.03710v1.pdf b/rtE2T4oBgHgl3EQfLAbq/content/2301.03710v1.pdf new file mode 100644 index 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b/tNE5T4oBgHgl3EQfmg9S/content/tmp_files/2301.05679v1.pdf.txt @@ -0,0 +1,1569 @@ +arXiv:2301.05679v1 [gr-qc] 13 Jan 2023 +A unified effective approach to cosmological perturbations +Antonio Enea Romano +Theoretical Physics Department, CERN, +CH-1211 Geneva 23, Switzerland and +ICRANet, Piazza della Repubblica 10, I–65122 Pescara, Italy +(Dated: January 16, 2023) +Abstract +A unified model independent effective description of cosmological perturbations is derived in terms +of two effective quantities, playing the role of effective propagation speeds of curvature perturbations +and gravitational waves, encoding the effects of the interaction of perturbations at any order, and +inducing a modification of the friction term of the perturbations propagation equation. The approach +can be applied to dark energy, modified gravity, dark matter, for fields of arbitrary number and spin, +and is particularly suitable for model independent analysis of observational data. The structure of the +effective actions and equations is the same for scalar and tensor perturbations. +The effective actions can be written as the Klein-Gordon action in terms of an appropriately defined +effective metric, dependent on the effective speed. In this geometrical interpretation, the effective +metric emerges as the result of the interaction and self-interaction of perturbations, hinting to possible +connections with emergent gravity. +As an example of an application we find that the effective speeds of curvature perturbations and +gravitational waves can be frequency and polarization dependent even for a minimally coupled scalar +field in general relativity, when higher order terms effects are computed, going beyond the quadratic +action, or in axion inflation. We discuss the relation between the effective speed and quantum corre- +lators. +1 + +I. +INTRODUCTION +Cosmological perturbations [1] play a fundamental role in modeling the Universe, since they +allow to study the origins of large structure, the anisotropies of the cosmic microwave back- +ground (CMB) radiation, and the propagation of gravitational waves on cosmological distances. +We derive a set of universal model independent effective equations and Lagrangians which +can describe the evolution of scalar and tensor perturbation of any system whose field equations +can be written in an Einstein like form. This includes for example multi-fields systems [2], or +modified gravity theories such as Horndeski’s theory[3], once they have been transformed to +the Einstein frame. Given the generality of this effective description it is particularly suitable +for model independent phenomenological analysis of observational data. +The approach predicts naturally that the speed of gravitational waves cT,A con depend on +frequency and polarization, due to the interactions of the graviton with itself or other fields. +This prediction allows to use gravitational waves observations to investigate the elusive nature +of dark matter and dark energy. +The equation and Lagrangian for scalar and tensor perturbations has the same universal +structure, and the effects of the interaction can be modeled at any order in perturbations by +a single effective quantity, playing the role of effective propagation speed. This is particularly +useful since it allows to compare different models in terms of the two effective quantities cs +and cT. Combining different set of observational data such as cosmic microwave background +radiation and gravitational waves, it will be possible to constrain cs and cT,A, to determine +possible deviations from general relativity and vanilla inflation. +The effective equations and Lagrangians are derived separately for scalar and tensor per- +turbations, both in physical and momentum space, and the consistency with previous results +derived in the literature is checked in different cases. Other predictions are then considered, +such as the frequency dependency of cs and cT in vanilla inflation due to the effects of higher +order interaction terms. +2 + +II. +SCALAR PERTURBATIONS +A. +Space dependent effective sound speed +In this section we will show that it is possible to define a space dependent effective sound +speed (SESS) in terms of which a model independent equation for comoving curvature pertur- +bations ζ can be derived. The SESS encodes the effects of the interaction of ζ with itself and +other fields, at any order in perturbations. +1. +Effective Lagrangian approach +In the action approach the EST on the r.h.s. of the Einstein’s equations originates from +the interaction of scalar perturbations with themselves or other fields. Based on this we can +obtain an effective Lagrangian corresponding to the perturbed field equations by introducing +higher order interaction terms as Lint(ζ, φi), where φi denotes abstractly all the other fields ζ +is coupled to. +We will use conformal time, and adopt the following notation for the Lagrangian density L +S = +� +dηdxL = +� +dηdxz2L +, +L = z2L +, +z2 = ǫa2 , +(1) +where we are assuming a Friedman background (FRW) background with scale factor a(η), and +ǫ denotes the first order slow-roll parameter, defined in terms of cosmic time dt = a−1dη. +In general relativity the quadratic Lagrangian of the comoving curvature perturbations of a +minimally coupled scalar field is +L(2) +ζ += z2� +ζ′2 − (∇ζ)2� +, z2 = ǫa2. +(2) +We will call the above model vanilla inflation. +Including higher order interaction and self interaction terms we can write a general model +independent Lagrangian +Lζ = L(2) +ζ ++ Lint +ζ += z2� +ζ′2 − (∇ζ)2 + Lint +ζ (ζ, φi) +� += z2� +ζ′2� +1 + Lint +ζ +ζ′2 +� +− (∇ζ)2� +, +(3) +where φi denotes collectively any other field. +From the above equation we can obtain the +effective Lagrangian +Leff +ζ += z2 +c2s +� +ζ′2 − c2 +s(∇ζ)2� += α2� +ζ′2 − c2 +s(∇ζ)2� +, +α2 = z2 +c2s += ǫa2 +c2s +, +(4) +3 + +where we have defined the space effective sound speed (SESS) according to +c2 +s(η, xi) = +� +1 + Lint +ζ +ζ′2 +�−1 +. +(5) +The variation of Leff +ζ +gives the model independent equation +ζ′′ + 2α′ +α ζ′ − c2 +s∇2ζ = 0 , +(6) +which can be also written as +ζ′′ + 2 +�z′ +z − c′ +s +cs +� +ζ′ − c2 +s∇2ζ = 0 . +(7) +Note that in deriving the equations of motion for ζ the SESS has been treated as a function +independent of ζ, since the SESS is an effective quantity determined by substituting the so- +lutions of the full theory, including the effects of interaction, into the interaction Lagrangian +Lint. For any system with a well defined Lagrangian it should be always possible to solve the +equations of motions, and then use the solutions to define the SESS. +Eq.(6) and eq.(4) show that α(η, xi) can be interpreted as an effective space dependent scale +factor, while eq.(7) shows explicitly the modification of the friction term induced by the SESS. +Since these equations are model independent, we can immediately conclude that the friction +terms cannot be modified if c′ +s = 0. +The effective Lagrangian can be obtained from the vanilla case in eq.(8) +L(2) +ζ += z2� +ζ′2 − c2(∇ζ)2� +. +(8) +by the transformation +z2 → α = z2 +c2s +, +c → cs , +(9) +where we are denoting with c the unity sound speed, to avoid ambiguity. This is in agreement +with eq.(6), which shows that α can be regarded as an effective scale factor. +The main advantage of using eq.(6) is that it is completely model independent, allowing to +study in a systematic way deviations from general relativity or the effects of the interaction +with different fields using a single function. Note that since Lint can be space dependent, also +cs(η, xi) depends on space, with the exception of Lint ∝ f(η)ζ′2, when it is only time dependent, +which corresponds to K-inflation [4]. +In presence of multiple scalar fields the space dependence is manifested already in the +quadratic Lagrangian [5], while the effects of anisotropic perturbations requires at least a cubic +Lagrangian, because scalar fields anisotropies appear at second order in the EST. +4 + +Note that even in vanilla inflation the sound speed can be space dependent when including +the cubic and higher order Lagrangians [6], as we will show explicitly later. This is consistent +with the EST approach, since a scalar field EST is anisotropic at second order, leading to source +terms in the perturbed field equations. +2. +Interpretation of the space dependency of the SESS +The space dependency of the SESS is a natural manifestation of the interaction and self- +interaction of scalar perturbations, and of the coupling of tensor and scalar perturbations. In +multiple scalar fields systems these effects are already present in the quadratic Lagrangian, +and are associated to entropy perturbations, while for a single scalar field they manifest only +starting from the cubic Lagrangian. +3. +Effective stress-energy tensor approach +Scalar perturbations of the metric and of the Effective stress-energy tensor (EST) can be +written as +ds2 = −(1 + 2A)dt2 + 2a∂iBdxidt + a2 {δij(1 + 2C) + 2∂i∂jE} dxidxj , +(10) +T 0 +0 = −(ρ + δρ) +, +T 0 +i = (ρ + P)∂i(v + B) , +T i +j = (P + δP)δi +j + δik∂k∂jΠ − 1 +3δi +j∇2Π . +(11) +where v is the velocity potential and ∇2 ≡ δkl∂k∂l. Note that any metric and EST can always +be written in the above form, making all the results obtained from it completely model inde- +pendent. The comoving slices gauge is defined by the condition (T 0i)c = 0, and we denote with +a subscript c quantities evaluated on comoving slices. In multiple fields systems the EST is +the sum of the stress-energy tensors of each field, which in the comoving gauge is associated to +entropy perturbations [7]. +In the comoving gauge entropy perturbations Γ are introduced [1] by +δPc(η, xi) = ca(η)2δρc(η, xi) + Γ(η, xi) , +(12) +where ca is interpreted as the adiabatic sound speed, and is by definition a function of time +only. Note that this definition can be ambiguous [7]. +The manipulation of the perturbed Einstein’s equation in the comoving gauge gives +5 + +ζ′′ + ∂ηz2 +a +z2a +ζ′ − c2 +a∇2ζ = a2S , +(13) +S = −c2 +a +ǫ ∇2Π − +1 +2a2z2a +� a3 +c2aH +� +Γ + 2 +3∇2Π +��′ +, z2 +a = ǫa2 +c2a +, +(14) +where we are denoting derivatives with respect to conformal time with a prime. The source +term in the above equation can be absorbed into the definition of the space effective sound +speed (SESS) following a similar approach to the one adopted for gravitational waves [8]. +We can first re-write eq.(14) as +(ζ′z2 +a)′ +z2 +a +− (gz2 +a)′ +z2 +a +− c2 +a∇2ζ = +� +ζ′z2 +a(1 − g/ζ′) +�′ +z2 +a +− c2 +a∇2ζ = 0 , +(15) +where we have defined +g = 1 +z2a +� +z2 +aa2S dη . +(16) +After introducing the quantities +1 + δ(η, xi) = +� +1 − g +ζ′ +�−1/2 +, +α2 = +z2 +a +(1 + δ)2 = +ǫa2 +c2 +a(1 + δ)2 , +(17) +we can rewrite eq.(15) as +1 +z2a +(α2ζ′)′ − c2 +a∇2ζ = 0 . +(18) +Defining the space effective sound speed (SESS) as +cs(η, xi) = ca(η) +� +1 + δ(η, xi) +� +, +(19) +and re-writing α in terms of cs as +α2 = ǫa2 +c2s +, +(20) +we finally obtain the model independent effective equation +ζ′′ + 2α′ +α ζ′ − c2 +s∇2ζ = 0 , +(21) +which shows that cs is the correct definition of effective sound speed. Eq.(21) is in agreement +with eq.(6), obtained using the effective Lagrangian approach, is completely general, and it can +be applied to study multi-fields models, modified gravity, dark energy or dark matter. +Note that the SESS definition given in eq.(19) is more general that the one given in [7], +which is including the effects of entropy, but not of anisotropy. Using an effective Langrangian +6 + +approach there is no distinction between entropy and anisotropy, since they are both associated +to interaction terms, and the effective description is more transparent than using the EST +approach, but both methods lead to the same conclusions, since the source terms in the field +equations are obtained from the variation of the interaction Lagrangian. +The SVT decomposition of the EST is valid at any order in perturbations, so eq.(21) is +including the effects of interaction at any order in perturbations, including self-interaction, and +for this reason is in agreement with eq.(6), obtained by encoding in cs the effects of all higher +order interaction terms. +B. +Momentum effective sound speed +In this section we will show that it is possible to define a momentum dependent effective +sound speed (MESS) in terms of which a model independent equation for comoving curvature +perturbations ζ can be derived. The MESS encodes the effects of the interaction of ζ with +itself and other fields, at any order in perturbations. The MESS is not the Fourier transform of +the SESS, but it is mathematically convenient, since it allows to obtain a model independent +equation involving minimal changes of the vanilla case. +1. +Effective Lagrangian approach +A model independent effective equation and Lagrangian can also be derived in momen- +tum space, using the two different methods adopted previously, the field equations and action +approach. The Lagrangian in momentum space can be written as +Lζk = L(2) +ζk + Lint +ζk = z2� +ζ′2 +k + k2ζ2 +k + Lint +ζk (ζk, φi +k) +� +. +(22) +(23) +The effective Lagrangian is +Leff +ζk += ˜α2� +ζ′2 +k + ˜c2 +sk2ζ2 +k +� +, ˜α2(η, k) = z2 +˜c2s += ǫa2 +˜c2s +, +(24) +where we have defined the momentum effective sound speed (MESS) ˜cs and effective scalar +factor as +˜c2 +s(η, k) = +� +1 + Lint +k +ζ′2 +k +�−1 +, ˜α2(η, k) = ǫa2 +˜c2s += z2 +˜c2s +. +(25) +7 + +which gives the equation +ζ′′ +k + 2 ˜α′ +˜α ζ′ +k + ˜c2 +sk2ζk = 0 , +(26) +which can be also be written as +ζ′′ +k + 2 +�z′ +z − ˜cs +′ +˜cs +� +ζ′ +k + ˜c2 +sk2ζk = 0 . +(27) +Eq.(26) and eq.(24) show that ˜α(η, k) can be interpreted as an effective momentum dependent +scale factor, while eq.(27) shows explicitly the modification of the friction term induced the +MESS. Since these equations are model independent, we can immediately conclude that the +friction terms cannot be modified if ˜c′ +s = 0. +The effective Lagrangian can be obtained from the vanilla inflation action +L(2) +ζk = z2� +ζ′2 +k + c2k2ζ2 +k +� +, +(28) +by the transformation +z2 → ˜α = z2 +˜c2s +, +c → ˜cs , +(29) +where we are denoting with c the unity sound speed, to avoid ambiguity. +This is in agreement with eq.(26), which shows that ˜α can be regarded as a momentum +dependent effective scale factor. Note that the quantities ˜cs and ˜α are not the Fourier transform +of cs and α. +2. +Field equations approach +Taking the Fourier transform of eq.(14) we get +ζ′′ +k + ∂ηz2 +z2 ζ′ +k + c2 +ak2ζk = a2Sk , +(30) +Sk = c2 +a +ǫ k2Πk − +1 +2z2aa +� a3 +c2aH +� +Γk − 2 +3k2Πk +��′ +, z2 +a = ǫa2/c2 +a , +(31) +After introducing the quantities +gk = 1 +z2a +� +z2 +aa2Sk dη +, +1 + δk(η) = +� +1 − gk +ζ′ +k +�−1/2 +, +˜α2 = +z2 +(1 + δk)2 = +ǫa2 +c2a(1 + δk)2 , (32) +we can rewrite eq.(31) as +1 +z2(˜α2ζ′ +k)′ + c2 +ak2ζk = 0 . +(33) +8 + +Defining the momentum effective sound speed (MESS) as +˜cs(η, k) = ca(η) +� +1 + δk(η) +� +, +(34) +and re-writing ˜α in terms of ˜cs as +˜α2 = ǫa2 +˜c2 +s +, +(35) +we finally obtain the model independent effective equation +ζ′′ +k + 2 ˜α′ +˜α ζ′ +k + ˜c2 +sk2ζk = 0 , +(36) +which shows that ˜cs is the correct definition of momentum effective sound speed, in agreement +with eq.(26). +C. +Effective metric description +In terms of the effective metric +ds2 +eff = ǫa2� +csdη2 − δij +cs +dxidxj� +, +(37) +the effective Lagrangian can be written as +Leff +ζ += √−g(∂µζ∂µζ) = gζ +µνdxµdxν . +(38) +for which the equation of motion is simply given by the convariant D’Alembert operator +□ζ = +1 +� +−gζ ∂µ( +� +−gζ∂µζ) = 0 . +(39) +In this geometrical description the perturbations propagate in an empty curved space, whose +geometry is determined by the interaction of the perturbations. This is conceptually analogous +to the general relativistic geometrical interpretation of the effects of gravity in terms of geodesics +in a curved space, whose geometry is determined by the EST. More about this geometrical +interpretation will be discussed in a future work. +D. +Consistency with previous calculations +1. +Minimally coupled scalar field in general relativity +The vanilla scenario corresponds to Lint = 0, leading to δ = 0, cs = ca = 1. The quantity +cw = P ′/ρ′ does not give the correct definition of sound speed, since it does not coincide with +[9] the SESS cs = 1 ̸= cw. +9 + +2. +K-inflation +When the interaction Lagrangian is of the form Lint ∝ f(η)ζ′2 the SESS is just a function +of time cs(η) = ca(η) and δ = 0. In this case the effective action in eq.(4) and effective eq.(6) +are in agreement with [4] +Leff +ζ += +z2 +cs(η) +� +ζ′2 − c2 +s(η)(∇ζ)2� +. +(40) +3. +Ultra-slow roll inflation and its generalizations +Ultra-slow roll inflation (USR) is a particular case of globally adiabatic system, in general +characterized by the vanishing of δPnad = δPud = δP −cwδρ on any scale [9], where cw = P ′/ρ′, +and the subscript ”ud” stands for uniform density gauge, defined by the condition δρ = 0. In +USR models the quantity cw = P/ρ′ coincides with [9] the SESS cs = cw = 1. In other globally +adiabatic models such as generalized USR and Lambert inflation [10] cs = cw ̸= 1. +4. +MESS with multiple scalar fields +The momentum dependency of the sound speed has been found in some specific multi-fields +systems [11] where entropy modes can be integrated out analytically. This was generalized in a +model independent framework in[2, 7], defining the MESS for an arbitrary multi-field system, +including those in which entropy modes cannot be easily integrated out analytically, and for an +arbitrary field space metric. +5. +MESS in modified gravity +In modified gravity theories an effective entropy and anisotropy can arise in the comoving +gauge perturbed field equations, leading to a MESS depending on the specific gravity theory +[5]. An equivalent definition can be obtained from the Lagrangian describing the perturbations, +using the effective action approach outlined in the previous section. +10 + +6. +Effective sound speed and entropy perturbations +The SESS was introduced for the first time in a model independent way in [7] as cs = δPc/δρc, +but that definition is only valid for systems with entropy perturbations, but no anisotropy, while +the correct generalization including anisotropy was given in this paper in eq.(19). +It is easy to check that in absence of anisotropy eq.(19) is in agreement with eq.(29) in [7]. +From the perturbations equation we can in fact obtain S, and the corresponding interaction +Lagrangian Lint +S = − +1 +2a2z2a +� a3 +c2aH Γ +�′ +, +(41) +Lint = z2 +aLint = z2 +a +aΓ +2ǫH ζ′ = a3Γ +2c2aH ζ′ , +(42) +g = +1 +z2a +� +z2 +aa2S dη = − aΓ +2ǫH , +(43) +c2 +s = c2 +a +� +1 − g +ζ′ +�−1 += c2 +a +� +1 + Lint +ζ′2 +�−1 += +� +1 + +aΓ +2ǫHζ′ +�−1 +, +(44) +which shows explicitly that entropy and curvature perturbations are coupled already at second +order in the term Lint ∝ Γζ′, explaining why a momentum dependent cs arises already from +the quadratic action, while the calculation of the momentum dependency due to the anisotropy +require the cubic action. +E. +New predictions and applications +1. +MESS in vanilla inflation due to self interaction +Even in the vanilla scenario higher order interaction terms are expected to induce a mo- +mentum dependency of the effective sound speed, associated to cubic and higher order terms. +These effects are ignored in leading order calculations, but arise naturally at higher order. +For example, for the scalar perturbations we have the interaction Lagrangian [6] +L(3) +int = a4 +� ǫ2 +a2ζ′2ζ + 1 +a2ǫ2(∂iζ)2ζ −2 ǫ +aζ′∂iζ∂iχ − 1 +2 +ǫ3 +a2ζ′2ζ + 1 +2ǫζ(∂i∂jχ)2 + 1 +2 +ǫ +a2η′ζ′ζ2 +� +, +(45) +where ∂2χ = ζ′ǫ/a. In momentum space we can compute the MESS +˜c2 +s(η, k) = +� +1 + L(3) +k,int +ζ′2 +k +�−1 +, +(46) +11 + +where L(3) +k,int = z2L(3) +k,int is the Fourier transform of L(3) +int. The MESS encodes the effects of self- +interaction on ζ, which are associated to loop corrections of the power spectrum [12], which +can become large when slow-roll is violated [10]. +2. +MESS in modified gravity +For a specific case of Horndeski theory [5] the MESS was computed in the comoving gauge, +showing explicitly that it is momentum dependent, as expected. +The same result can be +extended to other modified gravity theories once the cubic and higher order actions have been +computed. For example for Horndeski theory the cubic action was computed in [13], including +the coupling of tensor and scalar perturbations. +The model independent approach we have derived does not require any definition of entropy +perturbations, and it includes the effects of anisotropy, since they are both related to interactions +terms. +3. +f(R) theories +In the Einstein’s frame f(R) theories are mathematically equivalent to general relativity with +a minimally coupled scalar field. The anisotropic part of the EST of the scalar field arises only +at second order in scalar perturbations, is proportional to the space derivatives δφ,i, δφ,j [14], +and is associated to cubic terms in the action [13], coupling also tensor and scalar perturbations, +which are not included in the quadratic actions. +Cubic order calculations are expected to show the momentum dependency predicted by the +MESS approach, which can be interpreted as the effects of the anisotropy of the EST, which +corresponds to cubic self-interaction terms in the Lagrangian, and of the coupling of scalar and +tensor perturbations. +4. +MESS in axion inflation +The coupling of scalar perturbations with a gauge field induces a momentum dependency of +the MESS, which should arise already in the quadratic Lagrangian. For example the quadratic +interaction Lagrangian can contain terms of the form +L(2)int +ζ +∝ δAµ∂µζ , δAµ∂µh +(47) +12 + +where δAµ denotes perturbations of the gauge field Aµ, while at higher order other terms can +appear such as for example +L(3)int +ζ +⊃ ∂ν(δAµ)∂µζ∂νζ +, +δAµδAµζ +, +δFµν∂µζ∂νζ , +(48) +where Fµν denoted the perturbations of the Faraday tensor Fµν = ∂µAν − ∂νAµ. The effects +of these interaction terms are often ignored in the literature, but a priori there is no no gen- +eral argument to justify that they can be always neglected. These interaction terms give rise +to effects similar to those associated to entropy in multi-fields scalar systems, which can be +dominant [2], and are worth being studied systematically. +III. +GRAVITATIONAL WAVES +A. +Space effective gravitational wave speed +Adopting an approach similar to the one used for scalar perturbations, we derive an effective +action and propagation equation for gravitational waves. +1. +Effective Lagrangian approach +The effective Lagrangian for gravitational waves can be obtained with a method analogous +to the one used for scalar perturbations. In this section we will use a slightly different notation +for the Lagrangian density +S = +� +dηdxL = +� +dηdx√−gL = +� +dηdx a2L +, +L = a2L . +(49) +The Lagrangian for the polarization mode hA in general relativity is +LGR +h += a2� +h′2 +A − (∇hA)2� +, +(50) +and adding interaction terms we have +Lh = LGR +h ++ Lint +h += a2� +h′2 +A − (∇hA)2 + Lint +h (hA, φi) +� += a2� +h′2 +A +� +1 + Lint +h +h′2 +A +� +− (∇hA)2� +, +(51) +where φi denotes abstractly all the other fields the graviton is coupled to, including itself, or +another polarization. We can then obtain the effective Lagrangian [8] +Leff +h += a2 +c2 +T,A +� +h′2 +A − c2 +T,A(∇hA)2� += α2� +h′2 +A − c2 +T,A(∇hA)2� +, +(52) +13 + +by defining the space effective GW speed (SEGS) as +c2 +T,A(η, xi) = +� +1 + Lint +h +h′2 +A +�−1 +. +(53) +The effective Lagrangian for gravitational waves has the same structure of that for comoving +curvature perturbations in eq.(4), and one can in fact be obtained from the other by the +transformation +z ←→ a +, +cs ←→ cT,A , +, +ζ ←→ hA . +(54) +The effective description in terms of SESS and SEGS is the same, i.e. the effective action +and equation are universal, and can be used for both scalar and tensor perturbations. It is +convenient to organize the equations for scalar and tensor perturbations in a table, to show the +universality of the effective approach. +Gravitational waves +Curvature perturbations +Speed +c2 +T,A(η, xi) = +� +1+ +Lint +h +h′2 +A +�−1 += +� +1− gA +h′ +A +�−1 +c2 +s(η, xi) = +� +1 + +Lint +ζ +ζ′2 +�−1 += c2 +a(η) +� +1 − g +ζ′ +�−1 +Leff , α +a2 +c2 +T,A +� +h′2 +A − c2 +T,A(∇hA)2� +, αA = +a2 +c2 +T,A +z2 +c2s +� +ζ′2 − c2 +s(∇ζ)2� +, α = ǫa2 +c2s = z2 +c2s +Eq. +h′′ +A + 2 +α′ +A +αAh′ +A − c2 +T,A∇2hA = 0 +ζ′′ + 2 α′ +α ζ′ − c2 +s∇2ζ = 0 +Eq. +h′′ +A + 2 +� +a′ +a − +c′ +T,A +cT,A +� +h′ +A − c2 +T,A∇2hA = 0 +ζ′′ + 2 +� +z′ +z − c′ +s +cs +� +ζ′ − c2 +s∇2ζ = 0 +2. +Effective stress-energy tensor approach +The perturbed field equations [1] +h′′ +A + 2Hh′ +A + ∇2hA = a2Πeff +A +, +(55) +can be manipulated to get the same model independent equation for gravitational waves that +was obtained using the effective Lagrangian approach, but with the SEGS defined in terms of +the EST as [8] +c2 +T,A(η, xi) = +� +1 − gA +h′ +A +�−1 +, +gA = 1 +a2 +� +a4Πeff +A +dη . +(56) +B. +Momentum effective gravitational wave speed +Using a method similar to the one used in physical space, it is possible to derive a model +independent effective action and equation in momentum space. The results are summarized in +the table. We are denoting with ˜h the Fourier transform of h. +14 + +Gravitational waves +Curvature perturbations +Speed +˜c2 +T,A(η, k) = +� +1+ +Lint +˜h +˜h′2 +A +�−1 += +� +1− ˜gA +˜˜h′ +A +�−1 +˜c2 +s(η, k) = +� +1 + +Lint +ζk +ζ′2 +k +�−1 += ca(η)2� +1 − ˜g +ζ′ +k +�−1 +Leff , α +a2 +˜c2 +T,A +� +˜h′2 +A + k2˜c2 +T,A˜h2 +A +� +, ˜αA = +a2 +˜c2 +T,A +z2 +˜c2s +� +ζ′2 +k + k2˜c2 +sζ2 +k +� +, ˜α = ǫa2 +˜c2s = z2 +˜c2s +Eq. +˜h′′ +A + 2 +˜α′ +A +˜αA ˜h′ +A + k2˜c2 +T,A˜h2 +A = 0 +ζ′′ +k + 2 ˜α′ +˜α ζ′ +k + k2˜c2 +sζ2 +k = 0 +Eq. +˜h′′ +A + 2 +� +a′ +a − +˜c′ +T,A +˜cT,A +� +˜h′ +A + k2˜c2 +T,A˜h2 +A = 0 +ζ′′ +k + 2 +� +z′ +z − ˜c′ +s +˜cs +� +ζ′ +k + k2˜c2 +sζ2 +k = 0 +C. +Effective metric description +Similarly to curvature perturbations, the effective Lagrangian for gravitational waves can be +written as +Leff +h += √−gA(∂µhA∂µhA) , +(57) +in terms of the effective metric +ds2 +A = a2� +cT,Adη2 − δij +cT,A +dxidxj� +, +(58) +for which the GW propagation equation can be written in terms of the covariant d’Alembert +operator +□hA = +1 +√−gA +∂µ(√−gA∂µhA) = 0 . +(59) +As for scalar perturbations, the effects of the interaction of the graviton can be described as +the propagation in a curved space whose metric depends on the SEGS. A similar result can be +derived in momentum space. +D. +Consistency with previous calculations +The model independent action and Lagrangian derived in the previous sections are consistent +and extend previous calculation based on quadratic action calculations. +1. +Effective field theory of inflation +The quadratic order action for tensor modes obtained using the effective field theory of infla- +tion [15] is in agreement with eq.(52), but the latter includes also higher order interaction terms +neglected in the quadratic action, which induce the polarization and frequency dependency of +the speed. +15 + +2. +Horndeski’s theory +The quadratic action for Horndeski’s theory has been computed in [16, 17]. These calcula- +tions are in the Jordan frame, while in the previous section we have used the Einstein frame. +After performing the appropriate disformal transformation [8] it can be shown that the tensor +modes actions are in agreement at second order, while eq.(52) is including also the effects of +higher order interaction terms [13], associated to self interaction and tensor scalar coupling. +E. +New predictions and applications +In this section we consider some examples of application of the effective approach derived +previously. +1. +Scalar tensor feedback and effective speeds +As an example, let’s consider the interaction term +b L(3) +ζζh = b a2hij∂iζ∂jζ = b a2L(3) +ζζh = z2b +ǫ +� +h+(∂xζ∂xζ−∂yζ∂yζ)+2h×(∂xζ∂yζ) +� += z2b +ǫ +� +h+π++h×π× +� +, +(60) +which arises at cubic order in general relativity [6] and modified gravity theories [13]. The +Langrange equations give +h′′ +A + 2Hh′ +A + ∇2hA = a2 b πA = a2ΠA , +(61) +ζ′′ + 2z′ +z ζ′ − c2 +s∇2ζ = a2 b hij∂i∂jζ = a2S . +(62) +Contrary to the linear regime, tensor and scalar modes are coupled, and it is necessary to solve a +system coupled differential equations to compute the effects of the interactions. Using the field +equations approach the effective sound speed for scalar and tensor modes could be computed, +or we could also use the effective Lagrangian approach. +Using the notation introduced in the previous sections we have +Lζ = L(2) +ζ ++ Lint +ζ += z2� +ζ′2 − (∇ζ)2 + b +ǫL(3) +ζζh +� +, +(63) +Lh = L(2) +h + Lint +h += a2� +h′2 − (∇h)2 + b L(3) +ζζh +� +, +(64) +16 + +from which we obtain the SEGS and SESS +c2 +T,A(η, xi) = +� +1 + Lint +h +h′2 +A +�−1 += +� +1 + b L(3) +ζζh +h′2 +A +�−1 +, +(65) +c2 +s(η, xi) = +� +1 + Lint +ζ +ζ′2 +�−1 += +� +1 + b L(3) +ζζh +ǫζ′2 +�−1 +. +(66) +In terms of the SESS and SEGS, the system of coupled differential equations with source terms +reduces to three independent equations, without sources +h′′ +A + 2α′ +A +αA +h′ +A − c2 +T,A∇2hA = 0 +, +αA = a2 +c2 +T,A +, +(67) +ζ′′ + 2α′ +α ζ′ − c2 +s∇2ζ = 0 +, +α = ǫa2 +c2 +s += z2 +c2 +s +. +(68) +Note that +Lζ + Lh + Lint ̸= Leff +ζ ++ Leff +h +(69) +because the interaction Lagrangian Lint enters in the definition of both effective speeds, analo- +gously to the fact it produces different source terms Sζ and ΠA in the equations of motions. The +effects of the interaction induce a modification of both speeds, since the interaction produces +a source term in both equations, while in the literature often only the effects on gravitational +waves are considered, ignoring those on scalar perturbations, and their possible back-reaction +on tensor modes. +IV. +CHANGE OF FRAME AND PHYSICAL RELEVANCE OF THE EFFECTIVE +PLANCK MASS +All the result derived in the previous sections were in the Einstein frame, both using the +effective Lagrangian and the EST approach. +It can be convenient to find the conformal transformation taking to the Jordan frame, in +which many other calculations have been performed, especially in the context of modified +gravity [16, 17]. +A. +Curvature and tensor perturbations +Let’s start by writing the scalar and tensor quadratic Lagrangians in the Jordan frame [18] +LJ = a2qs[ζ′2 − c2 +s(∇ζ)2� ++ a2qt +� +h′2 − c2 +T(∇h)2� +, +(70) +17 + +where we are denoting with E and J quantities in the Jordan and Einstein frame. +The Lagrangian of perturbations in a new frame obtained via disformal transformation, such +that ˜a = Ωa, is of the form [18] +˜LJ = ˜a2˜qs[˜ζ′2 − ˜c2 +s(∇˜ζ)2� ++ a2˜qt +� +˜h′2 − ˜c2 +T(∇˜h)2� +, +(71) +where we are denoting with a tilde quantities in the new frame, and we are just defining +the conformal part of the disformal transformation, because this is the one relevant for the +coefficients of the Lagrangians for curvature and tensor modes. +Since scalar and tensor perturbations are invariant under disformal transformations [18], i.e. +˜ζ = ζ, ˜h = h, and L = ˜L by definition, by comparing the coefficients [18] in L and ˜L we obtain +the following transformations +˜a = Ωa +→ +˜qs = Ω−2qs , ˜qt = Ω−2qt , cs = ˜cs , cT = ˜cT , +(72) +Note that these relations are consistent, but different, from those derived in [18], due to the +use of conformal time. +The corresponding effective quadratic Lagrangian in the Einstein frame is +Leff +E += ˜ǫ˜a2 1 +c2s +� +ζ′2 − c2 +s(∇ζ)2� ++ ˜a2 1 +c2 +T +� +h′2 − c2 +T(∇h)2� += Leff +ζ ++ Leff +h +, +(73) +where we are using the conformal invariance of cs, cT, ζ, h, and we are denoting with ˜ǫ the slow +roll parameter in the Einstein frame, to distinguish it from that in the Jordan frame. We use this +notation only in this section, for consistency with the notation for conformal transformation +starting from the Jordan, while in other sections ǫ denotes the slow-roll parameter in the +Einstein frame. Note that the Lagrangian above is consistent with the scalar perturbations +action for K−inflation in the Einstein frame [4], and with the tensor action in the effective field +theory of inflation [15]. By comparing eq.(73) and eq.(72) we find that the Einstein frame and +Jordan frame are related by +˜qt = +1 +c2 +T +, +(74) +˜qs = +˜ǫ +c2 +s +. +(75) +which imply +Ω = cT +√qt , +(76) +˜ǫ = qsc2 +s +qtc2 +T +, +(77) +18 + +in agreement with [8]. +The parameter qt is sometimes denoted at M2 +∗ , and interpreted as effective Planck mass +[19]. The above equations show that in the Einstein frame the effective Planck mass is not +independent from cT , since √˜qt = ˜ +M∗ = M∗Ω−1 = 1/cT, and its effects on scalar perturbations +are encoded in Einstein frame slow-roll parameter ˜ǫ. +This is not surprising, since in the Einstein frame the only physically relevant quantities are +cs, cT, ˜ǫ, i.e. the so called Jordan frame effective Planck mass M∗ emerges only in the definition +of the conformal transformation taking to the Jordan frame, and as such does not really play +any physical role, due to the invariance of scalar and tensor perturbations under disformal +transformation [18]. +As explained earlier, the time dependency of the cs(η) and cT(η) is related to self interactions +terms of the form (1/c2 +s(η) − 1)ζ′2 and [15] h′2(1/c2 +T(η) − 1), but in this case, since ζ and h are +not coupled to each other, but only self-coupled, we have +LE = LE +ζ + LE +h + LE +int = ˜ǫ˜a2� +ζ′2 − (∇ζ)2� ++ ˜a2� +h′2 − (∇h)2� ++ Lint +h + Lint +ζ += Leff +ζ ++ Leff +h +, +Lint +h += ˜a2h′2� +1 +c2 +T (η) − 1 +� +, Lint +ζ += ˜ǫ˜a2� +1 +c2s(η) − 1 +� +ζ′2 +(78) +which was used to find the relation between the quantities in different from +LJ = LE = Leff +ζ ++ Leff +h +. +(79) +The first equality in the above equation just comes from the fact that Lagrangian densities in +different frames are equal, because they are obtained by simply re-writing the same object in +different ways. +The conformal transformation defined in eq.(77) can also be applied to the Lagrangians +including higher order interactions terms [8], and is consistent with the invariance of the coef- +ficients of the perturbations equations expected from the invariance of ζ, h, i.e. the invariance +of the solutions implies the invariance of the equations and of the Lagrangians. In other words +the equations and Lagrangians in the Jordan frame look different because of the conformal +transformation, but the coefficients of the equations and of the Lagrangians as functions of +space and time are the same, just written in a different way in terms of Ω. In the Einstein +frame the real number of independent degrees of freedom is more transparent. +Note we are assuming that the coupling to gravity is properly transformed from one frame +to the other. This implies that theories in which matter fields are minimally coupled to gravity +19 + +by the metric in the Einstein vs Jordan frame are different, i.e. they have different Lagrangians, +which cannot be simply related by a conformal transformation. +B. +Effective gravitational coupling +The perturbed Einstein equations give the useful equation +k2 +˜a2 ˜ψB = 1 +2δ˜ρc +(80) +where ˜ψB is one of the Bardeen potentials, and ˜δρc is the comoving energy density perturbation +[1], and the tilde denotes quantities in the Einstein frame. The above equation is also valid for +a modified gravity theory, once it has been transformed to the Einstein frame, in which the full +Lagrangian takes the Hilbert form +˜L = +� +−˜g ˜R + ˜Ltot = +� +−˜g ˜R + ˜Lm + ˜Lφ + ˜Lmφ , +(81) +k2 +˜a2 ˜ψB = 1 +2δ˜ρtot +c +, δ˜ρtot +c += δ˜ρm +c + δ˜ρφ +c + δ˜ρmφ +c +, +(82) +where ˜Ltot contains terms associated to matter and the modification of gravity in a single +object, and ˜Lφ, ˜Lm, ˜Lmφ are respectively the Einstein frame Lagrangians related to the gravity +modification, matter, and the non minimal coupling of gravity to matter, and a similar notation +is adopted for the comoving energy density. +In the Jordan frame the Poisson equation takes the form [20]. +k2 +a2 ψB = 1 +2GJδρc +(83) +The difference between the gravitational couplings in the two frames is expected, since in in our +units in the Einstein frame we have by definition 8πGtot +E = 1, and the effects of the modification +of gravity are encoded in the total effective energy density δ˜ρtot +c , which includes contributions +from ˜Lφ, ˜Lm, ˜Lmφ, not just from ˜Lm. The relation between the gravitational coupling Gm +E for +matter comoving energy density δ˜ρm +c will be investigated in more details in a future work, but +we can anticipate that by appropriately manipulating eq.(82) we can write an equation of the +form +k2 +˜a2 ˜ψB = 1 +2Gm +Eδ˜ρm +c . +(84) +Note that, contrary to curvature and tensor perturbations, the Bardeen potential and the +comoving density perturbations are not invariant under disformal transformations [21], and for +this reason we use a tilde to distinguish between them. +20 + +V. +RELATION TO OTHER EFFECTIVE APPROACHES +The effective approach formulated in this paper is completely general, and as such includes +all the effects of interaction at any order in a single effective quantity, and for an arbitrary +number fields. We can compare this with previous results to see how it includes and extend +them. +A. +Effective field theory of inflation +The effective approach we have derived can describe the evolution of curvature not only +for single field models, but also for multi-fields models [2], while in the EFT of inflation [22] +approach it is assumed only one scalar degree of freedom, i.e. entropy perturbations are ignored, +and no general effective action for curvature perturbations is derived for multi-fields systems +[23]. The MESS approach allows to compute the effects of entropy and anisotropy on curvature +perturbations for a generic system, including any number of fields, and predicts naturally the +momentum dependence of the effective sound speed of curvature perturbations. +B. +Effective field theory of dark energy +The effective field theory of dark energy [24] applies the same symmetry breaking idea of +the EFT of inflation to dark energy, but in the Jordan frame. +The action is expanded to +quadratic order, and for this reason is missing the frequency and polarization dependence of +the effective speed which arises naturally in the MESS and SEGS approach, due to the higher +order interactions terms. The general relation between Jordan and Einstein frame was derived +in the previous section. +VI. +QUANTUM FIELD THEORY IMPLICATIONS +The effective Lagrangians we have derived are based on classical calculations, but they can +be related [25] to the wavefunction of the scalar and tensor fields by the path integral +Ψ[ϕ] = +� +φ(t) = ϕ +φ(−∞) = 0 +Dφ eiS[φ] . +(85) +21 + +where φ is a generic field, which in our case could be ζ or h, and S denotes the action. At tree +level the path integral can be approximated by the action evaluated on the classical solution, +which is the way in which we define the effective Lagrangian. +From the wavefunction we can compute the equal-time correlators as +⟨φ1 · · · φN⟩ = +� +Dφ φ1 · · · φN |Ψ[φ]|2 . +(86) +The above method should give the same result obtained by using canonical quantization in +the in-in formalism [6]. +Following this method we can for example compute corrections to +the spectrum, arising from higher order interaction terms in the Lagrangian, in terms of the +effective speed we have defined previously. More details about this approach will be given in a +separate work. +VII. +CONCLUSIONS +We have derived a set of universal model independent effective equations and Lagrangians +which can describe the evolution of scalar and tensor perturbation of any system whose field +equations can be written in an Einstein like form. This includes for example multi-fields systems, +or modified gravity, once they have been transformed to the Einstein frame. Given the generality +of this effective description it is particularly suitable for model independent phenomenological +analysis of observational data. +This approach predicts naturally that the speed of gravitational waves con depend on fre- +quency and polarization, due to the interactions of the graviton with itself or other fields. This +prediction allows to use gravitational waves observations to investigate the elusive nature of +dark matter and dark energy. +The equation and Lagrangian for scalar and tensor perturbations has the same universal +structure, and the effects of the interaction can be modeled at any order in perturbations by +a single effective quantity, playing the role of effective propagation speed. This is particularly +useful since it allows to compare different models in terms of the two quantities cs and cT. +Combining different set of observational data such as cosmic microwave background radiation +and gravitational waves, it will be possible to constrain the cs and cT,A, to determine possible +deviations from general relativity and vanilla inflation. If a deviation is found, the theoretical +research can be focused on those models able to predict the cs and cT supported by observations. +In order to test specific models it will be important to perform higher order perturbations +22 + +calculations for different models, in order to compute the effects which are not included in +the quadratic action, and in the EST approach are treated as effective model independent +phenomenological quantities. +Acknowledgments +I thank Misao Sasaki, Tessa Baker, Sergio Vallejo, Riccardo Sturani, Rogerio Rosenfeld, +Nicola Tamanini and Suvodip Mukherjee for interesting discussions. I thank the ICTP-SAIFR +for the kind hospitality during the preparation of this paper. +[1] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1 (1984). +[2] A. E. Romano, S. A. Vallejo-Pe˜na, and K. Turzy´nski, (2020), arXiv:2006.00969. +[3] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974). +[4] J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999), arXiv:hep-th/9904176. +[5] S. A. Vallejo-Pena and A. E. Romano, (2019), arXiv:1911.03327. +[6] J. M. Maldacena, JHEP 05, 013 (2003), arXiv:astro-ph/0210603. +[7] A. E. Romano and S. A. Vallejo Pena, Phys. Lett. B 784, 367 (2018), arXiv:1806.01941. +[8] A. E. Romano, (2022), arXiv:2211.05760. +[9] A. E. Romano, S. Mooij, and M. Sasaki, Phys. Lett. B 755, 464 (2016), arXiv:1512.05757. +[10] A. E. Romano, S. Mooij, and M. Sasaki, Phys. Lett. B 761, 119 (2016), arXiv:1606.04906. +[11] A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma, and S. P. Patil, JHEP 05, 066 (2012), +arXiv:1201.6342. +[12] J. Kristiano and J. 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Senatore, JHEP 03, 014 (2008), +arXiv:0709.0293. +[23] L. Senatore and M. Zaldarriaga, JHEP 04, 024 (2012), arXiv:1009.2093. +[24] G. Gubitosi, F. Piazza, and F. Vernizzi, JCAP 02, 032 (2013), arXiv:1210.0201. +[25] J. Bonifacio, H. Goodhew, A. Joyce, E. Pajer, and D. Stefanyszyn, (2022), arXiv:2212.07370. +24 + diff --git a/tNE5T4oBgHgl3EQfmg9S/content/tmp_files/load_file.txt b/tNE5T4oBgHgl3EQfmg9S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89fea5a8b8b4f8827bf75c7b1b0463a8c768fb9d --- /dev/null +++ b/tNE5T4oBgHgl3EQfmg9S/content/tmp_files/load_file.txt @@ -0,0 +1,481 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf,len=480 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='05679v1 [gr-qc] 13 Jan 2023 A unified effective approach to cosmological perturbations Antonio Enea Romano Theoretical Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' CERN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' CH-1211 Geneva 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Switzerland and ICRANet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Piazza della Repubblica 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' I–65122 Pescara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Italy (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 2023) Abstract A unified model independent effective description of cosmological perturbations is derived in terms of two effective quantities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' playing the role of effective propagation speeds of curvature perturbations and gravitational waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' encoding the effects of the interaction of perturbations at any order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' and inducing a modification of the friction term of the perturbations propagation equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The approach can be applied to dark energy, modified gravity, dark matter, for fields of arbitrary number and spin, and is particularly suitable for model independent analysis of observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The structure of the effective actions and equations is the same for scalar and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effective actions can be written as the Klein-Gordon action in terms of an appropriately defined effective metric, dependent on the effective speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In this geometrical interpretation, the effective metric emerges as the result of the interaction and self-interaction of perturbations, hinting to possible connections with emergent gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' As an example of an application we find that the effective speeds of curvature perturbations and gravitational waves can be frequency and polarization dependent even for a minimally coupled scalar field in general relativity, when higher order terms effects are computed, going beyond the quadratic action, or in axion inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We discuss the relation between the effective speed and quantum corre- lators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' INTRODUCTION Cosmological perturbations [1] play a fundamental role in modeling the Universe, since they allow to study the origins of large structure, the anisotropies of the cosmic microwave back- ground (CMB) radiation, and the propagation of gravitational waves on cosmological distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We derive a set of universal model independent effective equations and Lagrangians which can describe the evolution of scalar and tensor perturbation of any system whose field equations can be written in an Einstein like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This includes for example multi-fields systems [2], or modified gravity theories such as Horndeski’s theory[3], once they have been transformed to the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Given the generality of this effective description it is particularly suitable for model independent phenomenological analysis of observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The approach predicts naturally that the speed of gravitational waves cT,A con depend on frequency and polarization, due to the interactions of the graviton with itself or other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This prediction allows to use gravitational waves observations to investigate the elusive nature of dark matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The equation and Lagrangian for scalar and tensor perturbations has the same universal structure, and the effects of the interaction can be modeled at any order in perturbations by a single effective quantity, playing the role of effective propagation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is particularly useful since it allows to compare different models in terms of the two effective quantities cs and cT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Combining different set of observational data such as cosmic microwave background radiation and gravitational waves, it will be possible to constrain cs and cT,A, to determine possible deviations from general relativity and vanilla inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effective equations and Lagrangians are derived separately for scalar and tensor per- turbations, both in physical and momentum space, and the consistency with previous results derived in the literature is checked in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Other predictions are then considered, such as the frequency dependency of cs and cT in vanilla inflation due to the effects of higher order interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' SCALAR PERTURBATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Space dependent effective sound speed In this section we will show that it is possible to define a space dependent effective sound speed (SESS) in terms of which a model independent equation for comoving curvature pertur- bations ζ can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The SESS encodes the effects of the interaction of ζ with itself and other fields, at any order in perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective Lagrangian approach In the action approach the EST on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' of the Einstein’s equations originates from the interaction of scalar perturbations with themselves or other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Based on this we can obtain an effective Lagrangian corresponding to the perturbed field equations by introducing higher order interaction terms as Lint(ζ, φi), where φi denotes abstractly all the other fields ζ is coupled to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We will use conformal time, and adopt the following notation for the Lagrangian density L S = � dηdxL = � dηdxz2L , L = z2L , z2 = ǫa2 , (1) where we are assuming a Friedman background (FRW) background with scale factor a(η), and ǫ denotes the first order slow-roll parameter, defined in terms of cosmic time dt = a−1dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In general relativity the quadratic Lagrangian of the comoving curvature perturbations of a minimally coupled scalar field is L(2) ζ = z2� ζ′2 − (∇ζ)2� , z2 = ǫa2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (2) We will call the above model vanilla inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Including higher order interaction and self interaction terms we can write a general model independent Lagrangian Lζ = L(2) ζ + Lint ζ = z2� ζ′2 − (∇ζ)2 + Lint ζ (ζ, φi) � = z2� ζ′2� 1 + Lint ζ ζ′2 � − (∇ζ)2� , (3) where φi denotes collectively any other field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' From the above equation we can obtain the effective Lagrangian Leff ζ = z2 c2s � ζ′2 − c2 s(∇ζ)2� = α2� ζ′2 − c2 s(∇ζ)2� , α2 = z2 c2s = ǫa2 c2s , (4) 3 where we have defined the space effective sound speed (SESS) according to c2 s(η, xi) = � 1 + Lint ζ ζ′2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (5) The variation of Leff ζ gives the model independent equation ζ′′ + 2α′ α ζ′ − c2 s∇2ζ = 0 , (6) which can be also written as ζ′′ + 2 �z′ z − c′ s cs � ζ′ − c2 s∇2ζ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (7) Note that in deriving the equations of motion for ζ the SESS has been treated as a function independent of ζ, since the SESS is an effective quantity determined by substituting the so- lutions of the full theory, including the effects of interaction, into the interaction Lagrangian Lint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' For any system with a well defined Lagrangian it should be always possible to solve the equations of motions, and then use the solutions to define the SESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (4) show that α(η, xi) can be interpreted as an effective space dependent scale factor, while eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (7) shows explicitly the modification of the friction term induced by the SESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Since these equations are model independent, we can immediately conclude that the friction terms cannot be modified if c′ s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effective Lagrangian can be obtained from the vanilla case in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (8) L(2) ζ = z2� ζ′2 − c2(∇ζ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (8) by the transformation z2 → α = z2 c2s , c → cs , (9) where we are denoting with c the unity sound speed, to avoid ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6), which shows that α can be regarded as an effective scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The main advantage of using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6) is that it is completely model independent, allowing to study in a systematic way deviations from general relativity or the effects of the interaction with different fields using a single function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that since Lint can be space dependent, also cs(η, xi) depends on space, with the exception of Lint ∝ f(η)ζ′2, when it is only time dependent, which corresponds to K-inflation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In presence of multiple scalar fields the space dependence is manifested already in the quadratic Lagrangian [5], while the effects of anisotropic perturbations requires at least a cubic Lagrangian, because scalar fields anisotropies appear at second order in the EST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 4 Note that even in vanilla inflation the sound speed can be space dependent when including the cubic and higher order Lagrangians [6], as we will show explicitly later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is consistent with the EST approach, since a scalar field EST is anisotropic at second order, leading to source terms in the perturbed field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Interpretation of the space dependency of the SESS The space dependency of the SESS is a natural manifestation of the interaction and self- interaction of scalar perturbations, and of the coupling of tensor and scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In multiple scalar fields systems these effects are already present in the quadratic Lagrangian, and are associated to entropy perturbations, while for a single scalar field they manifest only starting from the cubic Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective stress-energy tensor approach Scalar perturbations of the metric and of the Effective stress-energy tensor (EST) can be written as ds2 = −(1 + 2A)dt2 + 2a∂iBdxidt + a2 {δij(1 + 2C) + 2∂i∂jE} dxidxj , (10) T 0 0 = −(ρ + δρ) , T 0 i = (ρ + P)∂i(v + B) , T i j = (P + δP)δi j + δik∂k∂jΠ − 1 3δi j∇2Π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (11) where v is the velocity potential and ∇2 ≡ δkl∂k∂l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that any metric and EST can always be written in the above form, making all the results obtained from it completely model inde- pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The comoving slices gauge is defined by the condition (T 0i)c = 0, and we denote with a subscript c quantities evaluated on comoving slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In multiple fields systems the EST is the sum of the stress-energy tensors of each field, which in the comoving gauge is associated to entropy perturbations [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In the comoving gauge entropy perturbations Γ are introduced [1] by δPc(η, xi) = ca(η)2δρc(η, xi) + Γ(η, xi) , (12) where ca is interpreted as the adiabatic sound speed, and is by definition a function of time only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that this definition can be ambiguous [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The manipulation of the perturbed Einstein’s equation in the comoving gauge gives 5 ζ′′ + ∂ηz2 a z2a ζ′ − c2 a∇2ζ = a2S , (13) S = −c2 a ǫ ∇2Π − 1 2a2z2a � a3 c2aH � Γ + 2 3∇2Π ��′ , z2 a = ǫa2 c2a , (14) where we are denoting derivatives with respect to conformal time with a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The source term in the above equation can be absorbed into the definition of the space effective sound speed (SESS) following a similar approach to the one adopted for gravitational waves [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We can first re-write eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (14) as (ζ′z2 a)′ z2 a − (gz2 a)′ z2 a − c2 a∇2ζ = � ζ′z2 a(1 − g/ζ′) �′ z2 a − c2 a∇2ζ = 0 , (15) where we have defined g = 1 z2a � z2 aa2S dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (16) After introducing the quantities 1 + δ(η, xi) = � 1 − g ζ′ �−1/2 , α2 = z2 a (1 + δ)2 = ǫa2 c2 a(1 + δ)2 , (17) we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (15) as 1 z2a (α2ζ′)′ − c2 a∇2ζ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (18) Defining the space effective sound speed (SESS) as cs(η, xi) = ca(η) � 1 + δ(η, xi) � , (19) and re-writing α in terms of cs as α2 = ǫa2 c2s , (20) we finally obtain the model independent effective equation ζ′′ + 2α′ α ζ′ − c2 s∇2ζ = 0 , (21) which shows that cs is the correct definition of effective sound speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (21) is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6), obtained using the effective Lagrangian approach, is completely general, and it can be applied to study multi-fields models, modified gravity, dark energy or dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that the SESS definition given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (19) is more general that the one given in [7], which is including the effects of entropy, but not of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Using an effective Langrangian 6 approach there is no distinction between entropy and anisotropy, since they are both associated to interaction terms, and the effective description is more transparent than using the EST approach, but both methods lead to the same conclusions, since the source terms in the field equations are obtained from the variation of the interaction Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The SVT decomposition of the EST is valid at any order in perturbations, so eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (21) is including the effects of interaction at any order in perturbations, including self-interaction, and for this reason is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6), obtained by encoding in cs the effects of all higher order interaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Momentum effective sound speed In this section we will show that it is possible to define a momentum dependent effective sound speed (MESS) in terms of which a model independent equation for comoving curvature perturbations ζ can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The MESS encodes the effects of the interaction of ζ with itself and other fields, at any order in perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The MESS is not the Fourier transform of the SESS, but it is mathematically convenient, since it allows to obtain a model independent equation involving minimal changes of the vanilla case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective Lagrangian approach A model independent effective equation and Lagrangian can also be derived in momen- tum space, using the two different methods adopted previously, the field equations and action approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The Lagrangian in momentum space can be written as Lζk = L(2) ζk + Lint ζk = z2� ζ′2 k + k2ζ2 k + Lint ζk (ζk, φi k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (22) (23) The effective Lagrangian is Leff ζk = ˜α2� ζ′2 k + ˜c2 sk2ζ2 k � , ˜α2(η, k) = z2 ˜c2s = ǫa2 ˜c2s , (24) where we have defined the momentum effective sound speed (MESS) ˜cs and effective scalar factor as ˜c2 s(η, k) = � 1 + Lint k ζ′2 k �−1 , ˜α2(η, k) = ǫa2 ˜c2s = z2 ˜c2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (25) 7 which gives the equation ζ′′ k + 2 ˜α′ ˜α ζ′ k + ˜c2 sk2ζk = 0 , (26) which can be also be written as ζ′′ k + 2 �z′ z − ˜cs ′ ˜cs � ζ′ k + ˜c2 sk2ζk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (27) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (26) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (24) show that ˜α(η, k) can be interpreted as an effective momentum dependent scale factor, while eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (27) shows explicitly the modification of the friction term induced the MESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Since these equations are model independent, we can immediately conclude that the friction terms cannot be modified if ˜c′ s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effective Lagrangian can be obtained from the vanilla inflation action L(2) ζk = z2� ζ′2 k + c2k2ζ2 k � , (28) by the transformation z2 → ˜α = z2 ˜c2s , c → ˜cs , (29) where we are denoting with c the unity sound speed, to avoid ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (26), which shows that ˜α can be regarded as a momentum dependent effective scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that the quantities ˜cs and ˜α are not the Fourier transform of cs and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Field equations approach Taking the Fourier transform of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (14) we get ζ′′ k + ∂ηz2 z2 ζ′ k + c2 ak2ζk = a2Sk , (30) Sk = c2 a ǫ k2Πk − 1 2z2aa � a3 c2aH � Γk − 2 3k2Πk ��′ , z2 a = ǫa2/c2 a , (31) After introducing the quantities gk = 1 z2a � z2 aa2Sk dη , 1 + δk(η) = � 1 − gk ζ′ k �−1/2 , ˜α2 = z2 (1 + δk)2 = ǫa2 c2a(1 + δk)2 , (32) we can rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (31) as 1 z2(˜α2ζ′ k)′ + c2 ak2ζk = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (33) 8 Defining the momentum effective sound speed (MESS) as ˜cs(η, k) = ca(η) � 1 + δk(η) � , (34) and re-writing ˜α in terms of ˜cs as ˜α2 = ǫa2 ˜c2 s , (35) we finally obtain the model independent effective equation ζ′′ k + 2 ˜α′ ˜α ζ′ k + ˜c2 sk2ζk = 0 , (36) which shows that ˜cs is the correct definition of momentum effective sound speed, in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective metric description In terms of the effective metric ds2 eff = ǫa2� csdη2 − δij cs dxidxj� , (37) the effective Lagrangian can be written as Leff ζ = √−g(∂µζ∂µζ) = gζ µνdxµdxν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (38) for which the equation of motion is simply given by the convariant D’Alembert operator □ζ = 1 � −gζ ∂µ( � −gζ∂µζ) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (39) In this geometrical description the perturbations propagate in an empty curved space, whose geometry is determined by the interaction of the perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is conceptually analogous to the general relativistic geometrical interpretation of the effects of gravity in terms of geodesics in a curved space, whose geometry is determined by the EST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' More about this geometrical interpretation will be discussed in a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Consistency with previous calculations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Minimally coupled scalar field in general relativity The vanilla scenario corresponds to Lint = 0, leading to δ = 0, cs = ca = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The quantity cw = P ′/ρ′ does not give the correct definition of sound speed, since it does not coincide with [9] the SESS cs = 1 ̸= cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' K-inflation When the interaction Lagrangian is of the form Lint ∝ f(η)ζ′2 the SESS is just a function of time cs(η) = ca(η) and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In this case the effective action in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (4) and effective eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (6) are in agreement with [4] Leff ζ = z2 cs(η) � ζ′2 − c2 s(η)(∇ζ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (40) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Ultra-slow roll inflation and its generalizations Ultra-slow roll inflation (USR) is a particular case of globally adiabatic system, in general characterized by the vanishing of δPnad = δPud = δP −cwδρ on any scale [9], where cw = P ′/ρ′, and the subscript ”ud” stands for uniform density gauge, defined by the condition δρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In USR models the quantity cw = P/ρ′ coincides with [9] the SESS cs = cw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In other globally adiabatic models such as generalized USR and Lambert inflation [10] cs = cw ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' MESS with multiple scalar fields The momentum dependency of the sound speed has been found in some specific multi-fields systems [11] where entropy modes can be integrated out analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This was generalized in a model independent framework in[2, 7], defining the MESS for an arbitrary multi-field system, including those in which entropy modes cannot be easily integrated out analytically, and for an arbitrary field space metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' MESS in modified gravity In modified gravity theories an effective entropy and anisotropy can arise in the comoving gauge perturbed field equations, leading to a MESS depending on the specific gravity theory [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' An equivalent definition can be obtained from the Lagrangian describing the perturbations, using the effective action approach outlined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective sound speed and entropy perturbations The SESS was introduced for the first time in a model independent way in [7] as cs = δPc/δρc, but that definition is only valid for systems with entropy perturbations, but no anisotropy, while the correct generalization including anisotropy was given in this paper in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' It is easy to check that in absence of anisotropy eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (19) is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (29) in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' From the perturbations equation we can in fact obtain S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' and the corresponding interaction Lagrangian Lint S = − 1 2a2z2a � a3 c2aH Γ �′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (41) Lint = z2 aLint = z2 a aΓ 2ǫH ζ′ = a3Γ 2c2aH ζ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (42) g = 1 z2a � z2 aa2S dη = − aΓ 2ǫH ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (43) c2 s = c2 a � 1 − g ζ′ �−1 = c2 a � 1 + Lint ζ′2 �−1 = � 1 + aΓ 2ǫHζ′ �−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (44) which shows explicitly that entropy and curvature perturbations are coupled already at second order in the term Lint ∝ Γζ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' explaining why a momentum dependent cs arises already from the quadratic action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' while the calculation of the momentum dependency due to the anisotropy require the cubic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' New predictions and applications 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' MESS in vanilla inflation due to self interaction Even in the vanilla scenario higher order interaction terms are expected to induce a mo- mentum dependency of the effective sound speed, associated to cubic and higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' These effects are ignored in leading order calculations, but arise naturally at higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' For example, for the scalar perturbations we have the interaction Lagrangian [6] L(3) int = a4 � ǫ2 a2ζ′2ζ + 1 a2ǫ2(∂iζ)2ζ −2 ǫ aζ′∂iζ∂iχ − 1 2 ǫ3 a2ζ′2ζ + 1 2ǫζ(∂i∂jχ)2 + 1 2 ǫ a2η′ζ′ζ2 � , (45) where ∂2χ = ζ′ǫ/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In momentum space we can compute the MESS ˜c2 s(η, k) = � 1 + L(3) k,int ζ′2 k �−1 , (46) 11 where L(3) k,int = z2L(3) k,int is the Fourier transform of L(3) int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The MESS encodes the effects of self- interaction on ζ, which are associated to loop corrections of the power spectrum [12], which can become large when slow-roll is violated [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' MESS in modified gravity For a specific case of Horndeski theory [5] the MESS was computed in the comoving gauge, showing explicitly that it is momentum dependent, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The same result can be extended to other modified gravity theories once the cubic and higher order actions have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' For example for Horndeski theory the cubic action was computed in [13], including the coupling of tensor and scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The model independent approach we have derived does not require any definition of entropy perturbations, and it includes the effects of anisotropy, since they are both related to interactions terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' f(R) theories In the Einstein’s frame f(R) theories are mathematically equivalent to general relativity with a minimally coupled scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The anisotropic part of the EST of the scalar field arises only at second order in scalar perturbations, is proportional to the space derivatives δφ,i, δφ,j [14], and is associated to cubic terms in the action [13], coupling also tensor and scalar perturbations, which are not included in the quadratic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Cubic order calculations are expected to show the momentum dependency predicted by the MESS approach, which can be interpreted as the effects of the anisotropy of the EST, which corresponds to cubic self-interaction terms in the Lagrangian, and of the coupling of scalar and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' MESS in axion inflation The coupling of scalar perturbations with a gauge field induces a momentum dependency of the MESS, which should arise already in the quadratic Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' For example the quadratic interaction Lagrangian can contain terms of the form L(2)int ζ ∝ δAµ∂µζ , δAµ∂µh (47) 12 where δAµ denotes perturbations of the gauge field Aµ, while at higher order other terms can appear such as for example L(3)int ζ ⊃ ∂ν(δAµ)∂µζ∂νζ , δAµδAµζ , δFµν∂µζ∂νζ , (48) where Fµν denoted the perturbations of the Faraday tensor Fµν = ∂µAν − ∂νAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effects of these interaction terms are often ignored in the literature, but a priori there is no no gen- eral argument to justify that they can be always neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' These interaction terms give rise to effects similar to those associated to entropy in multi-fields scalar systems, which can be dominant [2], and are worth being studied systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' GRAVITATIONAL WAVES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Space effective gravitational wave speed Adopting an approach similar to the one used for scalar perturbations, we derive an effective action and propagation equation for gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective Lagrangian approach The effective Lagrangian for gravitational waves can be obtained with a method analogous to the one used for scalar perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In this section we will use a slightly different notation for the Lagrangian density S = � dηdxL = � dηdx√−gL = � dηdx a2L , L = a2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (49) The Lagrangian for the polarization mode hA in general relativity is LGR h = a2� h′2 A − (∇hA)2� , (50) and adding interaction terms we have Lh = LGR h + Lint h = a2� h′2 A − (∇hA)2 + Lint h (hA, φi) � = a2� h′2 A � 1 + Lint h h′2 A � − (∇hA)2� , (51) where φi denotes abstractly all the other fields the graviton is coupled to, including itself, or another polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We can then obtain the effective Lagrangian [8] Leff h = a2 c2 T,A � h′2 A − c2 T,A(∇hA)2� = α2� h′2 A − c2 T,A(∇hA)2� , (52) 13 by defining the space effective GW speed (SEGS) as c2 T,A(η, xi) = � 1 + Lint h h′2 A �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (53) The effective Lagrangian for gravitational waves has the same structure of that for comoving curvature perturbations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (4), and one can in fact be obtained from the other by the transformation z ←→ a , cs ←→ cT,A , , ζ ←→ hA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (54) The effective description in terms of SESS and SEGS is the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' the effective action and equation are universal, and can be used for both scalar and tensor perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' It is convenient to organize the equations for scalar and tensor perturbations in a table, to show the universality of the effective approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Gravitational waves Curvature perturbations Speed c2 T,A(η, xi) = � 1+ Lint h h′2 A �−1 = � 1− gA h′ A �−1 c2 s(η, xi) = � 1 + Lint ζ ζ′2 �−1 = c2 a(η) � 1 − g ζ′ �−1 Leff , α a2 c2 T,A � h′2 A − c2 T,A(∇hA)2� , αA = a2 c2 T,A z2 c2s � ζ′2 − c2 s(∇ζ)2� , α = ǫa2 c2s = z2 c2s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' h′′ A + 2 α′ A αAh′ A − c2 T,A∇2hA = 0 ζ′′ + 2 α′ α ζ′ − c2 s∇2ζ = 0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' h′′ A + 2 � a′ a − c′ T,A cT,A � h′ A − c2 T,A∇2hA = 0 ζ′′ + 2 � z′ z − c′ s cs � ζ′ − c2 s∇2ζ = 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective stress-energy tensor approach The perturbed field equations [1] h′′ A + 2Hh′ A + ∇2hA = a2Πeff A , (55) can be manipulated to get the same model independent equation for gravitational waves that was obtained using the effective Lagrangian approach, but with the SEGS defined in terms of the EST as [8] c2 T,A(η, xi) = � 1 − gA h′ A �−1 , gA = 1 a2 � a4Πeff A dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (56) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Momentum effective gravitational wave speed Using a method similar to the one used in physical space, it is possible to derive a model independent effective action and equation in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The results are summarized in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We are denoting with ˜h the Fourier transform of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 14 Gravitational waves Curvature perturbations Speed ˜c2 T,A(η, k) = � 1+ Lint ˜h ˜h′2 A �−1 = � 1− ˜gA ˜˜h′ A �−1 ˜c2 s(η, k) = � 1 + Lint ζk ζ′2 k �−1 = ca(η)2� 1 − ˜g ζ′ k �−1 Leff , α a2 ˜c2 T,A � ˜h′2 A + k2˜c2 T,A˜h2 A � , ˜αA = a2 ˜c2 T,A z2 ˜c2s � ζ′2 k + k2˜c2 sζ2 k � , ˜α = ǫa2 ˜c2s = z2 ˜c2s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' ˜h′′ A + 2 ˜α′ A ˜αA ˜h′ A + k2˜c2 T,A˜h2 A = 0 ζ′′ k + 2 ˜α′ ˜α ζ′ k + k2˜c2 sζ2 k = 0 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' ˜h′′ A + 2 � a′ a − ˜c′ T,A ˜cT,A � ˜h′ A + k2˜c2 T,A˜h2 A = 0 ζ′′ k + 2 � z′ z − ˜c′ s ˜cs � ζ′ k + k2˜c2 sζ2 k = 0 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective metric description Similarly to curvature perturbations, the effective Lagrangian for gravitational waves can be written as Leff h = √−gA(∂µhA∂µhA) , (57) in terms of the effective metric ds2 A = a2� cT,Adη2 − δij cT,A dxidxj� , (58) for which the GW propagation equation can be written in terms of the covariant d’Alembert operator □hA = 1 √−gA ∂µ(√−gA∂µhA) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (59) As for scalar perturbations, the effects of the interaction of the graviton can be described as the propagation in a curved space whose metric depends on the SEGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' A similar result can be derived in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Consistency with previous calculations The model independent action and Lagrangian derived in the previous sections are consistent and extend previous calculation based on quadratic action calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective field theory of inflation The quadratic order action for tensor modes obtained using the effective field theory of infla- tion [15] is in agreement with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (52), but the latter includes also higher order interaction terms neglected in the quadratic action, which induce the polarization and frequency dependency of the speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Horndeski’s theory The quadratic action for Horndeski’s theory has been computed in [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' These calcula- tions are in the Jordan frame, while in the previous section we have used the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' After performing the appropriate disformal transformation [8] it can be shown that the tensor modes actions are in agreement at second order, while eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (52) is including also the effects of higher order interaction terms [13], associated to self interaction and tensor scalar coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' New predictions and applications In this section we consider some examples of application of the effective approach derived previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Scalar tensor feedback and effective speeds As an example, let’s consider the interaction term b L(3) ζζh = b a2hij∂iζ∂jζ = b a2L(3) ζζh = z2b ǫ � h+(∂xζ∂xζ−∂yζ∂yζ)+2h×(∂xζ∂yζ) � = z2b ǫ � h+π++h×π× � , (60) which arises at cubic order in general relativity [6] and modified gravity theories [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The Langrange equations give h′′ A + 2Hh′ A + ∇2hA = a2 b πA = a2ΠA , (61) ζ′′ + 2z′ z ζ′ − c2 s∇2ζ = a2 b hij∂i∂jζ = a2S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (62) Contrary to the linear regime, tensor and scalar modes are coupled, and it is necessary to solve a system coupled differential equations to compute the effects of the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Using the field equations approach the effective sound speed for scalar and tensor modes could be computed, or we could also use the effective Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Using the notation introduced in the previous sections we have Lζ = L(2) ζ + Lint ζ = z2� ζ′2 − (∇ζ)2 + b ǫL(3) ζζh � , (63) Lh = L(2) h + Lint h = a2� h′2 − (∇h)2 + b L(3) ζζh � , (64) 16 from which we obtain the SEGS and SESS c2 T,A(η, xi) = � 1 + Lint h h′2 A �−1 = � 1 + b L(3) ζζh h′2 A �−1 , (65) c2 s(η, xi) = � 1 + Lint ζ ζ′2 �−1 = � 1 + b L(3) ζζh ǫζ′2 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (66) In terms of the SESS and SEGS, the system of coupled differential equations with source terms reduces to three independent equations, without sources h′′ A + 2α′ A αA h′ A − c2 T,A∇2hA = 0 , αA = a2 c2 T,A , (67) ζ′′ + 2α′ α ζ′ − c2 s∇2ζ = 0 , α = ǫa2 c2 s = z2 c2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (68) Note that Lζ + Lh + Lint ̸= Leff ζ + Leff h (69) because the interaction Lagrangian Lint enters in the definition of both effective speeds, analo- gously to the fact it produces different source terms Sζ and ΠA in the equations of motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The effects of the interaction induce a modification of both speeds, since the interaction produces a source term in both equations, while in the literature often only the effects on gravitational waves are considered, ignoring those on scalar perturbations, and their possible back-reaction on tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' CHANGE OF FRAME AND PHYSICAL RELEVANCE OF THE EFFECTIVE PLANCK MASS All the result derived in the previous sections were in the Einstein frame, both using the effective Lagrangian and the EST approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' It can be convenient to find the conformal transformation taking to the Jordan frame, in which many other calculations have been performed, especially in the context of modified gravity [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Curvature and tensor perturbations Let’s start by writing the scalar and tensor quadratic Lagrangians in the Jordan frame [18] LJ = a2qs[ζ′2 − c2 s(∇ζ)2� + a2qt � h′2 − c2 T(∇h)2� , (70) 17 where we are denoting with E and J quantities in the Jordan and Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The Lagrangian of perturbations in a new frame obtained via disformal transformation, such that ˜a = Ωa, is of the form [18] ˜LJ = ˜a2˜qs[˜ζ′2 − ˜c2 s(∇˜ζ)2� + a2˜qt � ˜h′2 − ˜c2 T(∇˜h)2� , (71) where we are denoting with a tilde quantities in the new frame, and we are just defining the conformal part of the disformal transformation, because this is the one relevant for the coefficients of the Lagrangians for curvature and tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Since scalar and tensor perturbations are invariant under disformal transformations [18], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' ˜ζ = ζ, ˜h = h, and L = ˜L by definition, by comparing the coefficients [18] in L and ˜L we obtain the following transformations ˜a = Ωa → ˜qs = Ω−2qs , ˜qt = Ω−2qt , cs = ˜cs , cT = ˜cT , (72) Note that these relations are consistent, but different, from those derived in [18], due to the use of conformal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The corresponding effective quadratic Lagrangian in the Einstein frame is Leff E = ˜ǫ˜a2 1 c2s � ζ′2 − c2 s(∇ζ)2� + ˜a2 1 c2 T � h′2 − c2 T(∇h)2� = Leff ζ + Leff h , (73) where we are using the conformal invariance of cs, cT, ζ, h, and we are denoting with ˜ǫ the slow roll parameter in the Einstein frame, to distinguish it from that in the Jordan frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We use this notation only in this section, for consistency with the notation for conformal transformation starting from the Jordan, while in other sections ǫ denotes the slow-roll parameter in the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note that the Lagrangian above is consistent with the scalar perturbations action for K−inflation in the Einstein frame [4], and with the tensor action in the effective field theory of inflation [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' By comparing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (73) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (72) we find that the Einstein frame and Jordan frame are related by ˜qt = 1 c2 T , (74) ˜qs = ˜ǫ c2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (75) which imply Ω = cT √qt , (76) ˜ǫ = qsc2 s qtc2 T , (77) 18 in agreement with [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The parameter qt is sometimes denoted at M2 ∗ , and interpreted as effective Planck mass [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The above equations show that in the Einstein frame the effective Planck mass is not independent from cT , since √˜qt = ˜ M∗ = M∗Ω−1 = 1/cT, and its effects on scalar perturbations are encoded in Einstein frame slow-roll parameter ˜ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is not surprising, since in the Einstein frame the only physically relevant quantities are cs, cT, ˜ǫ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' the so called Jordan frame effective Planck mass M∗ emerges only in the definition of the conformal transformation taking to the Jordan frame, and as such does not really play any physical role, due to the invariance of scalar and tensor perturbations under disformal transformation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' As explained earlier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' the time dependency of the cs(η) and cT(η) is related to self interactions terms of the form (1/c2 s(η) − 1)ζ′2 and [15] h′2(1/c2 T(η) − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' but in this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' since ζ and h are not coupled to each other,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' but only self-coupled,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' we have LE = LE ζ + LE h + LE int = ˜ǫ˜a2� ζ′2 − (∇ζ)2� + ˜a2� h′2 − (∇h)2� + Lint h + Lint ζ = Leff ζ + Leff h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Lint h = ˜a2h′2� 1 c2 T (η) − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Lint ζ = ˜ǫ˜a2� 1 c2s(η) − 1 � ζ′2 (78) which was used to find the relation between the quantities in different from LJ = LE = Leff ζ + Leff h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (79) The first equality in the above equation just comes from the fact that Lagrangian densities in different frames are equal, because they are obtained by simply re-writing the same object in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The conformal transformation defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (77) can also be applied to the Lagrangians including higher order interactions terms [8], and is consistent with the invariance of the coef- ficients of the perturbations equations expected from the invariance of ζ, h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' the invariance of the solutions implies the invariance of the equations and of the Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In other words the equations and Lagrangians in the Jordan frame look different because of the conformal transformation, but the coefficients of the equations and of the Lagrangians as functions of space and time are the same, just written in a different way in terms of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In the Einstein frame the real number of independent degrees of freedom is more transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Note we are assuming that the coupling to gravity is properly transformed from one frame to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This implies that theories in which matter fields are minimally coupled to gravity 19 by the metric in the Einstein vs Jordan frame are different, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' they have different Lagrangians, which cannot be simply related by a conformal transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective gravitational coupling The perturbed Einstein equations give the useful equation k2 ˜a2 ˜ψB = 1 2δ˜ρc (80) where ˜ψB is one of the Bardeen potentials, and ˜δρc is the comoving energy density perturbation [1], and the tilde denotes quantities in the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The above equation is also valid for a modified gravity theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' once it has been transformed to the Einstein frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' in which the full Lagrangian takes the Hilbert form ˜L = � −˜g ˜R + ˜Ltot = � −˜g ˜R + ˜Lm + ˜Lφ + ˜Lmφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (81) k2 ˜a2 ˜ψB = 1 2δ˜ρtot c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' δ˜ρtot c = δ˜ρm c + δ˜ρφ c + δ˜ρmφ c ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (82) where ˜Ltot contains terms associated to matter and the modification of gravity in a single object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' and ˜Lφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' ˜Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' ˜Lmφ are respectively the Einstein frame Lagrangians related to the gravity modification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' and the non minimal coupling of gravity to matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' and a similar notation is adopted for the comoving energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In the Jordan frame the Poisson equation takes the form [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' k2 a2 ψB = 1 2GJδρc (83) The difference between the gravitational couplings in the two frames is expected, since in in our units in the Einstein frame we have by definition 8πGtot E = 1, and the effects of the modification of gravity are encoded in the total effective energy density δ˜ρtot c , which includes contributions from ˜Lφ, ˜Lm, ˜Lmφ, not just from ˜Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The relation between the gravitational coupling Gm E for matter comoving energy density δ˜ρm c will be investigated in more details in a future work, but we can anticipate that by appropriately manipulating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (82) we can write an equation of the form k2 ˜a2 ˜ψB = 1 2Gm Eδ˜ρm c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (84) Note that, contrary to curvature and tensor perturbations, the Bardeen potential and the comoving density perturbations are not invariant under disformal transformations [21], and for this reason we use a tilde to distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' 20 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' RELATION TO OTHER EFFECTIVE APPROACHES The effective approach formulated in this paper is completely general, and as such includes all the effects of interaction at any order in a single effective quantity, and for an arbitrary number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' We can compare this with previous results to see how it includes and extend them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective field theory of inflation The effective approach we have derived can describe the evolution of curvature not only for single field models, but also for multi-fields models [2], while in the EFT of inflation [22] approach it is assumed only one scalar degree of freedom, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' entropy perturbations are ignored, and no general effective action for curvature perturbations is derived for multi-fields systems [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The MESS approach allows to compute the effects of entropy and anisotropy on curvature perturbations for a generic system, including any number of fields, and predicts naturally the momentum dependence of the effective sound speed of curvature perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Effective field theory of dark energy The effective field theory of dark energy [24] applies the same symmetry breaking idea of the EFT of inflation to dark energy, but in the Jordan frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The action is expanded to quadratic order, and for this reason is missing the frequency and polarization dependence of the effective speed which arises naturally in the MESS and SEGS approach, due to the higher order interactions terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The general relation between Jordan and Einstein frame was derived in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' QUANTUM FIELD THEORY IMPLICATIONS The effective Lagrangians we have derived are based on classical calculations, but they can be related [25] to the wavefunction of the scalar and tensor fields by the path integral Ψ[ϕ] = � φ(t) = ϕ φ(−∞) = 0 Dφ eiS[φ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (85) 21 where φ is a generic field, which in our case could be ζ or h, and S denotes the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' At tree level the path integral can be approximated by the action evaluated on the classical solution, which is the way in which we define the effective Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' From the wavefunction we can compute the equal-time correlators as ⟨φ1 · · · φN⟩ = � Dφ φ1 · · · φN |Ψ[φ]|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' (86) The above method should give the same result obtained by using canonical quantization in the in-in formalism [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Following this method we can for example compute corrections to the spectrum, arising from higher order interaction terms in the Lagrangian, in terms of the effective speed we have defined previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' More details about this approach will be given in a separate work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' CONCLUSIONS We have derived a set of universal model independent effective equations and Lagrangians which can describe the evolution of scalar and tensor perturbation of any system whose field equations can be written in an Einstein like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This includes for example multi-fields systems, or modified gravity, once they have been transformed to the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Given the generality of this effective description it is particularly suitable for model independent phenomenological analysis of observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This approach predicts naturally that the speed of gravitational waves con depend on fre- quency and polarization, due to the interactions of the graviton with itself or other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This prediction allows to use gravitational waves observations to investigate the elusive nature of dark matter and dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' The equation and Lagrangian for scalar and tensor perturbations has the same universal structure, and the effects of the interaction can be modeled at any order in perturbations by a single effective quantity, playing the role of effective propagation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' This is particularly useful since it allows to compare different models in terms of the two quantities cs and cT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Combining different set of observational data such as cosmic microwave background radiation and gravitational waves, it will be possible to constrain the cs and cT,A, to determine possible deviations from general relativity and vanilla inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' If a deviation is found, the theoretical research can be focused on those models able to predict the cs and cT supported by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' In order to test specific models it will be important to perform higher order perturbations 22 calculations for different models, in order to compute the effects which are not included in the quadratic action, and in the EST approach are treated as effective model independent phenomenological quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' Acknowledgments I thank Misao Sasaki, Tessa Baker, Sergio Vallejo, Riccardo Sturani, Rogerio Rosenfeld, Nicola Tamanini and Suvodip Mukherjee for interesting discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' I thank the ICTP-SAIFR for the kind hospitality during the preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tNE5T4oBgHgl3EQfmg9S/content/2301.05679v1.pdf'} +page_content=' [1] H.' 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/dev/null +++ b/u9E0T4oBgHgl3EQfbgDu/content/tmp_files/2301.02351v1.pdf.txt @@ -0,0 +1,366 @@ +arXiv:2301.02351v1 [nlin.PS] 6 Jan 2023 +Coupled multi-quartic and multi-dipole solitons in highly dispersive optical fibers +Vladimir I. Kruglov1 and Houria Triki2 +1Centre for Engineering Quantum Systems, School of Mathematics and Physics, +The University of Queensland, Brisbane, Queensland 4072, Australia +2Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, +Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria +Multiple-hump soliton modes which can be potentially utilized for transmission are discovered in +a highly dispersive optical fiber. Such wave packets are very well described by an extended nonlinear +Schr¨odinger equation involving both cubic and quartic dispersion terms. It is found that the third- +and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole +solitons into double-, triple-, and N-hump solitons. +We provide the initial conditions necessary +for the formation of coupled multi-hump quartic and dipole solitons. Numerical results illustrate +the propagation of multi-quartic and multi-dipole solitons in highly dispersive optical fibers. It is +important for applications that described multiple-hump soliton modes are stable to small noise +perturbation that was confirmed by numerical simulations. +PACS numbers: 05.45.Yv, 42.65.Tg +Propagation of quartic solitons in optical waveguides is one of central research topics of nonlinear optical dynamics +[1–7]. The formation of these ultrashort localized pulses is essentially mediated by the interplay between self-phase +modulation and anomalous second- and fourth-order dispersions. The recent availability of silicon photonics has +opened a new way to generate waveguide structures exhibiting a wide range of dispersion profiles which allow possible +quartic soliton creation [8–12]. In this regard, impressive results obtained recently indicate the possibility of observing +localized quartic solitons in specially designed silicon-based slot waveguides [13]. Additionally, a detailed analysis has +demonstrated the existence of a stable quartic soliton envelope taking the form of sech2 in optical fibers exhibiting +all orders of dispersion up to the order four [14, 15]. It is interesting to note that quartic solitons can be generated +not only in presence of higher-order dispersions but also under the influence of self-steepening process [16]. +More recently, we reported the formation of stable dipole solitons in an optical fiber medium exhibiting all orders +of dispersion up to the order four [17]. The obtained results showed that such dipole-mode solitons characteristically +exist due to a balance among the effects of second-, third-, and fourth-order dispersions and self-phase modulation. +Notice that a dipole soliton is a localized structure possessing two symmetrical humps with a zero intensity value in +the middle of the pulse [18]. It is also interesting to note that dipole-mode solitons have been recently observed in a +three-level cascade atomic system [19]. +In this Letter, we present the first demonstration of the existence of multi-hump quartic and dipole solitons in +a highly dispersive optical fiber system. We will show analytically and numerically that considering the combined +influence of third- and fourth-order dispersion effects may pave the way to generate multiple soliton pulses in the +form of two, three and four coupled quartic or dipole solitons. Importantly, such wave packets are very well described +by the so-called extended nonlinear Schr¨odinger equation (NLSE) incorporating high order of dispersion and Kerr +nonlinearity. +An optical fiber medium when influenced by higher-order dispersion effects is described by the extended NLSE +model as follows [7, 14, 15, 17]: +i∂ψ +∂z = α∂2ψ +∂τ 2 + iσ ∂3ψ +∂τ 3 − ǫ∂4ψ +∂τ4 − γ |ψ|2 ψ, +(1) +where ψ(z, τ) is the complex field envelope, z represents the distance along the direction of propagation, and τ = t−β1z +is the retarded time in the frame moving with the group velocity of wave packets. Also α = β2/2, σ = β3/6, and +ǫ = β4/24, with βk = (dkβ/dωk)ω=ω0 denotes the k-order dispersion of the optical fiber with β(ω) is the propagation +constant depending on the optical frequency. The parameter γ represents the cubic nonlinearity coefficient. +We now present the first analysis of the existence of multi-soliton solutions of Eq. (1), which models the ultrashort +pulse propagation in highly dispersive optical fibers. To this end we consider the case when U and V are the localized +solutions of Eq. (1) and the sum U + V is also the solution of NSLE. The substitution of these functions to Eq. (1) +leads to three NSLE with ψ = U, ψ = V and ψ = U + V . Using some transformations with these three NLSE one +can find the condition that U + V is the solution of Eq. (1) assuming the localized waves U and V are the solution +of this NLSE as well. This condition is given by the following algebraic equation, +U 2V ⋆ + 2V UU ⋆ + 2UV V ⋆ + V 2U ⋆ = 0, +(2) + +2 +where the star means complex conjugation. +Let the traveling localized waves U and V have the form as U = +f(ξ − ξ1)eiφ(z,τ) and V = g(ξ − ξ2)eiθ(z,τ) where f(ξ − ξ1) and g(ξ − ξ2) are real function of the variable ξ = τ − qz. +Here the parameter q = 1/v is inverse velocity given by q = σ(σ2−4αǫ)/8ǫ2. The condition in Eq. (2) is satisfied when +the following relation f(ξ − ξ1)g(ξ − ξ2) = 0 holds. However, for interacting solitons forming multi-soliton pulses the +above condition is satisfied approximately only. Nevertheless the above consideration leads to approach for coupled +multi-soliton pulses based on summation of the soliton solutions of NLSE (1) with appropriate constant phases. +The soliton solution of Eq. (1) can be written as ψ(x, τ) = F(ξ − ξ0) exp(iΦ(z, τ)) and the coupled multi-soliton +waves (N-solitons) have the approximate form, +ψN(z, τ) = +N +� +n=1 +F(ξ − ξn) exp[i(Φ(z, τ) + Φn)], +(3) +where ξn = ξ0 +(n−1)a and Φn are the constant phases. The parameter a can be found using appropriate variational +procedure. +The coupled N-soliton solution given in Eq. +(3) has a good precision when the following condition +max F 2(ξ) ≫ wnm (with n ̸= m) is satisfied. The numbers wnm in this inequality are given by +wnm = max |F(ξ − ξn)F(ξ − ξm)|. +(4) +Note that the above condition, without loss of generality, can also be written as max F 2(ξ) ≫ w12. +The wave function of coupled multi-soliton solution can be written as ψN(z, τ) = UN(z, τ) exp(iΦ(z, τ)) where +UN(z, τ) is a real function. Moreover, this function is changing the sign at some point between two neighbouring +solitons of the coupled multi-soliton. Hence, the wave function ψN(z, τ) of coupled N-soliton solution is equal to zero +at such points. Thus, the coupled N-soliton solution has N − 1 “zero points”. We emphasis that these “zero points” +are connected with stability of the coupled N-soliton solutions of Eq. (1). We show below that the existence of these +“zero points” allow us to find the constant phases Φn for coupled multi-quartic and multi-dipole N-solitons. +The approximate wave function of multi-quartic N-soliton is given in Eq. (3) where the wave function of quartic +soliton [14, 15, 17] is +ψ(x, τ) = A0sech2(wξ) exp[i(κz − δτ + θ)]. +(5) +The amplitude A0 and inverse width w of quartic soliton are +A0 = ± +� −3 +10γǫ +�3σ2 − 8αǫ +8ǫ +� +, +w = 1 +4 +� +8αǫ − 3σ2 +10ǫ2 +. +(6) +The multi-quartic N-soliton is the coupling of quartic solitons ψ(x, τ) into N-soliton with constant phases Φn = (n−1)π. +Eq. (3) yields in this case the multi-quartic N-soliton as +ψN(z, τ) = +N +� +n=1 +(−1)n−1A0sech2(w(ξ − ξn)) exp[i(κz − δτ + θ)], +(7) +where ξn = ξ0 + (n − 1)a. The parameter a = a0w−1 is period of intensity (I = |ψN|2) for quartic N-soliton where a0 +is the dimensionless constant which can be found theoretically or numerically. Note that the parameter a is defined by +interaction of solitons which constitute the coupled multi-quartic soliton given in Eq. (7). We emphasis that the found +constant phases Φn = (n − 1)π follow from the condition that real function UN(z, τ) = ψN(z, τ) exp[−i(κz − δτ + θ)] +in Eq. (7) has N − 1 “zero points” located between neighbouring solitons of the coupled multi-quartic N-soliton. The +frequency shift in the phase Φ(z, τ) = κz − δτ + θ of multi-quartic N-soliton is δ = −σ/4ǫ and the wave number κ is +given as +κ = − 4 +25ǫ3 +�3σ2 +8 +− αǫ +�2 +− σ2 +16ǫ3 +�3σ2 +16 − αǫ +� +. +(8) +The approximate wave function of multi-dipole N-soliton is presented in Eq. (3) where the wave function of dipole +soliton [17] is +ψ(x, τ) = A0sech(wξ)th(wξ) exp[i(κz − δτ + θ)]. +(9) +The amplitude A0 and inverse width w of dipole soliton are +A0 = ± +� 6 +5γǫ +�3σ2 − 8αǫ +8ǫ +� +, +w = 1 +4 +� +3σ2 − 8αǫ +5ǫ2 +. +(10) + +3 +The multi-dipole N-soliton is the coupling dipole solitons ψ(x, τ) into N-soliton with constant phases Φn = 0. Eq. (3) +yields in this case the multi-dipole N-soliton as +ψN(z, τ) = +N +� +n=1 +A0sech(w(ξ − ξn))th(w(ξ − ξn)) exp[i(κz − δτ + θ)], +(11) +where ξn = ξ0 + (n − 1)a. The parameter a = a0w−1 is period of intensity for dipole N-soliton where a0 is the +dimensionless constant. Here the parameter a is defined by interaction of solitons which constitute the coupled multi- +dipole soliton given in Eq. (11). The constant phases Φn = 0 in this solution follow from the condition that real +function UN(z, τ) = ψN(z, τ) exp[−i(κz − δτ + θ)] in Eq. (11) has N − 1 “zero points” located between neighbouring +solitons of the coupled multi-dipole N-soliton. The frequency shift in the phase Φ(z, τ) = κz − δτ + θ of multi-dipole +N-soliton is δ = −σ/4ǫ and the wave number κ is given as +κ = +11 +100ǫ3 +�3σ2 +8 +− αǫ +�2 +− σ2 +16ǫ3 +�3σ2 +16 − αǫ +� +. +(12) +The variable ξ = τ − qz in these quartic and dipole N-soliton solutions depends on velocity v = 1/q which is +v = +8ǫ2 +σ(σ2 − 4αǫ). +(13) +We now find the numerical multi-hump solutions to the extended NLSE model (1) by using the split-step Fourier +method [20]. Here, we show that the cubic and quartic dispersions when acting in combination lead to the coupling +of quartic and dipole solitons into multiple-modes. We also show that the shape of the newly found solutions depends +crucially on the choice of the initial condition. +First, we analyze the existence of multiple-hump quartic soliton +FIG. 1: Propagation of the double-hump quartic soliton solution of Eq. (1) with parameters α = −2.0375, γ = 2.6, σ = 0.1, +and ǫ = −0.1. +solutions of Eq. +(1) and demonstrate that waveforms of double-, triple-, and quadruple-hump form can readily +be generated in the fiber system. +To perform a numerical study of multi-hump quartic solitons, we numerically +integrated the full underlying equation (1) using the analytic soliton solution (7) with N = 2 as an initial condition. +The propagation of the double-hump quartic solitons calculated within the framework of Eq. (1) is shown in Fig. 1 +for the parameter values: α = −2.0375, γ = 2.6, σ = 0.1, ǫ = −0.1, and ξ0 = 0. It should be noted that here, we +have evaluated the dimensionless constant a0 numerically as a0 = 3. From this figure, we see that the two humps are +well separated and both of them have the same shape (width and maximum intensity). We also see that the wave +profile remains unchanged after propagating a distance of forty normalized lengths. Assuming initial conditions in +the form of the analytic soliton solution (7) containing three (N = 3) or four (N = 4) superimposed single-hump +sech2 solitons, we observe that both tripole and quadrupole quartic solitons are obtained, as shown in Figs. 2 and 3 +respectively. +This also indicates that these numerical findings agree excellently with our analytic solution (7). This +physically important result implies that such multi-quartic solitons can be observed experimentally as long as the + +4.6 +40 +30 +2 +2.3 +20 +N +10 +0.0 +0 +1 +-30 +-15 +0 +15 +30 +T4 +FIG. 2: Propagation of the triple-hump quartic soliton solution of Eq. (1). The parameters are the same as in Fig. 1. +FIG. 3: Propagation of the quadruple-hump quartic soliton solution of Eq. (1). The parameters are the same as in Fig. 1. +model (1) applies, thus implying that the solutions obtained here can be utilized for transmission. +Now we focus +on the formation of the multiple-dipole soliton families in the fiber medium. To obtain the numerical multi-hump +solutions, we solved Eq. (1) by means of the split-step Fourier method for an input condition taking the form of the +analytic soliton solution (11) with N = 2. Here we have taken the following parameter values: α = −1, γ = −2.6, +σ = 1.53, ǫ = −0.25, and ξ0 = 0. We have also calculated the dimensionless constant a0 numerically as a0 = 3.5. +Figure 4 demonstrates that two dipole-type solitons may exist in the fiber medium. As seen, this double dipole-type +soliton keeps its profile over a distance of forty normalized lengths. In addition to double dipole-type soliton modes, +we find that families of multiple-hump solutions can also be generated in the system, including three, four, and N +dipole-type soliton pulses. +A distinguishing property of localized pulses is their stability to perturbations, as only stable solitons can be +observed experimentally and utilized in physical applications. It is therefore important to analyze the stability of +the obtained multiple-hump solutions with respect to small perturbations. +Here, we take the numerically found +double-hump quartic and dipole soliton solutions as examples to perform numerical experiments of the model (1). +The numerical evolutions of double-hump quartic and doubl dipole soliton solutions under the perturbation of 10% +white noise are depicted in Figs. 5 and 6, respectively. These results show that the multiple-hump soliton modes can + +40 +30 +z +20 +10 +0 +04.6 +2.3 +0.0 +-30 +-15 +0 +15 +3 +T40 +30 +一 +20 +Z +-10 +04.6 +2 +#2.3 +0.0 +-30 +-20 +-10 +0 +10 +20 +30 +T5 +FIG. 4: Propagation of the double-hump dipole soliton solution of Eq. (1) with parameters α = −1, γ = −2.6, σ = 1.53, and +ǫ = −0.25. +FIG. 5: The numerical evolution of double-hump quartic solitons under the perturbation of white noise whose maximal value +is 0.1. The parameters are the same as those used in Fig. 1. +propagate stably in the fiber system under the initial perturbation of the additive white noise. Hence we can conclude +that the novel multi-soliton modes we obtained are stable. +In conclusion, we have reported the discovery of multi-soliton pulses in highly dispersive optical fibers. We revealed +that in such media, the presence of cubic and quartic dispersions may lead to the coupling of quartic or dipole solitons +into localized multi-hump pulses. The dynamics of the newly found solutions in the fiber material have been found +to be very well modeled by the extended nonlinear Schr¨odinger equation incorporating the contributions of second-, +third-, and fourth-order dispersions and self-phase modulation. The analytical and numerical results indicate that +both quartic and dipole types of double-, triple-, and N-soliton solutions can be formed in the fiber system. We have +also demonstrated numerically that such soliton modes are stable with respect to small perturbations, thus implying +that they can be utilized for transmission. +The discovery of such coupled multi-quartic and multi-dipole solitons represent an important advance in nonlinear +optics. It should be mentioned that the newly found localized multi-solutions could find important application not +only in optical fibers but also in other physical media and systems. In future research problems, we shall take into + +5.0 +40 +2 +2.5 +20 +Z +0.0 +0 +-30 +-15 +0 +15 +30 +T40 +20 +Z +03 +2 +2 +1 +0 +-30 +-15 +0 +15 +30 +T6 +FIG. 6: The numerical evolution of double dipole solitons under the perturbation of white noise whose maximal value is 0.1. +The parameters are the same as those used in Fig. 4. +consideration the absorption or amplification effects to expand the applicability of obtained multi-soliton solutions. +[1] A. H¨o¨ok and M. Karlsson, Opt. Lett. 18, 1388 (1993). +[2] M. Karlsson and A. H¨o¨ok, Opt. Commun. 104, 303 (1994). +[3] N. N. Akhmediev, A. V. Buryak, and M. Karlsson, Opt. Commun. 110, 540 (1994). +[4] N. N. Akhmediev and A. V. Buryak, Opt. Commun. 121, 109 (1995). +[5] A. V. Buryak and N. N. Akhmediev, Phys. Rev. E 51, 3572 (1995). +[6] V. E. Zakharov and E. A. Kuznetsov, J. Exp. Theor. Phys. 86, 1035 (1998). +[7] M. Pich´e, J.-F. Cormier, and X. Zhu, Opt. Lett. 21, 845 (1996). +[8] Silicon Photonics, edited by L. Pavesi and D. J. Lockwood (Springer, New York, 2004). +[9] B. Jalali, J. Lightwave Technol. 24, 4600 (2006). +[10] Q. Lin, O. J. Painter, and G. P. Agrawal, Opt. Express 15, 16604 (2007). +[11] A. D. Bristow, N. Rotenberg, and H. M. van Driel, Appl. Phys. Lett. 90, 191104 (2007). +[12] J. Leuthold, C. Koos, and W. Freude, Nat. Photonics 4(8), 535 (2010). +[13] S. Roy and F. Biancalana, Phys.Rev.A 87, 025801 (2013). +[14] V. I. Kruglov and J. D. Harvey, Phys. Rev. A 98, 063811 (2018). +[15] V. I. Kruglov, Opt. Commun. 472, 125866 (2020). +[16] V. I. Kruglov and H. Triki,Phys. Rev. A 102, 043509 (2020). +[17] H. Triki and V. I. Kruglov, Phys. Rev. E 101, 042220 (2020). +[18] A. Choudhuri and K. Porsezian, Opt. Commun. 285, 364 (2012). +[19] Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011). +[20] G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, Boston, 2006). + +Z40 +3 +2 +2 +20 +1 +0 +0 +-30 +-15 +0 +15 +30 +T \ No newline at end of file diff --git a/u9E0T4oBgHgl3EQfbgDu/content/tmp_files/load_file.txt b/u9E0T4oBgHgl3EQfbgDu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..618f48e0411e2352ec889630cc68385d93221222 --- /dev/null +++ b/u9E0T4oBgHgl3EQfbgDu/content/tmp_files/load_file.txt @@ -0,0 +1,305 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf,len=304 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='02351v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='PS] 6 Jan 2023 Coupled multi-quartic and multi-dipole solitons in highly dispersive optical fibers Vladimir I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Kruglov1 and Houria Triki2 1Centre for Engineering Quantum Systems, School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia 2Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Box 12, 23000 Annaba, Algeria Multiple-hump soliton modes which can be potentially utilized for transmission are discovered in a highly dispersive optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Such wave packets are very well described by an extended nonlinear Schr¨odinger equation involving both cubic and quartic dispersion terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It is found that the third- and fourth-order dispersion effects in the fiber material may lead to the coupling of quartic or dipole solitons into double-, triple-, and N-hump solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We provide the initial conditions necessary for the formation of coupled multi-hump quartic and dipole solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Numerical results illustrate the propagation of multi-quartic and multi-dipole solitons in highly dispersive optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It is important for applications that described multiple-hump soliton modes are stable to small noise perturbation that was confirmed by numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' PACS numbers: 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='Yv, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='Tg Propagation of quartic solitons in optical waveguides is one of central research topics of nonlinear optical dynamics [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The formation of these ultrashort localized pulses is essentially mediated by the interplay between self-phase modulation and anomalous second- and fourth-order dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The recent availability of silicon photonics has opened a new way to generate waveguide structures exhibiting a wide range of dispersion profiles which allow possible quartic soliton creation [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' In this regard, impressive results obtained recently indicate the possibility of observing localized quartic solitons in specially designed silicon-based slot waveguides [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Additionally, a detailed analysis has demonstrated the existence of a stable quartic soliton envelope taking the form of sech2 in optical fibers exhibiting all orders of dispersion up to the order four [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It is interesting to note that quartic solitons can be generated not only in presence of higher-order dispersions but also under the influence of self-steepening process [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' More recently, we reported the formation of stable dipole solitons in an optical fiber medium exhibiting all orders of dispersion up to the order four [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The obtained results showed that such dipole-mode solitons characteristically exist due to a balance among the effects of second-, third-, and fourth-order dispersions and self-phase modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Notice that a dipole soliton is a localized structure possessing two symmetrical humps with a zero intensity value in the middle of the pulse [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It is also interesting to note that dipole-mode solitons have been recently observed in a three-level cascade atomic system [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' In this Letter, we present the first demonstration of the existence of multi-hump quartic and dipole solitons in a highly dispersive optical fiber system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We will show analytically and numerically that considering the combined influence of third- and fourth-order dispersion effects may pave the way to generate multiple soliton pulses in the form of two, three and four coupled quartic or dipole solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Importantly, such wave packets are very well described by the so-called extended nonlinear Schr¨odinger equation (NLSE) incorporating high order of dispersion and Kerr nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' An optical fiber medium when influenced by higher-order dispersion effects is described by the extended NLSE model as follows [7, 14, 15, 17]: i∂ψ ∂z = α∂2ψ ∂τ 2 + iσ ∂3ψ ∂τ 3 − ǫ∂4ψ ∂τ4 − γ |ψ|2 ψ, (1) where ψ(z, τ) is the complex field envelope, z represents the distance along the direction of propagation, and τ = t−β1z is the retarded time in the frame moving with the group velocity of wave packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Also α = β2/2, σ = β3/6, and ǫ = β4/24, with βk = (dkβ/dωk)ω=ω0 denotes the k-order dispersion of the optical fiber with β(ω) is the propagation constant depending on the optical frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameter γ represents the cubic nonlinearity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We now present the first analysis of the existence of multi-soliton solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1), which models the ultrashort pulse propagation in highly dispersive optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' To this end we consider the case when U and V are the localized solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) and the sum U + V is also the solution of NSLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The substitution of these functions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) leads to three NSLE with ψ = U, ψ = V and ψ = U + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Using some transformations with these three NLSE one can find the condition that U + V is the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) assuming the localized waves U and V are the solution of this NLSE as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' This condition is given by the following algebraic equation, U 2V ⋆ + 2V UU ⋆ + 2UV V ⋆ + V 2U ⋆ = 0, (2) 2 where the star means complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Let the traveling localized waves U and V have the form as U = f(ξ − ξ1)eiφ(z,τ) and V = g(ξ − ξ2)eiθ(z,τ) where f(ξ − ξ1) and g(ξ − ξ2) are real function of the variable ξ = τ − qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Here the parameter q = 1/v is inverse velocity given by q = σ(σ2−4αǫ)/8ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (2) is satisfied when the following relation f(ξ − ξ1)g(ξ − ξ2) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' However, for interacting solitons forming multi-soliton pulses the above condition is satisfied approximately only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Nevertheless the above consideration leads to approach for coupled multi-soliton pulses based on summation of the soliton solutions of NLSE (1) with appropriate constant phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) can be written as ψ(x, τ) = F(ξ − ξ0) exp(iΦ(z, τ)) and the coupled multi-soliton waves (N-solitons) have the approximate form, ψN(z, τ) = N � n=1 F(ξ − ξn) exp[i(Φ(z, τ) + Φn)], (3) where ξn = ξ0 +(n−1)a and Φn are the constant phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameter a can be found using appropriate variational procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The coupled N-soliton solution given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (3) has a good precision when the following condition max F 2(ξ) ≫ wnm (with n ̸= m) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The numbers wnm in this inequality are given by wnm = max |F(ξ − ξn)F(ξ − ξm)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (4) Note that the above condition, without loss of generality, can also be written as max F 2(ξ) ≫ w12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The wave function of coupled multi-soliton solution can be written as ψN(z, τ) = UN(z, τ) exp(iΦ(z, τ)) where UN(z, τ) is a real function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Moreover, this function is changing the sign at some point between two neighbouring solitons of the coupled multi-soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Hence, the wave function ψN(z, τ) of coupled N-soliton solution is equal to zero at such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Thus, the coupled N-soliton solution has N − 1 “zero points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We emphasis that these “zero points” are connected with stability of the coupled N-soliton solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We show below that the existence of these “zero points” allow us to find the constant phases Φn for coupled multi-quartic and multi-dipole N-solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The approximate wave function of multi-quartic N-soliton is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (3) where the wave function of quartic soliton [14, 15, 17] is ψ(x, τ) = A0sech2(wξ) exp[i(κz − δτ + θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (5) The amplitude A0 and inverse width w of quartic soliton are A0 = ± � −3 10γǫ �3σ2 − 8αǫ 8ǫ � , w = 1 4 � 8αǫ − 3σ2 10ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (6) The multi-quartic N-soliton is the coupling of quartic solitons ψ(x, τ) into N-soliton with constant phases Φn = (n−1)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (3) yields in this case the multi-quartic N-soliton as ψN(z, τ) = N � n=1 (−1)n−1A0sech2(w(ξ − ξn)) exp[i(κz − δτ + θ)], (7) where ξn = ξ0 + (n − 1)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameter a = a0w−1 is period of intensity (I = |ψN|2) for quartic N-soliton where a0 is the dimensionless constant which can be found theoretically or numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Note that the parameter a is defined by interaction of solitons which constitute the coupled multi-quartic soliton given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We emphasis that the found constant phases Φn = (n − 1)π follow from the condition that real function UN(z, τ) = ψN(z, τ) exp[−i(κz − δτ + θ)] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (7) has N − 1 “zero points” located between neighbouring solitons of the coupled multi-quartic N-soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The frequency shift in the phase Φ(z, τ) = κz − δτ + θ of multi-quartic N-soliton is δ = −σ/4ǫ and the wave number κ is given as κ = − 4 25ǫ3 �3σ2 8 − αǫ �2 − σ2 16ǫ3 �3σ2 16 − αǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (8) The approximate wave function of multi-dipole N-soliton is presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (3) where the wave function of dipole soliton [17] is ψ(x, τ) = A0sech(wξ)th(wξ) exp[i(κz − δτ + θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (9) The amplitude A0 and inverse width w of dipole soliton are A0 = ± � 6 5γǫ �3σ2 − 8αǫ 8ǫ � , w = 1 4 � 3σ2 − 8αǫ 5ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (10) 3 The multi-dipole N-soliton is the coupling dipole solitons ψ(x, τ) into N-soliton with constant phases Φn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (3) yields in this case the multi-dipole N-soliton as ψN(z, τ) = N � n=1 A0sech(w(ξ − ξn))th(w(ξ − ξn)) exp[i(κz − δτ + θ)], (11) where ξn = ξ0 + (n − 1)a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameter a = a0w−1 is period of intensity for dipole N-soliton where a0 is the dimensionless constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Here the parameter a is defined by interaction of solitons which constitute the coupled multi- dipole soliton given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The constant phases Φn = 0 in this solution follow from the condition that real function UN(z, τ) = ψN(z, τ) exp[−i(κz − δτ + θ)] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (11) has N − 1 “zero points” located between neighbouring solitons of the coupled multi-dipole N-soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The frequency shift in the phase Φ(z, τ) = κz − δτ + θ of multi-dipole N-soliton is δ = −σ/4ǫ and the wave number κ is given as κ = 11 100ǫ3 �3σ2 8 − αǫ �2 − σ2 16ǫ3 �3σ2 16 − αǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (12) The variable ξ = τ − qz in these quartic and dipole N-soliton solutions depends on velocity v = 1/q which is v = 8ǫ2 σ(σ2 − 4αǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (13) We now find the numerical multi-hump solutions to the extended NLSE model (1) by using the split-step Fourier method [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Here, we show that the cubic and quartic dispersions when acting in combination lead to the coupling of quartic and dipole solitons into multiple-modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We also show that the shape of the newly found solutions depends crucially on the choice of the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' First, we analyze the existence of multiple-hump quartic soliton FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 1: Propagation of the double-hump quartic soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) with parameters α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0375, γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1, and ǫ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) and demonstrate that waveforms of double-, triple-, and quadruple-hump form can readily be generated in the fiber system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' To perform a numerical study of multi-hump quartic solitons, we numerically integrated the full underlying equation (1) using the analytic soliton solution (7) with N = 2 as an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The propagation of the double-hump quartic solitons calculated within the framework of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 1 for the parameter values: α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0375, γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1, ǫ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1, and ξ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It should be noted that here, we have evaluated the dimensionless constant a0 numerically as a0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' From this figure, we see that the two humps are well separated and both of them have the same shape (width and maximum intensity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We also see that the wave profile remains unchanged after propagating a distance of forty normalized lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Assuming initial conditions in the form of the analytic soliton solution (7) containing three (N = 3) or four (N = 4) superimposed single-hump sech2 solitons, we observe that both tripole and quadrupole quartic solitons are obtained, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 2 and 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' This also indicates that these numerical findings agree excellently with our analytic solution (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' This physically important result implies that such multi-quartic solitons can be observed experimentally as long as the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6 40 30 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='3 20 N 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0 0 1 30 15 0 15 30 T4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 2: Propagation of the triple-hump quartic soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 3: Propagation of the quadruple-hump quartic soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' model (1) applies, thus implying that the solutions obtained here can be utilized for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Now we focus on the formation of the multiple-dipole soliton families in the fiber medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' To obtain the numerical multi-hump solutions, we solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) by means of the split-step Fourier method for an input condition taking the form of the analytic soliton solution (11) with N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Here we have taken the following parameter values: α = −1, γ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6, σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='53, ǫ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='25, and ξ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We have also calculated the dimensionless constant a0 numerically as a0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Figure 4 demonstrates that two dipole-type solitons may exist in the fiber medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' As seen, this double dipole-type soliton keeps its profile over a distance of forty normalized lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' In addition to double dipole-type soliton modes, we find that families of multiple-hump solutions can also be generated in the system, including three, four, and N dipole-type soliton pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' A distinguishing property of localized pulses is their stability to perturbations, as only stable solitons can be observed experimentally and utilized in physical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It is therefore important to analyze the stability of the obtained multiple-hump solutions with respect to small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Here, we take the numerically found double-hump quartic and dipole soliton solutions as examples to perform numerical experiments of the model (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The numerical evolutions of double-hump quartic and doubl dipole soliton solutions under the perturbation of 10% white noise are depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' These results show that the multiple-hump soliton modes can 40 30 z 20 10 0 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0 30 15 0 15 3 T40 30 一 20 Z 10 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6 2 #2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0 30 20 10 0 10 20 30 T5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 4: Propagation of the double-hump dipole soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' (1) with parameters α = −1, γ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='6, σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='53, and ǫ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 5: The numerical evolution of double-hump quartic solitons under the perturbation of white noise whose maximal value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameters are the same as those used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' propagate stably in the fiber system under the initial perturbation of the additive white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Hence we can conclude that the novel multi-soliton modes we obtained are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' In conclusion, we have reported the discovery of multi-soliton pulses in highly dispersive optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We revealed that in such media, the presence of cubic and quartic dispersions may lead to the coupling of quartic or dipole solitons into localized multi-hump pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The dynamics of the newly found solutions in the fiber material have been found to be very well modeled by the extended nonlinear Schr¨odinger equation incorporating the contributions of second-, third-, and fourth-order dispersions and self-phase modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The analytical and numerical results indicate that both quartic and dipole types of double-, triple-, and N-soliton solutions can be formed in the fiber system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' We have also demonstrated numerically that such soliton modes are stable with respect to small perturbations, thus implying that they can be utilized for transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The discovery of such coupled multi-quartic and multi-dipole solitons represent an important advance in nonlinear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' It should be mentioned that the newly found localized multi-solutions could find important application not only in optical fibers but also in other physical media and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' In future research problems, we shall take into 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0 40 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='5 20 Z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='0 0 30 15 0 15 30 T40 20 Z 03 2 2 1 0 30 15 0 15 30 T6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 6: The numerical evolution of double dipole solitons under the perturbation of white noise whose maximal value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' The parameters are the same as those used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' consideration the absorption or amplification effects to expand the applicability of obtained multi-soliton solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' H¨o¨ok and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Karlsson, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 18, 1388 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Karlsson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' H¨o¨ok, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 104, 303 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Akhmediev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Buryak, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Karlsson, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 110, 540 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Akhmediev and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Buryak, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 121, 109 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Buryak and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Akhmediev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' E 51, 3572 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Zakharov and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Kuznetsov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9E0T4oBgHgl3EQfbgDu/content/2301.02351v1.pdf'} +page_content=' 86, 1035 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100644 index 0000000000000000000000000000000000000000..14f72e0b15f1706afc4312cccac1f1762719384f --- /dev/null +++ b/ytFLT4oBgHgl3EQfnC-z/content/tmp_files/2301.12126v1.pdf.txt @@ -0,0 +1,883 @@ +arXiv:2301.12126v1 [math.GN] 28 Jan 2023 +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +NAOKI KITAZAWA +Abstract. In our paper, we introduce special-generic-like maps or SGL maps +as smooth maps and study their several algebraic topological and differential +topological properties. +The new class generalize the class of so-called special generic maps. Special +generic maps are smooth maps which are locally projections or the product +maps of Morse functions and the identity maps on disks. Morse functions with +exactly two singular points on spheres or Morse functions in Reeb’s theorem +are simplest examples. Special generic maps and the manifolds of their do- +mains have been studied well. Their structures are simple and this help us to +study explicitly. As important properties, they have been shown to restrict +the topologies and the differentiable structures of the manifolds strongly by +Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author. To cover +wider classes of manifolds as the domains, the author previously introduced a +class generalizing the class of special generic maps and smaller than our class: +simply generalized special generic maps. +1. Introduction. +Special generic maps are smooth maps which are locally projections or the prod- +uct maps of Morse functions and the identity maps on disks. Morse functions with +exactly two singular points on spheres or Morse functions in Reeb’s theorem are +simplest examples. Canonical projections of so-called unit spheres are also simplest +examples. +Pioneering studies are [3, 4, 6] for example. +Since the 1990s, their algebraic +topological and differential topological properties have been studied by Saeki and +Sakuma ([27, 28, 29, 30]), followed by Nishioka ([26]), Wrazidlo ([32, 33]) and the +author ([10, 11, 12, 13, 14, 15, 16, 17, 19]). +Some elementary manifolds such as ones represented as connected sums of the +products of two spheres admit natural special generic maps in considerable cases. +On the contrary, the differentiable structures of spheres and some elementary man- +ifolds admitting special generic maps are restricted strongly. Homology groups of +the manifolds are also restricted in considerable cases. +The author has started +studies on the cohomology rings of the manifolds. These studies are due to the fact +that special generic maps have simple and nice structures. +We introduce some fundamental notions, terminologies and notation. Rk is the +k-dimensional Euclidean space, which is a simplest k-dimensional smooth manifold +for an arbitrary positive integer k. It is, by considering a standard Euclidean metric, +a Riemannian manifold. ||x|| ≥ 0 denotes th distance between x ∈ Rk and the origin +0. R := R1 and Z denotes the ring of all integers. +Key words and phrases. Special generic maps. Morse-Bott functions. Homology and cohomol- +ogy. +2020 Mathematics Subject Classification: Primary 57R45. Secondary 57R19. +1 + +2 +NAOKI KITAZAWA +Sk := {x ∈ Rk+1 | ||x|| = 1} denotes the k-dimensional unit sphere. It is a +k-dimensional smooth compact and connected submanifold if k ≥ 2 and the two- +point set with the discrete topology if k = 1. Dk := {x ∈ Rk | ||x|| ≤ 1} denotes +the k-dimensional unit disk for an arbitrary integer k ≥ 1. It is a k-dimensional +smooth compact and connected submanifold. +For a non-empty topological space X having the structure of a cell complex, we +can define the dimension uniquely, denoted by dim X. A topological manifold is +known to have the structure of a CW complex. A smooth manifold is known to +have the structure of a polyhedron and more precisely, in some canonical way we +can define the unique PL structure. This is seen as a PL manifold canonically. +It is well-known that topological manifolds whose dimensions are at most 3 have +the unique structures of polyhedra and that topological spaces homeomorphic to +polyhedra whose dimensions are at most 2 have the unique structures of polyhedra. +Theory related to such uniqueness is discussed in [25] for example. +For a differentiable map c : X → Y , a point x ∈ X is a singular point if the +rank of the differential dcx there is smaller than the minimum between dim X and +dim Y . The singular set of c is the set of all singular points of c and let S(c) denote +the set. +We define and discuss special generic maps later. However, we explain about +these maps shortly. A special generic map is a smooth map from an m-dimensional +manifold with no boundary into n-dimensional one with m ≥ n. If the manifold +of the domain is closed, then the image is regarded as an n-dimensional smoothly +immersed compact manifold. +The preimage of each point in the interior of the +immersed manifold is diffeomorphic to Sm−n. +This gives a projection over the +interior of the manifold. The preimage of each point in the boundary is a point. +Around each point in the boundary, it is the product map of a Morse function with +exactly one singular point on a disk of dimension m − n + 1 and the identity map +on a disk of dimension n − 1. Propositions 1 and 2, presented in the next section, +are on this. +Problem 1. Can you formulate new nice classes of smooth maps having simple and +nice structures and properties special generic maps have? +Problem 2. Are the classes of Problem 1 can cover wider classes of manifolds to +study as the domains of these maps +Essentially same problems are also asked in the previous preprint of the author +[18]. There the author introduced simply generalized special generic maps. In our +paper, we generalize this class as the class of special-generic-like maps or SGL maps. +We explain about simply generalized special generic maps shortly. The preim- +age of a point in the interior of the n-dimensional smoothly immersed manifold is +replaced by the product of manifolds diffeomorphic to unit spheres. Around each +point in the boundary, a Morse function on a disk is replaced by the composition +of a projection onto the disk with a Morse function with exactly one singular point +there. Each preimage of the projection is the product of manifolds diffeomorphic +to unit spheres. The Morse functions are replaced by Morse-Bott functions. For +Morse-Bott functions, see a related pioneering study [1] for example. +Note that this class respects local structures of so-called moment maps on so- +called (symplectic) toric manifolds. The class of moment maps and that of sym- +plectic toric manifolds are important in symplectic geometry and toric geometry. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +3 +In most cases, special generic maps are not admitted by such manifolds. This has +been conjectured by us and the author has shown results seeming to be related to +this. [13, 15] show the non-existence of special generic maps on complex projec- +tive spaces except for the 1-dimensional case or 2-dimensional spheres. They are +simplest (symplectic) toric manifolds. [5] is a pioneering study on symlectic toric +manifolds. See also [2] for example. Exposition related to this history is also in [18] +For our new class, the preimage of each point in the interior of the immersed +manifold and Morse(-Bott) functions used around the boundary are generalized. +We present our main results as Main Theorems. Hereafter, elementary algebraic +topology, more precisely, elementary (co)homology theory is important. We do not +explain about this rigorously or systematically and we also expect we have related +knowledge. For related studies, see [8] and we also abuse notions, terminologies +and notation here or ones which seem to be generally used or familiar to us. +Main Theorem 1 (Theorem 2). Let k > 0 be a positive integer. Let m > n be +an arbitrary positive integers satisfying m − n ≥ k. Let f : M → N be an SGL +map from an m-dimensional closed and connected manifold into an n-dimensional +connected manifold N with no boundary which is represented as the composition +of a smooth surjection qf : M → Wf onto a compact and k-connected connected +manifold Wf with a smooth immersion ¯f : Wf → N. Suppose that a family {pj} ⊂ +Int Wf of finitely many points satisfying the following conditions exists . +(1) The preimage qf −1(pj) for pj ∈ Int Wf is diffeomorphic to a closed and +(k − 1)-connected manifold F. +(2) For a smooth curve αpj : [0, 1] → Wf satisfying αpj(0) = pj, αpj((0, 1)) ⊂ +Int Wf and αpj(1) ∈ ∂Wf and a so-called ”transversality”, presented later +in several situations, the inclusion αpj +−1(0) ⊂ αpj +−1([0, 1]) gives the kernel +Ker αpj ∗ of the naturally defined homomorphism between the k-th homotopy +groups πk(αpj +−1(0)) and πk(αpj +−1([0, 1])). +(3) The union � +jKer αpj ∗ generates πk(F). +Then M is k-connected. +Main Theorem 2 (Theorem 3). Let n be an arbitrary positive integer. Suppose +that an n-dimensional connected manifold N with no boundary and a smooth im- +mersion ¯f : ¯N → N of an n-dimensional compact and simply-connected manifold ¯N +which has at least two boundary components are given. Let F be a closed, connected +and orientable surface which is not a sphere. +Then we have an SGL map f : M → N from some m-dimensional closed and +simply-connected manifold M into N satisfying the conditions of Main Theorem 1 +with the notation being abused and Wf and ¯N being identified suitably as smooth +manifolds. +These results are presented again in revised versions in the third section. +We explain about the content of our paper. The second section is for preliminar- +ies. There we also review special generic maps for example as fundamental objects. +We define our new class rigorously in the third section. The fourth section proves +Main Theorems and present some exmaples. Main Theorem 3 is also presented as +another new result of us. +Conflict of interest. +The author is a member of the project JSPS KAKENHI Grant Number JP22K18267 + +4 +NAOKI KITAZAWA +”Visualizing twists in data through monodromy” (Principal Investigator: Osamu +Saeki). Our present study is supported by the project. +Data availability. +Data essentially supporting our present study are all contained in our present paper. +2. Preliminaries. +2.1. Diffeomorphisms. A diffeomorphism between smooth manifolds means a +smooth map which has no no singular points and which is a homeomorphism. A +diffeomorphism on a manifold is a diffeomorphism from the (smooth) manifold onto +itself. Two manifolds are diffeomorphic if and only if there exists a diffeomorphism +between these manifolds. This naturally gives an equivalence relation on the family +of all smooth manifolds with their corners being eliminated in a well-known canon- +ical way. These operations always give mutually diffeomorphic manifolds with no +corners for a fixed manifold. We can define PL homeomorphic manifolds using PL +homeomorphisms or piecewise smooth homeomorphisms similarly. +The diffeomorphism group of a manifold is the space consisting of all diffeomor- +phisms on the manifold where the so-called Whitney C∞ topology is given as its +topology. It is also a topological group and a so-called infinite dimensional Lie +groups. Whitney C∞ topologies on the spaces of smooth maps between smooth +manifolds are natural and important topologies. Such spaces are fundamental and +important spaces in the singularity theory of differentiable maps and (applications +to) differential topology of manifolds. For this see [7]. +2.2. Smooth bundles and linear bundles. A smooth bundle means a bundle +whose fiber is a smooth manifold and whose structure group is regarded as some +subgroup of the diffeomorphism group. Linear bundles form an important subclass. +A linear bundle is a bundle whose fiber is a Euclidean space, a unit disk, or a unit +sphere, and whose structure group consists of linear transformations. Note that we +can define linear transformations here naturally. +To know general theory of bundles systematically, see [31] for example. For linear +bundles and so-called characteristic classes of them, see [24] for example. +2.3. Special generic maps. +Definition 1. A smooth map c : X → Y between two smooth manifolds with no +boundaries is special generic if at some small neighborhood of each singular point +p ∈ X, we can choose suitable local coordinates around p and c(p) and c can be +represented by (x1, · · · , xdim X) → (x1, · · · , xdim Y −1, Σdim X−dim Y +1 +j=1 +xdim Y +j−12) +locally for the local coordinates. +A canonical projection of a unit sphere, mapping (x1, x2) ∈ Sk ⊂ Rk+1 = +Rk1 × Rk2 to x1 ∈ Rk1, is a special generic map where k ≥ 1, k1, k2 ≥ 1 and +k = k1 + k2. To check that this is special generic maps is a kind of elementary +exercises on smooth manifolds, smooth maps, Morse functions and differentiable +maps. +Proposition 1 ([27, 28]). Let m ≥ n ≥ 1 be integers. Given a special generic +map f : M → N on an m-dimensional closed and connected manifold M into an +n-dimensional connected manifold N with no boundary. This enjoys the following +properties. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +5 +(1) A suitable n-dimensional compact and connected smooth manifold Wf, a +suitable smooth surjection qf : M → Wf and a suitable smooth immersion +¯f : Wf → N exist and we have a relation f = ¯f ◦ qf. Furthermore, qf can +be chosen as a smooth map whose restriction to the singular set S(f) of f +is a diffeomorphism onto the boundary ∂Wf ⊂ Wf where the manifold of +the target is restricted. +(2) We have some small collar neighborhood N(∂Wf) of the boundary ∂Wf ⊂ +Wf with the following two properties. +(a) The composition of the restriction of qf to the preimage qf −1(N(∂Wf)) +with the canonical projection to ∂Wf is the projection of some linear +bundle whose fiber is the unit disk Dm−n+1. +(b) Suppose that ∂Wf is not closed. The restriction of qf to the preimage +of Wf − Int N(∂Wf) is the projection of some smooth bundle whose +fiber is the unit sphere Sm−n. In some specific cases, the bundle is +regarded as a linear one and the case m − n = 0, 1, 2, 3 satisfies the +conditions. +Definition 2 (E. g. [17]). In Proposition 1, we call the bundle in (2a) the boundary +linear bundle (of f). We call the bundle of (2b) the internal smooth bundle (of it). +The following gives simplest special generic maps. +Proposition 2 ([27]). Let m ≥ n ≥ 1 be integers. Let ¯N be an n-dimensional +smooth, compact and connected manifold whose boundary is not empty and N an +n-dimensional smooth connected manifold with no boundary. Assume also that a +smooth immersion ¯fN : ¯N → N is given. +Then we have a suitable m-dimensional closed and connected manifold M some +special generic map f : M → N and have the following two. +(1) The property (1) of Proposition 1 are enjoyed where Wf and ¯N are identified +as smooth manifolds in a suitable way with the relation ¯fN = ¯f. +(2) A boundary linear bundle and an internal smooth bundle of f are trivial. +This is also regarded as an elementary exercise on smooth maps and differential +topology of manifolds. Remark 1 with Example 2 gives related exposition. +Canonical projections of unit spheres are simplest special generic maps. +We +present another simplest example. +Example 1. Let l be an arbitrary positive integer. Let m and n be integers satisfying +the condition m ≥ n ≥ 2. Assume that an integer 1 ≤ nj ≤ n − 1 is defined for +each integer 1 ≤ j ≤ l. We take a connected sum of l > 0 manifolds in the smooth +category where the j-th manifold is Snj ×Sm−nj. Thus we have a smooth manifold +M0. We have a special generic map f0 : M0 → Rn as in Proposition 2. More +precisely, we have f0 in such a way that the image is represented as a boundary +connected sum of l manifolds taken in the smooth category with the j-th manifold +diffeomorphic to Snj × Dn−nj +Hereafter, a homotopy sphere is a smooth manifold homeomorphic to a (unit) +sphere whose dimension is positive. A standard sphere is a homotopy sphere dif- +feomorphic to some unit sphere. An exotic sphere is a homotopy sphere which is +not diffeomorphic to any unit sphere. It is well-known that 4-dimensional exotic +spheres are still undiscovered. Except these 4-dimensional cases, homotopy spheres +are known to be PL homeomorphic to standard spheres where they are seen as the + +6 +NAOKI KITAZAWA +PL manifolds defined canonically. In this philosophy, 4-dimensional exotic spheres +are known to be not PL homeomorphic to standard spheres. +As a kind of appendices, we present known results on special generic maps and +manifolds admitting them in several situations. +Theorem 1 ([27, 28]). +(1) Let m be an arbitrary integer satisfying m ≥ 2. An +m-dimensional closed and connected manifold M admits a special generic +map f : M → R2 if and only if M is either of the following two. +(a) A homotopy sphere which is not a 4-dimensional exotic sphere. +(b) A manifold represented as a connected sum of smooth manifolds each of +which is the total space of some smooth bundle over S1. Furthermore, +the connected sum is taken in the smooth category and the fiber of each +bundle here is a homotopy sphere which is not a 4-dimensional exotic +sphere. +(2) Let m be an arbitrary integer greater than or equal to 4. Let M be an m- +dimensional closed and simply-connected manifold M. If a special generic +map f : M → R3 exists, then M is either of the following two. +(a) A homotopy sphere which is not a 4-dimensional exotic sphere. +(b) A manifold represented as a connected sum of smooth manifolds each +of which is the total space of a smooth bundle over S2. Furthermore, +the connected sum is taken in the smooth category and the fiber of each +bundle here is a homotopy sphere which is not a 4-dimensional exotic +sphere. +In the case m = 4, 5, 6, the converse is also true. In such a case, a fiber of +each bundle is an (m − 2)-dimensional standard sphere and the total spaces +of the bundles are replaced by the total spaces of linear bundles without +changing the fibers and the base spaces. +(3) Both in the cases (1) and (2), consider the manifold M which is not a +homotopy sphere. M admits a special generic map as in Example 1 such +that an internal smooth bundle and a boundary linear bundle of it may not +be trivial. Let nj = 1 and nj = 2 in Example 1, respectively. +In addition, [26] and [14] have solved variants of problems of Theorem 1 (2) and +obtained answers in the cases (m, n) = (5, 4), (6, 4) where n is the dimension of the +Euclidean space of the target, respectively. +2.4. Some exposition on elementary algebraic topology and differential +topology. We omit systematic and rigorous exposition on elementary algebraic +topology as we have said in the first section. +However, we need exposition for +some. For systematic studies, consult [8] again for example. +One of such exposition is on fundamental classes of connected and compact +(oriented) manifolds. +Let A be a commutative ring having a unique identity element 1A which is +different from the zero element 0A. 1A and −1A are generators of A. A is also +seen as a module over A canonically. For any compact and connected oriented +manifold X, Hdim X(X, ∂X; A) is isomorphic to A as the module. A generator is +given according to the orientation of X. Note that for example, orientations are +not needed in the case A := Z/2Z or the commutative ring of order 2. +For a manifold Y , consider an embedding iX : X → Y satisfying suitable con- +ditions according to the category where we argue. For example, in the smooth + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +7 +category, the embedding is smooth and in the PL or equivalently, in the piecewise +smooth category, this is defined to be piecewise smooth. Furthermore, X is em- +bedded properly. In other words, the boundary is embedded into the boundary and +the interior is embedded into the interior and (in the smooth category) X must +be embedded in a so-called generic way. More precisely, we need ”transversality”. +If a ∈ Hj(Y, ∂Y ; A) is the value of the homomorphism iX∗ : Hdim X(X, ∂X; A) → +Hdim X(Y, ∂Y ; A) induced canonically from the embedding at the fundamental class +[X] ∈ Hdim X(X, ∂X; A), then a is represented by the submanifold iX(X). +We add exposition on ”transversality” related to our smooth embedding iX : +X → Y . We generally consider a smooth embedding satisfying a nice condition on +the dimensions of subspaces of tangent vector spaces and the images of differentials. +More rigorously, the dimension of the intersection of the image of the differential +diXp of the embedding iX : X → Y at each point p ∈ ∂X and the tangent space at +iX(p) ∈ ∂Y must be calculated as dim X + dim ∂Y − dim Y = dim X − 1. This is +also fundamental and important in the singularity theory of differentiable maps and +applications to differential topology of manifolds. Consult [7] again for systematic +studies for example. +For compact, connected and oriented manifolds, so-called Poincar´e duals to ele- +ments of the (co)homology groups and Poincar´e duality (theorem) are important. +We explain about duals in modules and cohomology duals. Let BA be a module +over A having a unique maximal free submodule and let the rank of this submodule +be finite and l. Suppose that we have a basis BA := {ej}l +j=1 of it consisting of +elements which are not divisible by elements which are not units of A. We can define +a homomorphism ej∗,BA from BA into A uniquely by the relation: ej1 +∗,BA(ej2) := 1A +in the case j1 = j2 ej1 +∗,BA(ej2) := 0A in the case j1 ̸= j2. This is the dual to ej +respecting the basis BA := {ej}l +j=1. If BA is a homology group of some topological +space, then, the element is the cohomology dual respecting the basis. +3. Our new class of smooth maps like special generic maps. +Definition 3. Let m ≥ n ≥ 1 be integers. A special-generic-like map or an SGL +map is a smooth map f : M → N on an m-dimensional closed and connected +manifold M into an n-dimensional smooth manifold N with no boundary enjoying +the following properties. +(1) The image f(M) is the image of some smooth immersion ¯fN : ¯N → N of +some n-dimensional compact and connected manifold ¯N. +(2) As in Proposition 1, we have some smooth surjection qf : M → Wf with +the manifold ¯N being identified in a suitable way with Wf as a smooth +manifold and have the relation f = ¯fN ◦qf. Furthermore, we can choose qf +as a map whose restriction to the singular set S(f) gives a diffeomorphism +onto the boundary ∂Wf. +(3) We have some small collar neighborhood N(∂Wf) of the boundary ∂Wf ⊂ +Wf and the composition of the restriction of qf to the preimage with the +canonical projection to the boundary is the projection of some smooth +bundle over ∂Wf. This is also as in Proposition 1. +(4) On the collar neighborhood N(∂Wf) and the preimage qf −1(N(∂Wf)), it +is represented as the product map of the following two smooth maps for +suitable local coordinates around each point p of ∂Wf ⊂ N(∂Wf). + +8 +NAOKI KITAZAWA +(a) A smooth function ˜fp,∂Wf on an (m−n+1)-dimensional smooth com- +pact and connected manifold Ep. +(i) The image of the function ˜fp,∂Wf can be denoted by [minp, maxp]. +We have ˜f −1 +p,∂Wf (minp) = ∂Ep or ˜f −1 +p,∂Wf (maxp) = ∂Ep. The sin- +gular set of the function ˜fp,∂Wf is in the interior Int Ep. +(ii) For values of the function, the preimage of a value contains +some singular points if and only if it is the maximum or the +minimum, which is defined uniquely. If ˜f −1 +p,∂Wf (minp) = ∂Ep, +then ˜f −1 +p,∂Wf (maxp) is a subpolyhedron of dimension at most +m − n where Ep, the outer manifold M and related smooth +manifolds are regarded as the canonically defined PL manifolds. +If ˜f −1 +p,∂Wf (maxp) = ∂Ep, then ˜f −1 +p,∂Wf (minp) is a subpolyhedron +of dimension at most m − n where Ep, the outer manifold M +and related smooth manifolds are regarded as the canonically +defined PL manifolds. +(b) The identity map on a small open neighborhood of p where the neigh- +borhood is considered in ∂Wf +(5) The restriction of qf to the preimage qf −1(Wf − Int N(∂Wf)) is the pro- +jection of some smooth bundle whose fiber is some smooth closed manifold +F. The boundary of the manifold Ep of the domain of the function used +for the product maps before is diffeomorphic to F. +We discuss Definition 3 further by assuming m > n and the manifold F and the +preimage qf −1(p′) of any point in p′ ∈ ∂Wf to be connected. Fix the manifold F. +We fix an arbitrary point q in the interior of Wf. +In Definition 3, for each +connected component C of the boundary ∂Wf, we can define a smooth embedding +α : [0, 1] → Wf enjoying the following properties. +• α(0) = q +• α([0, 1)) ⊂ Int Wf +• α(1) ∈ ∂Wf +• The intersection of the image of the differential of α at 1 and the tangent +vector space of ∂Wf at p := α(1) is of dimension dim ∂Wf − 1. It is a +condition on ”transversality”. +The preimage qf −1(α([0, 1])) is regarded as the manifold Ep of the domain of a +smooth function ˜fp,∂Wf . More rigorously, it is regarded as a function C∞ equivalent +to the function ˜fp,∂Wf and see [7] for such a notion. We have the inclusion of F into +Ep. Let A be a commutative ring (having the unique identity element which is not +the zero element). We can define the homomorphism from Hj(F; A) into Hj(Ep; A) +and the homomorphism πj(F) into πj(Ep) induced by the inclusion uniquely. +Definition 4. We call a homomorphism between the groups above a reference ho- +momorphism along the curve α. +In considerable situations, such homomorphisms are defined uniquely (modulo +suitable equivalence relation). However, we do not consider such arguments pre- +cisely. +Example 2. We present examples by abusing terminologies and notation in Defini- +tions 3 and 4. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +9 +(1) Special generic maps (whose singular sets are not empty on closed and +connected manifolds) are seen as specific SGL maps. We explain about +this. We explain about a height function of a unit sphere. This gives a +specific case of canonical projections of unit spheres and a Morse function. +A height function of a unit disk is defined as the restriction of the height +function of the unit sphere to the preimage of the half-line, the intersection +of the line R and {t ≥ 0 | t ∈ R}. F is taken as the unit sphere Sm−n, +Ep is taken as the unit disk Dm−n+1, and the function ˜fp,∂Wf is a height +function here. Reference homomorphisms along these curves are always the +zero homomorphisms except the case of degree 0 in Definition 4. +(2) Simply generalized special generic maps (whose singular sets are not empty +on closed and connected manifolds) give cases where F is the product of +standard spheres in Definitions 3 and 4. We explain about local functions +used for the product maps on the boundary. Ep is defined as the product of +a copy of some unit disk Dkp+1 and finitely many standard spheres where +kp ≥ 1 is an integer. Here the family of the spheres here are obtained by +removing exactly one sphere in the family of the spheres for the product +F. The function ˜fp,∂Wf is represented as the projection of a trivial bundle +over the copy of Dkp+1 with a height function. The fiber is diffeomorphic +to the product of standard spheres where the family of the spheres here are +obtained by removing the exactly one sphere in the family of the spheres +for the product F as before. The function is a Morse-Bott function. Let +A be a principal ideal domain having the identity element different from +the zero element. In simplest cases, our reference homomorphisms along +our curves are homomorphisms enjoying the following properties where the +coefficient ring is A. +• The kernels are free and of rank 1. +• The kernel is generated by an element represented by a subspace of +the product of standard spheres where the product of the spheres and +the subspace are presented in the following. +– The product of the spheres is canonically identified with F, dif- +feomorphic to and identified with the product �l+1 +j=1Skj, and the +dimension is m − n where l ≥ 0 is an integer: if l = 0, then the +Morse-Bott function is a height function on the copy of Dkp+1 +with kp = k1. This does not depend on which point we choose +in ∂Wf. +– The subspace is represented as �l +j=1Sp(Skj) ⊂ �l +j=1Skj satis- +fying the following conditions: Sp(Skj) = Skj for exactly one +integer j = jp and suitable one-point set {∗j,p} for remaining +integer j ̸= jp. This does not depend on which point we choose +in the connected component C ∋ p of ∂Wf. +Note that in [18], simply generalized special generic maps in simplest cases +here and the homology groups and the cohomology rings of the manifolds +are studied. +Of course, special generic maps are also simply generalized special generic +maps. +We give an additional remark. + +10 +NAOKI KITAZAWA +Remark 1. If the following are given, then we have a natural SGL map with some +simple structure easily by using the canonically obtained trivial bundles over the +complementary set of a suitable collar neighborhood of ∂ ¯N ⊂ ¯N and the product +maps around the collar neighborhood. Proposition 2 is a specific case. +• Integers m ≥ n ≥ 1. +• A smooth immersion ¯fN : ¯N → N of some n-dimensional compact and +connected manifold ¯N into an n-dimensional smooth manifold N with no +boundary. +• A suitable smooth function ˜fp,∂Wf on an (m − n + 1)-dimensional smooth +compact and connected manifold Ep. +We first construct these bundles, projections and these product maps and gluing +them suitably via isomorphisms of the trivial smooth bundles defined canonically +on the boundaries. Simplest examples are given by considering the product maps +of the identity maps between the base spaces and the fibers. Note here that for the +identity maps here, identifications are given suitably and naturally beforehand. +This is important in construction and used in the proof of Main Theorem 2. +Related to Example 2 (2), one of our main work of [18] is regarded as a work +constructing simply generalized special generic maps with prescribed reference ho- +momorphisms along given curves and studying the homology groups and the co- +homology rings of the obtained manifolds of the domains. +Of course reference +homomorphisms along curves are not introduced there. Note also that this gen- +eralizes our previous work on the homology groups and the cohomology rings of +the manifolds in the cases of special generic maps [19]. More precisely, [19] studies +the case where the isomorphisms of the trivial smooth bundles defined canonically +on the boundaries of the canonically obtained trivial bundles are regarded as the +product maps of the identity maps between the base spaces and the fibers. +4. On Main Theorems. +We present Main Theorems 1 and 2 again in terms of Definitions 3 and 4. +Theorem 2 (Main Theorem 1). Let m > n be an arbitrary positive integer satisfy- +ing m − n > 0. Let f : M → N be an SGL map from an m-dimensional closed and +connected manifold into an n-dimensional connected manifold N with no boundary +which is represented as the composition of a smooth surjection qf : M → Wf onto +an n-dimensional compact and k-connected connected manifold Wf with a smooth +immersion ¯f : Wf → N. +We also assume the following conditions. +• k < m − n. +• The preimage qf −1(p) of a point p ∈ Int Wf is diffeomorphic to a closed +and (k − 1)-connected manifold F. +• For any point p ∈ ∂Wf, qf −1(p) is connected and at most k′-dimensional +with k′ ≤ m − n. +• k + k′ + n − 1 < m. +• There exist finitely many smooth curves in Int ¯N and that the reference +homomorphisms along these curves between the k-th homotopy groups are +defined. +Furthermore, the union of their kernels, uniquely defined, and +πk(F) coincide. +Then M is k-connected. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +11 +Proof. We can take a smooth submanifold Sk0 diffeomorphic to Sk in M. qf −1(∂Wf) +is a polyhedron of dimension at most n − 1 + k′. k + (n − 1 + k′) < m implies that +we can smoothly isotope the submanifold Sk0 apart from qf −1(∂Wf). +Sk is, thus, homotopically, regarded as in the total space of the trivial smooth +bundle over a smoothly embedded copy of the unit disk Dn in Int Wf defined by +considering the preimage by qf. Its fiber is (k − 1)-connected and diffeomorphic +to F. By the assumption on our reference homomorphisms, Sk is shown to be +null-homotopic. This completes the proof. +□ +Example 3. +(1) According to [27], for a special generic map into the Euclidean +space or a connected manifold which is not compact and has no boundary, +qf induces isomorphisms between the homology groups and the homotopy +groups whose degrees are at most m − n. Theorem 2 generalizes this. +(2) A map represented as the composition of the projection of a trivial smooth +bundle Sk1 ×Sk2 over Sk1 whose fiber is diffeomorphic to Sk2 with a canon- +ical projection into Rk1−1 where the conditions k1 ≥ 2 and k2 ≥ 2 are +assumed. This is for Theorem 2 where m = k1 + k2, n = k1 − 1, N := Rn, +k = 1, Wf := Dn, ¯f : Wf → N is the canonical smooth embedding, and +F = S1 × Sk2. +Definition 5. An SGL map in Theorem 2 is said to be a k-connected-SGL map. +Theorem 3 (Main Theorem 2). Let n be an arbitrary positive integer. Suppose +that an n-dimensional connected manifold N with no boundary and a smooth im- +mersion ¯f : ¯N → N of an n-dimensional compact and simply-connected manifold +¯N having at least two boundary components are given. Let m = n + 2. Let F be +a closed, connected and orientable surface which is not a sphere. Then we have +a 1-connected-SGL map f : M → N on some m-dimensional closed and simply- +connected manifold M into N satisfying the conditions of Theorem 2 with the no- +tation being abused and ¯N and Wf being identified suitably as smooth manifolds. +Proof. We review our main ingredient of [9]. +We consider a Morse function ˜fS2,l on a surface S2(l) obtained by removing the +interiors of l > 1 copies of the unit disk D2 smoothly embedded in a 2-dimensional +standard sphere enjoying the following properties. This is a fundamental exercise +on Morse functions. See [22, 23] for example. +• The image is denoted by [−a, a] ⊂ R for a real number a. +• ˜f −1 +S2,l(a) and one of connected components of the boundary coincide. +• ˜f −1 +S2,l(−a) and l − 1 of connected components of the boundary coincide. +• Let {sj}l−2 +j=1 denote the set of all singular points of ˜fS2,l. For these points, +we have ˜fS2,l(sj) = −a + 2a 1 +l−1j. +We can consider a smooth deformation ˜fS2,l,[−1,1] : S2(l) × [−1, 1] → [−a, a] × +[−1, 1] of the function enjoying the following properties by fundamental arguments +on the theory of Morse functions, smooth functions, and differential topology of +manifolds, for example. +• Around each singular point of the deformation, it is represented as the +product map of a Morse function and the identity map on a line for suitable +local coordinates. It is a singular point of a so-called fold map and the class + +12 +NAOKI KITAZAWA +of fold maps generalizes the class of Morse functions and that of special +generic maps. See [7] for fold singularity. +• The restriction to the singular set of the deformation is a smooth embedding +such that around the boundaries our condition on transversality is satisfied. +• The image is [−a, a] × [−1, 1]. +• ˜fS2,l,[−1,1]((p, −1)) = ˜fS2,l(p). +• Outside the union of small neighborhoods of singular points sj, the values +of ˜fS2,l,[−1,1] are constant in the interval [−1, 1] in S2(l) × [−1, 1]. +• In the interval [−1, 1] in S2(l) × [−1, 1], the obtained functions are Morse +functions whose singular sets are invariant. Furthermore, at t ∈ [−1, 1], sj +is mapped to ((−a + 2a 1 +l−1j)(−t), t) ∈ [−a, a] × [−1, 1]. +We restrict ˜fS2,l,[−1,1] : S2(l) × [−1, 1] → [−a, a] × [−1, 1] to the preimage of a +disk {x ∈ R2 | ||x|| ≤ 1 +2} and compose this with the height function mapping x to +±|||x||2+c where c is some real number. We adopt a manifold diffeomorphic to the 3- +dimensional manifold of the domain as Ep and a function ˜fp,∂Wf as the obtained one: +more rigorously a smooth function being C∞ equivalent to the obtained function. +We construct our desired map as presented in Remark 1 with Proposition 2. We +investigate ∂Ep and its fundamental group. +It is essentially same to investigate S2(l) × [−1, 1] and its boundary. They are +diffeomorphic to the original 3-dimensional manifold Ep and the surface ∂Ep after +the corners are eliminated. ∂Ep is, by the original argument or fundamental ar- +guments from Morse functions, a closed, connected and orientable surface of genus +l − 1. More precisely, we can consider a smooth isotopy from S2(l) × [−1, 1] to Ep +by deforming the map in a natural way. We can argue and have smooth maps, +1-dimensional manifolds diffeomorphic to circles or closed intervals, and other geo- +metric objects as follows. +• In S2(l) × {−1}, we have exactly l − 1 smoothly embedded curves, each of +which is denoted by cj : [−1, 1] → S2(l) × {−1} for an integer 1 ≤ j ≤ l − 1. +Furthermore, we can have the family enjoying the following properties. Here +the connected component ˜f −1 +S2,l,[−1,1]((a, −1)) is denoted by C0 and a circle +of course. ˜f −1 +S2,l,[−1,1]((−a, −1)) consists of exactly l−1 disjoint circles, each +of which is denoted by Cj for an integer 1 ≤ j ≤ l − 1. The disjoint union +⊔l−1 +j=1Cj−1 and the boundary of the surface S2(l) × {−1} coincide of course. +– cj((−1, 1)) ⊂ Int S2(l) × {−1}. +– cj(−1) ∈ Cj−1 and cj(1) ∈ Cj. +– At boundaries, the condition on transversality is satisfied. +– The images of distinct curves are mutually disjoint. +– The images are sufficiently far from the singular set of the function +˜fS2,l,[−1,1]|S2(l)×{−1}. +• We consider the product of the image of each curve before and [−1, 1]. +The boundary is regarded as a circle smoothly embedded in the boundary +∂ (S2(l) × [−1, 1]) and null-homotopic in S2(l) × [−1, 1]. +• The l − 1 circles in the boundary ∂ (S2(l) × [−1, 1]) are mutually disjoint +and (by choosing the curves cj suitably first) we can obtain a situation such +that after cutting ∂ (S2(l) × [−1, 1]) along the circles, we have a compact, +connected and orientable surface whose Euler number is 4 − 2l and which + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +13 +has exactly 2(l − 1) boundary connected components. In terms of [21], the +obtained l − 1 circles form an independent and regular system of hyper- +surfaces in the closed, connected and orientable surface ∂ (S2(l) × [−1, 1]) +of genus l − 1. Let {Sj}l−1 +j=1 denote the family of the smoothly embedded +circles. +• (By considering suitable situations), we can have another family {Sj +′}l−1 +j=1 +of disjointly and smoothly embedded circles in ∂ (S2(l) × [−1, 1]) enjoying +the following properties due to fundamental topological theory of surfaces. +– Sj1 and Sj2 +′ do not intersect if j1 ̸= j2 and intersect at some one-point +set if j1 = j2. +– In the one-point set, these distinct circles intersect satisfying a condi- +tion on the transversality: the sum of the images of the differentials +there is 2. They give a so-called normal-crossing. +– There exists a diffeomorphism which preserves the orientation and +maps the disjoint union ⊔jSj onto ⊔jSj +′ and the disjoint union ⊔jSj +′ +onto ⊔jSj. +– We can choose an arbitrary point in the surface outside the union of +these l−1+l−1 = 2(l−1) circles and connect the point and some point +in each of the 2(l−1) circles by a smooth curve. For each circle, we can +have an element of π1(∂ (S2(l) × [−1, 1])) and the set of the resulting +2l − 2 elements generates the fundamental group of the surface. +Related to these arguments, [21] applies methods closely related to us. This +preprint is independent of our study. However methods presented there must be +important for us. There, to study homomorphisms from the fundamental group +of a compact and connected manifold into free groups, closed intervals or circles +disjointly embedded into surfaces, or more generally, such systems of hypersurfaces +are studied, for example. +We go back to our proof. Remember that S2(l) × [−1, 1] and Ep are essentially +same in our argument. We apply methods in Remark 1. By the obtained property +on the surface, its fundamental group and nice systems of circles, for distinct two +connected components of the boundary ∂ ¯N, we can choose suitable distinct diffeo- +morphisms on ∂ ¯N for the two connected components and use the product map of +the identity map on the base space and the diffeomorphism on the fiber for each +connected components of the boundary ∂ ¯N to have a situation for Theorem 2. +This completes the proof. +□ +Example 4. In the case F := S1 × S1 in Theorem 3, our construction gives simply +generalized special genric maps. +For example, we consider the situation in the +following. +• A smooth immersion ¯f : ¯N → N of an n-dimensional compact and simply- +connected manifold ¯N is an embedding. +• ¯N := Sk × D1 and N := Rk+1 with k ≥ 2 and n = k + 1. +We easily have a smooth embedding of ¯N into N here. For our desired map here, +first consider a smooth function on S3 whose image is a closed interval in R and +whose singular set is mapped onto the boundary of the image. Furthermore, the +function is chosen as a Morse-Bott function such that the preimage of a point in +the interior of the image is diffeomorphic to F. We consider the product map of + +14 +NAOKI KITAZAWA +this function and the identity map on Sk. After that, we embed the image via the +embedding ¯f to have our desired map. +We present another phenomenon on SGL maps. We explain about simply gen- +eralized special generic maps in Example 2 (2) and Remark 1 again. Hereafter, as +the coefficient ring A, we take a principal ideal domain having the unique iden- +tity element different from the zero element. According to our main work of [18], +as Remark 1 presents shortly, we can easily construct a natural simply generalized +special generic map on some suitable m-dimensional closed and connected manifold +M such that in Example 2 (2), we can choose jp as an arbitrary integer 1 ≤ j ≤ l+1 +for some point p ∈ ∂ ¯N (if the boundary of the manifold ¯N has sufficiently many +connected components). According to our construction, the resulting reference ho- +momorphisms along the curves between the homology groups of degrees at least +1 are always the zero homomorphisms. This is also due to the structure of the +(co)homology group of the product of spheres, understood by so-called K¨unneth +theorem for example. See [8] again. K¨unneth theorem also shows the following. +• We can choose the cohomology duals to the elements of the homology +groups being also the generators of the kernels of the presented reference +homomorphisms along the curves and represented by the spheres in the +product of the spheres, identified with F, in Example 2 (2), respecting +some suitable bases. For this, universal coefficient theorem is important for +example. See [8] again. +• The set of these cohomology duals respecting the bases generates the co- +homology ring of F. +In other words, we may expect the following phenomenon: these two conditions on +the reference homomorphisms along the curves and the structure of the cohomol- +ogy ring of F induce the fact that the reference homomorphisms along the curves +between the homology groups of degrees at least 1 are always the zero homomor- +phisms. +Problem 3. Are phenomena as presented here observed in general? +Example 5. Maps in Theorem 3 support Problem 3. +We give a counterexample. +Main Theorem 3. The answer to Problem 3 is false in general. +Proof. Let k > 1 be an integer. Let A := Z/kZ, which is the naturally defined +quotient ring of the ring Z and also a cyclic group of order k under the products for +the quotient ring. We need the notions of the Euler number and, more generally, +the Euler class of a linear bundle and for them, see [24] for example. This explains +about classical theory of linear bundles systematically and essentially same notions +are presented. In our case or the case of linear bundles over closed, connected and +orientable surfaces whose fibers are S1 or D2, Euler numbers and Euler classes are +essentially same by fundamental algebraic topology such as Poincar´e duality for the +surfaces. +We can consider a linear bundle ˜ +M 4,k over S2 whose fiber is the unit disk D2 +and whose subbundle ∂ ˜ +M 4,k obtained by restricting the fiber to ∂D2 ⊂ D2 is of +Euler number k. By some fundamental theory on construction of special genric +maps of [27], we have a smooth map ˜f4,k : ˜ +M 4,k → S2 × R enjoying the following +properties. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +15 +• The image is denoted by S2 × [a, b] ⊂ S2 × R. +• The singular set is mapped onto S2 × {b}. +• The preimage of S2 × {a} and the boundary coincide. +• Around each point of S2 × {b}, it is locally represented as the product +map of a height function on a copy of the unit disk D2 and the identity +map on a 2-dimensional disk. More rigorously, on an open neighborhood +U ⊂ S2 diffeomorphic to the interior of the unit disk D2, the restriction +of ˜f4,k to the preimage of U × [a, b], we have some diffeomorphism ΦU : +˜f −1 +4,k(U × [a, b]) → U × D2 and the relation (idU × ˜h2,[a,b]) ◦ ΦU = ˜f4,k : +˜f −1 +4,k(U × [a, b]) → U × [a, b] where the notation is as follows. +– idU is the identity map on U. +– ˜h2,[a,b] is a (suitably scaled) height function on (a copy of) D2. +An argument essentially same as this is also important in Theorem 5.7 (3) of our +preprint [20] for example. +By composing the resulting map into S2 × R with the projection to R gives a +function suitable for ˜f,∂Wf in Definition 3. In Definition 3, Ep = ˜ +M 4,k and F is +diffeomorphic to ∂ ˜ +M 4,k of course. We can construct an SGL map as in Remark 1. +We have the homology exact sequence for the pair ( ˜ +M 4,k, ∂ ˜ +M 4,k). See [8] again +for the sequence. By using some theory such as homology groups of closed, con- +nected and orientable manifolds here and Poincar´e duality, we can find isomor- +phisms represented via ”∼=”in the sequence +→ H3(∂ ˜ +M 4,k; A) ∼= A → H3( ˜ +M 4,k; A) ∼= {0} → H3( ˜ +M 4,k, ∂ ˜ +M 4,k; A) ∼= H1( ˜ +M 4,k; A) ∼= {0} +→ H2(∂ ˜ +M 4,k; A) ∼= A → H2( ˜ +M 4,k; A) ∼= A → H2( ˜ +M 4,k, ∂ ˜ +M 4,k; A) ∼= H2( ˜ +M 4,k; A) ∼= A +→ H1(∂ ˜ +M 4,k; A) ∼= A → H1( ˜ +M 4,k; A) ∼= {0} +and the inclusion induces an isomorphism between H2(∂ ˜ +M 4,k; A) and H2( ˜ +M 4,k; A). +The cohomology ring H∗(∂ ˜ +M 4,k; A) of ∂ ˜ +M 4,k is known to be generated by an el- +ement of H1(∂ ˜ +M 4,k; A). Furthermore, Hj(∂ ˜ +M 4,k; A) and Hj(∂ ˜ +M 4,k; A) are known +to be free for j = 0, 1, 2, 3 and of rank 1. +We can see that this completes the +proof. +□ +Remark 2. In our maps in our proof of Main Theorem 3, π1(F) is also known to be +isomorphic to A. So we easily have cases for Theorem 2 with k = 1 there if other +objects are suitably given. +5. Acknowledgement +The author would like to thank Takahiro Yamamoto for related discussions on +our previous study [18], especially, on the terminology ”generalized special generic +maps” and meanings of our studies in the singularity theory of differentiable maps +and applications to geometry of manifolds. These discussions have motivated the +author to study further and contributed to our present study. + +16 +NAOKI KITAZAWA +References +[1] R. Bott, Nondegenerate critical manifolds, Ann. of Math. 60 (1954), 248–261. +[2] V. M. Buchstaber and T. E. Panov, Toric topology, Mathematical Surveys and Monographs, +Vol. 204, American Mathematical Society, Providence, RI, 2015. +[3] O. Burlet and G. de Rham, Sur certaines applications g´en´eriques d’une vari´et´e close a 3 +dimensions dans le plan, Enseign. Math. 20 (1974). 275–292. +[4] E. 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Kitazawa, Closed manifolds admitting no special generic maps whose codimensions are +negative and their cohomology rings, submitted to a refereed journal, arxiv:2008.04226v5. +[11] N. Kitazawa, Notes on explicit special generic maps into Euclidean spaces whose dimensions +are greater than 4, a revised version is submitted based on positive comments (major revision) +by referees and editors after the first submission to a refereed journal, arxiv:2010.10078. +[12] N. Kitazawa, Restrictions on special generic maps on 6-dimensional or higher dimensional +closed and simply-connected manifolds, submitted to a refereed journal, arxiv:2201.09437v4. +[13] N. Kitazawa, Proofs of the non-existence of special generic maps on the 3-dimensional complex +projective space, submitted to a refereed journal, arxiv:2202.00883. +[14] N. Kitazawa, Characterizing certain classes of 6-dimensional closed and simply-connected +manifolds via special generic maps, arxiv:2205.04048. +[15] N. 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Michalak, Relations between Reeb graphs, systems of hypersur- +faces and epimorphisms onto free groups, arXiv:2002.02388. +[22] J. Milnor, Morse Theory, Annals of Mathematic Studies AM-51, Princeton University Press; +1st Edition (1963.5.1). +[23] J. Milnor, Lectures on the h-cobordism theorem, Math. Notes, Princeton Univ. Press, Prince- +ton, N.J. 1965. +[24] J. Milnor and J. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76. +Princeton, N. J; Princeton University Press (1974). +[25] E. E. Moise, Affine Structures in 3-Manifold: V. The Triangulation Theorem and Hauptver- +mutung, Ann. of Math., Second Series, Vol. 56, No. 1 (1952), 96–114. +[26] M. Nishioka, +Special generic maps of 5-dimensional manifolds, +Revue Roumaine de +Math‘ematiques Pures et Appliqu`ees, Volume LX No.4 (2015), 507–517. +[27] O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology +Appl. 49 (1993), 265–293. + +SMOOTH MAPS LIKE SPECIAL GENERIC MAPS +17 +[28] O. Saeki, Topology of special generic maps into R3, Workshop on Real and Complex Singu- +larities (Sao Carlos, 1992), Mat. Contemp. 5 (1993), 161–186. +[29] O. Saeki and K. Sakuma, On special generic maps into R3, Pacific J. Math. 184 (1998), +175–193. +[30] O. Saeki and K. Sakuma, Special generic maps of 4-manifolds and compact complex analytic +surfaces, Math. Ann. 313, 617–633, 1999. +[31] N. Steenrod, The topology of fibre bundles, Princeton University Press (1951). +[32] D. J. Wrazidlo, Standard special generic maps of homotopy spheres into Eucidean spaces, +Topology Appl. 234 (2018), 348–358, arxiv:1707.08646. +[33] D. J. Wrazidlo, On special generic maps of rational homology spheres into Euclidean spaces, +arxiv:2009.05928. +Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku +Fukuoka 819-0395, Japan, TEL (Office): ++81-92-802-4402, FAX (Office): ++81-92-802- +4405, +Email address: n-kitazawa@imi.kyushu-u.ac.jp +Webpage: https://naokikitazawa.github.io/NaokiKitazawa.html + diff --git a/ytFLT4oBgHgl3EQfnC-z/content/tmp_files/load_file.txt b/ytFLT4oBgHgl3EQfnC-z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9380c003d32613245bd6730a80fca1ecaf0c995e --- /dev/null +++ b/ytFLT4oBgHgl3EQfnC-z/content/tmp_files/load_file.txt @@ -0,0 +1,630 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf,len=629 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='12126v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='GN] 28 Jan 2023 SMOOTH MAPS LIKE SPECIAL GENERIC MAPS NAOKI KITAZAWA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In our paper, we introduce special-generic-like maps or SGL maps as smooth maps and study their several algebraic topological and differential topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The new class generalize the class of so-called special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Special generic maps are smooth maps which are locally projections or the product maps of Morse functions and the identity maps on disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Morse functions with exactly two singular points on spheres or Morse functions in Reeb’s theorem are simplest examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Special generic maps and the manifolds of their do- mains have been studied well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Their structures are simple and this help us to study explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' As important properties, they have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' To cover wider classes of manifolds as the domains, the author previously introduced a class generalizing the class of special generic maps and smaller than our class: simply generalized special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Special generic maps are smooth maps which are locally projections or the prod- uct maps of Morse functions and the identity maps on disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Morse functions with exactly two singular points on spheres or Morse functions in Reeb’s theorem are simplest examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Canonical projections of so-called unit spheres are also simplest examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Pioneering studies are [3, 4, 6] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Since the 1990s, their algebraic topological and differential topological properties have been studied by Saeki and Sakuma ([27, 28, 29, 30]), followed by Nishioka ([26]), Wrazidlo ([32, 33]) and the author ([10, 11, 12, 13, 14, 15, 16, 17, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Some elementary manifolds such as ones represented as connected sums of the products of two spheres admit natural special generic maps in considerable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' On the contrary, the differentiable structures of spheres and some elementary man- ifolds admitting special generic maps are restricted strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Homology groups of the manifolds are also restricted in considerable cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The author has started studies on the cohomology rings of the manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' These studies are due to the fact that special generic maps have simple and nice structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We introduce some fundamental notions, terminologies and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Rk is the k-dimensional Euclidean space, which is a simplest k-dimensional smooth manifold for an arbitrary positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is, by considering a standard Euclidean metric, a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ||x|| ≥ 0 denotes th distance between x ∈ Rk and the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' R := R1 and Z denotes the ring of all integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Morse-Bott functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Homology and cohomol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2020 Mathematics Subject Classification: Primary 57R45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Secondary 57R19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 1 2 NAOKI KITAZAWA Sk := {x ∈ Rk+1 | ||x|| = 1} denotes the k-dimensional unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is a k-dimensional smooth compact and connected submanifold if k ≥ 2 and the two- point set with the discrete topology if k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Dk := {x ∈ Rk | ||x|| ≤ 1} denotes the k-dimensional unit disk for an arbitrary integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is a k-dimensional smooth compact and connected submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For a non-empty topological space X having the structure of a cell complex, we can define the dimension uniquely, denoted by dim X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A topological manifold is known to have the structure of a CW complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A smooth manifold is known to have the structure of a polyhedron and more precisely, in some canonical way we can define the unique PL structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is seen as a PL manifold canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is well-known that topological manifolds whose dimensions are at most 3 have the unique structures of polyhedra and that topological spaces homeomorphic to polyhedra whose dimensions are at most 2 have the unique structures of polyhedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Theory related to such uniqueness is discussed in [25] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For a differentiable map c : X → Y , a point x ∈ X is a singular point if the rank of the differential dcx there is smaller than the minimum between dim X and dim Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The singular set of c is the set of all singular points of c and let S(c) denote the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We define and discuss special generic maps later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' However, we explain about these maps shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A special generic map is a smooth map from an m-dimensional manifold with no boundary into n-dimensional one with m ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If the manifold of the domain is closed, then the image is regarded as an n-dimensional smoothly immersed compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preimage of each point in the interior of the immersed manifold is diffeomorphic to Sm−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This gives a projection over the interior of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preimage of each point in the boundary is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Around each point in the boundary, it is the product map of a Morse function with exactly one singular point on a disk of dimension m − n + 1 and the identity map on a disk of dimension n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Propositions 1 and 2, presented in the next section, are on this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Can you formulate new nice classes of smooth maps having simple and nice structures and properties special generic maps have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Are the classes of Problem 1 can cover wider classes of manifolds to study as the domains of these maps Essentially same problems are also asked in the previous preprint of the author [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' There the author introduced simply generalized special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In our paper, we generalize this class as the class of special-generic-like maps or SGL maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about simply generalized special generic maps shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preim- age of a point in the interior of the n-dimensional smoothly immersed manifold is replaced by the product of manifolds diffeomorphic to unit spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Around each point in the boundary, a Morse function on a disk is replaced by the composition of a projection onto the disk with a Morse function with exactly one singular point there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Each preimage of the projection is the product of manifolds diffeomorphic to unit spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The Morse functions are replaced by Morse-Bott functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For Morse-Bott functions, see a related pioneering study [1] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note that this class respects local structures of so-called moment maps on so- called (symplectic) toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The class of moment maps and that of sym- plectic toric manifolds are important in symplectic geometry and toric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 3 In most cases, special generic maps are not admitted by such manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This has been conjectured by us and the author has shown results seeming to be related to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' [13, 15] show the non-existence of special generic maps on complex projec- tive spaces except for the 1-dimensional case or 2-dimensional spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' They are simplest (symplectic) toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' [5] is a pioneering study on symlectic toric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See also [2] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Exposition related to this history is also in [18] For our new class, the preimage of each point in the interior of the immersed manifold and Morse(-Bott) functions used around the boundary are generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We present our main results as Main Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Hereafter, elementary algebraic topology, more precisely, elementary (co)homology theory is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We do not explain about this rigorously or systematically and we also expect we have related knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For related studies, see [8] and we also abuse notions, terminologies and notation here or ones which seem to be generally used or familiar to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Main Theorem 1 (Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let k > 0 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m > n be an arbitrary positive integers satisfying m − n ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let f : M → N be an SGL map from an m-dimensional closed and connected manifold into an n-dimensional connected manifold N with no boundary which is represented as the composition of a smooth surjection qf : M → Wf onto a compact and k-connected connected manifold Wf with a smooth immersion ¯f : Wf → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Suppose that a family {pj} ⊂ Int Wf of finitely many points satisfying the following conditions exists .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (1) The preimage qf −1(pj) for pj ∈ Int Wf is diffeomorphic to a closed and (k − 1)-connected manifold F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) For a smooth curve αpj : [0, 1] → Wf satisfying αpj(0) = pj, αpj((0, 1)) ⊂ Int Wf and αpj(1) ∈ ∂Wf and a so-called ”transversality”, presented later in several situations, the inclusion αpj −1(0) ⊂ αpj −1([0, 1]) gives the kernel Ker αpj ∗ of the naturally defined homomorphism between the k-th homotopy groups πk(αpj −1(0)) and πk(αpj −1([0, 1])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (3) The union � jKer αpj ∗ generates πk(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Then M is k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Main Theorem 2 (Theorem 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let n be an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Suppose that an n-dimensional connected manifold N with no boundary and a smooth im- mersion ¯f : ¯N → N of an n-dimensional compact and simply-connected manifold ¯N which has at least two boundary components are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let F be a closed, connected and orientable surface which is not a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Then we have an SGL map f : M → N from some m-dimensional closed and simply-connected manifold M into N satisfying the conditions of Main Theorem 1 with the notation being abused and Wf and ¯N being identified suitably as smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' These results are presented again in revised versions in the third section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about the content of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The second section is for preliminar- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' There we also review special generic maps for example as fundamental objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We define our new class rigorously in the third section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The fourth section proves Main Theorems and present some exmaples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Main Theorem 3 is also presented as another new result of us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The author is a member of the project JSPS KAKENHI Grant Number JP22K18267 4 NAOKI KITAZAWA ”Visualizing twists in data through monodromy” (Principal Investigator: Osamu Saeki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Our present study is supported by the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Data essentially supporting our present study are all contained in our present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A diffeomorphism between smooth manifolds means a smooth map which has no no singular points and which is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A diffeomorphism on a manifold is a diffeomorphism from the (smooth) manifold onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Two manifolds are diffeomorphic if and only if there exists a diffeomorphism between these manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This naturally gives an equivalence relation on the family of all smooth manifolds with their corners being eliminated in a well-known canon- ical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' These operations always give mutually diffeomorphic manifolds with no corners for a fixed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can define PL homeomorphic manifolds using PL homeomorphisms or piecewise smooth homeomorphisms similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The diffeomorphism group of a manifold is the space consisting of all diffeomor- phisms on the manifold where the so-called Whitney C∞ topology is given as its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is also a topological group and a so-called infinite dimensional Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Whitney C∞ topologies on the spaces of smooth maps between smooth manifolds are natural and important topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Such spaces are fundamental and important spaces in the singularity theory of differentiable maps and (applications to) differential topology of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For this see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Smooth bundles and linear bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A smooth bundle means a bundle whose fiber is a smooth manifold and whose structure group is regarded as some subgroup of the diffeomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Linear bundles form an important subclass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A linear bundle is a bundle whose fiber is a Euclidean space, a unit disk, or a unit sphere, and whose structure group consists of linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note that we can define linear transformations here naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' To know general theory of bundles systematically, see [31] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For linear bundles and so-called characteristic classes of them, see [24] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A smooth map c : X → Y between two smooth manifolds with no boundaries is special generic if at some small neighborhood of each singular point p ∈ X, we can choose suitable local coordinates around p and c(p) and c can be represented by (x1, · · · , xdim X) → (x1, · · · , xdim Y −1, Σdim X−dim Y +1 j=1 xdim Y +j−12) locally for the local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A canonical projection of a unit sphere, mapping (x1, x2) ∈ Sk ⊂ Rk+1 = Rk1 × Rk2 to x1 ∈ Rk1, is a special generic map where k ≥ 1, k1, k2 ≥ 1 and k = k1 + k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' To check that this is special generic maps is a kind of elementary exercises on smooth manifolds, smooth maps, Morse functions and differentiable maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Proposition 1 ([27, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m ≥ n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Given a special generic map f : M → N on an m-dimensional closed and connected manifold M into an n-dimensional connected manifold N with no boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This enjoys the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 5 (1) A suitable n-dimensional compact and connected smooth manifold Wf, a suitable smooth surjection qf : M → Wf and a suitable smooth immersion ¯f : Wf → N exist and we have a relation f = ¯f ◦ qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, qf can be chosen as a smooth map whose restriction to the singular set S(f) of f is a diffeomorphism onto the boundary ∂Wf ⊂ Wf where the manifold of the target is restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) We have some small collar neighborhood N(∂Wf) of the boundary ∂Wf ⊂ Wf with the following two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (a) The composition of the restriction of qf to the preimage qf −1(N(∂Wf)) with the canonical projection to ∂Wf is the projection of some linear bundle whose fiber is the unit disk Dm−n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (b) Suppose that ∂Wf is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The restriction of qf to the preimage of Wf − Int N(∂Wf) is the projection of some smooth bundle whose fiber is the unit sphere Sm−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In some specific cases, the bundle is regarded as a linear one and the case m − n = 0, 1, 2, 3 satisfies the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Definition 2 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In Proposition 1, we call the bundle in (2a) the boundary linear bundle (of f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We call the bundle of (2b) the internal smooth bundle (of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The following gives simplest special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Proposition 2 ([27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m ≥ n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let ¯N be an n-dimensional smooth, compact and connected manifold whose boundary is not empty and N an n-dimensional smooth connected manifold with no boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Assume also that a smooth immersion ¯fN : ¯N → N is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Then we have a suitable m-dimensional closed and connected manifold M some special generic map f : M → N and have the following two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (1) The property (1) of Proposition 1 are enjoyed where Wf and ¯N are identified as smooth manifolds in a suitable way with the relation ¯fN = ¯f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) A boundary linear bundle and an internal smooth bundle of f are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is also regarded as an elementary exercise on smooth maps and differential topology of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Remark 1 with Example 2 gives related exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Canonical projections of unit spheres are simplest special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We present another simplest example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let l be an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m and n be integers satisfying the condition m ≥ n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Assume that an integer 1 ≤ nj ≤ n − 1 is defined for each integer 1 ≤ j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We take a connected sum of l > 0 manifolds in the smooth category where the j-th manifold is Snj ×Sm−nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Thus we have a smooth manifold M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We have a special generic map f0 : M0 → Rn as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More precisely, we have f0 in such a way that the image is represented as a boundary connected sum of l manifolds taken in the smooth category with the j-th manifold diffeomorphic to Snj × Dn−nj Hereafter, a homotopy sphere is a smooth manifold homeomorphic to a (unit) sphere whose dimension is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A standard sphere is a homotopy sphere dif- feomorphic to some unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' An exotic sphere is a homotopy sphere which is not diffeomorphic to any unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is well-known that 4-dimensional exotic spheres are still undiscovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Except these 4-dimensional cases, homotopy spheres are known to be PL homeomorphic to standard spheres where they are seen as the 6 NAOKI KITAZAWA PL manifolds defined canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In this philosophy, 4-dimensional exotic spheres are known to be not PL homeomorphic to standard spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' As a kind of appendices, we present known results on special generic maps and manifolds admitting them in several situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Theorem 1 ([27, 28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (1) Let m be an arbitrary integer satisfying m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' An m-dimensional closed and connected manifold M admits a special generic map f : M → R2 if and only if M is either of the following two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (a) A homotopy sphere which is not a 4-dimensional exotic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (b) A manifold represented as a connected sum of smooth manifolds each of which is the total space of some smooth bundle over S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, the connected sum is taken in the smooth category and the fiber of each bundle here is a homotopy sphere which is not a 4-dimensional exotic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) Let m be an arbitrary integer greater than or equal to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let M be an m- dimensional closed and simply-connected manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If a special generic map f : M → R3 exists, then M is either of the following two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (a) A homotopy sphere which is not a 4-dimensional exotic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (b) A manifold represented as a connected sum of smooth manifolds each of which is the total space of a smooth bundle over S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, the connected sum is taken in the smooth category and the fiber of each bundle here is a homotopy sphere which is not a 4-dimensional exotic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In the case m = 4, 5, 6, the converse is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In such a case, a fiber of each bundle is an (m − 2)-dimensional standard sphere and the total spaces of the bundles are replaced by the total spaces of linear bundles without changing the fibers and the base spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (3) Both in the cases (1) and (2), consider the manifold M which is not a homotopy sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' M admits a special generic map as in Example 1 such that an internal smooth bundle and a boundary linear bundle of it may not be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let nj = 1 and nj = 2 in Example 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In addition, [26] and [14] have solved variants of problems of Theorem 1 (2) and obtained answers in the cases (m, n) = (5, 4), (6, 4) where n is the dimension of the Euclidean space of the target, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Some exposition on elementary algebraic topology and differential topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We omit systematic and rigorous exposition on elementary algebraic topology as we have said in the first section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' However, we need exposition for some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For systematic studies, consult [8] again for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' One of such exposition is on fundamental classes of connected and compact (oriented) manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let A be a commutative ring having a unique identity element 1A which is different from the zero element 0A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 1A and −1A are generators of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A is also seen as a module over A canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For any compact and connected oriented manifold X, Hdim X(X, ∂X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) is isomorphic to A as the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A generator is given according to the orientation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note that for example, orientations are not needed in the case A := Z/2Z or the commutative ring of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For a manifold Y , consider an embedding iX : X → Y satisfying suitable con- ditions according to the category where we argue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For example, in the smooth SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 7 category, the embedding is smooth and in the PL or equivalently, in the piecewise smooth category, this is defined to be piecewise smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, X is em- bedded properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In other words, the boundary is embedded into the boundary and the interior is embedded into the interior and (in the smooth category) X must be embedded in a so-called generic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More precisely, we need ”transversality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If a ∈ Hj(Y, ∂Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) is the value of the homomorphism iX∗ : Hdim X(X, ∂X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) → Hdim X(Y, ∂Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) induced canonically from the embedding at the fundamental class [X] ∈ Hdim X(X, ∂X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A), then a is represented by the submanifold iX(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We add exposition on ”transversality” related to our smooth embedding iX : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We generally consider a smooth embedding satisfying a nice condition on the dimensions of subspaces of tangent vector spaces and the images of differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More rigorously, the dimension of the intersection of the image of the differential diXp of the embedding iX : X → Y at each point p ∈ ∂X and the tangent space at iX(p) ∈ ∂Y must be calculated as dim X + dim ∂Y − dim Y = dim X − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is also fundamental and important in the singularity theory of differentiable maps and applications to differential topology of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Consult [7] again for systematic studies for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For compact, connected and oriented manifolds, so-called Poincar´e duals to ele- ments of the (co)homology groups and Poincar´e duality (theorem) are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about duals in modules and cohomology duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let BA be a module over A having a unique maximal free submodule and let the rank of this submodule be finite and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Suppose that we have a basis BA := {ej}l j=1 of it consisting of elements which are not divisible by elements which are not units of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can define a homomorphism ej∗,BA from BA into A uniquely by the relation: ej1 ∗,BA(ej2) := 1A in the case j1 = j2 ej1 ∗,BA(ej2) := 0A in the case j1 ̸= j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is the dual to ej respecting the basis BA := {ej}l j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If BA is a homology group of some topological space, then, the element is the cohomology dual respecting the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Our new class of smooth maps like special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m ≥ n ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A special-generic-like map or an SGL map is a smooth map f : M → N on an m-dimensional closed and connected manifold M into an n-dimensional smooth manifold N with no boundary enjoying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (1) The image f(M) is the image of some smooth immersion ¯fN : ¯N → N of some n-dimensional compact and connected manifold ¯N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) As in Proposition 1, we have some smooth surjection qf : M → Wf with the manifold ¯N being identified in a suitable way with Wf as a smooth manifold and have the relation f = ¯fN ◦qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, we can choose qf as a map whose restriction to the singular set S(f) gives a diffeomorphism onto the boundary ∂Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (3) We have some small collar neighborhood N(∂Wf) of the boundary ∂Wf ⊂ Wf and the composition of the restriction of qf to the preimage with the canonical projection to the boundary is the projection of some smooth bundle over ∂Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is also as in Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (4) On the collar neighborhood N(∂Wf) and the preimage qf −1(N(∂Wf)), it is represented as the product map of the following two smooth maps for suitable local coordinates around each point p of ∂Wf ⊂ N(∂Wf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 8 NAOKI KITAZAWA (a) A smooth function ˜fp,∂Wf on an (m−n+1)-dimensional smooth com- pact and connected manifold Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (i) The image of the function ˜fp,∂Wf can be denoted by [minp, maxp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We have ˜f −1 p,∂Wf (minp) = ∂Ep or ˜f −1 p,∂Wf (maxp) = ∂Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The sin- gular set of the function ˜fp,∂Wf is in the interior Int Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (ii) For values of the function, the preimage of a value contains some singular points if and only if it is the maximum or the minimum, which is defined uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If ˜f −1 p,∂Wf (minp) = ∂Ep, then ˜f −1 p,∂Wf (maxp) is a subpolyhedron of dimension at most m − n where Ep, the outer manifold M and related smooth manifolds are regarded as the canonically defined PL manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If ˜f −1 p,∂Wf (maxp) = ∂Ep, then ˜f −1 p,∂Wf (minp) is a subpolyhedron of dimension at most m − n where Ep, the outer manifold M and related smooth manifolds are regarded as the canonically defined PL manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (b) The identity map on a small open neighborhood of p where the neigh- borhood is considered in ∂Wf (5) The restriction of qf to the preimage qf −1(Wf − Int N(∂Wf)) is the pro- jection of some smooth bundle whose fiber is some smooth closed manifold F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The boundary of the manifold Ep of the domain of the function used for the product maps before is diffeomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We discuss Definition 3 further by assuming m > n and the manifold F and the preimage qf −1(p′) of any point in p′ ∈ ∂Wf to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Fix the manifold F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We fix an arbitrary point q in the interior of Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In Definition 3, for each connected component C of the boundary ∂Wf, we can define a smooth embedding α : [0, 1] → Wf enjoying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' α(0) = q α([0, 1)) ⊂ Int Wf α(1) ∈ ∂Wf The intersection of the image of the differential of α at 1 and the tangent vector space of ∂Wf at p := α(1) is of dimension dim ∂Wf − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is a condition on ”transversality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preimage qf −1(α([0, 1])) is regarded as the manifold Ep of the domain of a smooth function ˜fp,∂Wf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More rigorously, it is regarded as a function C∞ equivalent to the function ˜fp,∂Wf and see [7] for such a notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We have the inclusion of F into Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let A be a commutative ring (having the unique identity element which is not the zero element).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can define the homomorphism from Hj(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) into Hj(Ep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) and the homomorphism πj(F) into πj(Ep) induced by the inclusion uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We call a homomorphism between the groups above a reference ho- momorphism along the curve α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In considerable situations, such homomorphisms are defined uniquely (modulo suitable equivalence relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' However, we do not consider such arguments pre- cisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We present examples by abusing terminologies and notation in Defini- tions 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 9 (1) Special generic maps (whose singular sets are not empty on closed and connected manifolds) are seen as specific SGL maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about a height function of a unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This gives a specific case of canonical projections of unit spheres and a Morse function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A height function of a unit disk is defined as the restriction of the height function of the unit sphere to the preimage of the half-line, the intersection of the line R and {t ≥ 0 | t ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' F is taken as the unit sphere Sm−n, Ep is taken as the unit disk Dm−n+1, and the function ˜fp,∂Wf is a height function here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Reference homomorphisms along these curves are always the zero homomorphisms except the case of degree 0 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) Simply generalized special generic maps (whose singular sets are not empty on closed and connected manifolds) give cases where F is the product of standard spheres in Definitions 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about local functions used for the product maps on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Ep is defined as the product of a copy of some unit disk Dkp+1 and finitely many standard spheres where kp ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Here the family of the spheres here are obtained by removing exactly one sphere in the family of the spheres for the product F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The function ˜fp,∂Wf is represented as the projection of a trivial bundle over the copy of Dkp+1 with a height function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The fiber is diffeomorphic to the product of standard spheres where the family of the spheres here are obtained by removing the exactly one sphere in the family of the spheres for the product F as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The function is a Morse-Bott function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let A be a principal ideal domain having the identity element different from the zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In simplest cases, our reference homomorphisms along our curves are homomorphisms enjoying the following properties where the coefficient ring is A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The kernels are free and of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The kernel is generated by an element represented by a subspace of the product of standard spheres where the product of the spheres and the subspace are presented in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – The product of the spheres is canonically identified with F, dif- feomorphic to and identified with the product �l+1 j=1Skj, and the dimension is m − n where l ≥ 0 is an integer: if l = 0, then the Morse-Bott function is a height function on the copy of Dkp+1 with kp = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This does not depend on which point we choose in ∂Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – The subspace is represented as �l j=1Sp(Skj) ⊂ �l j=1Skj satis- fying the following conditions: Sp(Skj) = Skj for exactly one integer j = jp and suitable one-point set {∗j,p} for remaining integer j ̸= jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This does not depend on which point we choose in the connected component C ∋ p of ∂Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note that in [18], simply generalized special generic maps in simplest cases here and the homology groups and the cohomology rings of the manifolds are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Of course, special generic maps are also simply generalized special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We give an additional remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 10 NAOKI KITAZAWA Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' If the following are given, then we have a natural SGL map with some simple structure easily by using the canonically obtained trivial bundles over the complementary set of a suitable collar neighborhood of ∂ ¯N ⊂ ¯N and the product maps around the collar neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Proposition 2 is a specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Integers m ≥ n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A smooth immersion ¯fN : ¯N → N of some n-dimensional compact and connected manifold ¯N into an n-dimensional smooth manifold N with no boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A suitable smooth function ˜fp,∂Wf on an (m − n + 1)-dimensional smooth compact and connected manifold Ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We first construct these bundles, projections and these product maps and gluing them suitably via isomorphisms of the trivial smooth bundles defined canonically on the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Simplest examples are given by considering the product maps of the identity maps between the base spaces and the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note here that for the identity maps here, identifications are given suitably and naturally beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is important in construction and used in the proof of Main Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Related to Example 2 (2), one of our main work of [18] is regarded as a work constructing simply generalized special generic maps with prescribed reference ho- momorphisms along given curves and studying the homology groups and the co- homology rings of the obtained manifolds of the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Of course reference homomorphisms along curves are not introduced there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Note also that this gen- eralizes our previous work on the homology groups and the cohomology rings of the manifolds in the cases of special generic maps [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More precisely, [19] studies the case where the isomorphisms of the trivial smooth bundles defined canonically on the boundaries of the canonically obtained trivial bundles are regarded as the product maps of the identity maps between the base spaces and the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' On Main Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We present Main Theorems 1 and 2 again in terms of Definitions 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Theorem 2 (Main Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m > n be an arbitrary positive integer satisfy- ing m − n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let f : M → N be an SGL map from an m-dimensional closed and connected manifold into an n-dimensional connected manifold N with no boundary which is represented as the composition of a smooth surjection qf : M → Wf onto an n-dimensional compact and k-connected connected manifold Wf with a smooth immersion ¯f : Wf → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We also assume the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' k < m − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preimage qf −1(p) of a point p ∈ Int Wf is diffeomorphic to a closed and (k − 1)-connected manifold F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For any point p ∈ ∂Wf, qf −1(p) is connected and at most k′-dimensional with k′ ≤ m − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' k + k′ + n − 1 < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' There exist finitely many smooth curves in Int ¯N and that the reference homomorphisms along these curves between the k-th homotopy groups are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, the union of their kernels, uniquely defined, and πk(F) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Then M is k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can take a smooth submanifold Sk0 diffeomorphic to Sk in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' qf −1(∂Wf) is a polyhedron of dimension at most n − 1 + k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' k + (n − 1 + k′) < m implies that we can smoothly isotope the submanifold Sk0 apart from qf −1(∂Wf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Sk is, thus, homotopically, regarded as in the total space of the trivial smooth bundle over a smoothly embedded copy of the unit disk Dn in Int Wf defined by considering the preimage by qf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Its fiber is (k − 1)-connected and diffeomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' By the assumption on our reference homomorphisms, Sk is shown to be null-homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (1) According to [27], for a special generic map into the Euclidean space or a connected manifold which is not compact and has no boundary, qf induces isomorphisms between the homology groups and the homotopy groups whose degrees are at most m − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Theorem 2 generalizes this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (2) A map represented as the composition of the projection of a trivial smooth bundle Sk1 ×Sk2 over Sk1 whose fiber is diffeomorphic to Sk2 with a canon- ical projection into Rk1−1 where the conditions k1 ≥ 2 and k2 ≥ 2 are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is for Theorem 2 where m = k1 + k2, n = k1 − 1, N := Rn, k = 1, Wf := Dn, ¯f : Wf → N is the canonical smooth embedding, and F = S1 × Sk2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' An SGL map in Theorem 2 is said to be a k-connected-SGL map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Theorem 3 (Main Theorem 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let n be an arbitrary positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Suppose that an n-dimensional connected manifold N with no boundary and a smooth im- mersion ¯f : ¯N → N of an n-dimensional compact and simply-connected manifold ¯N having at least two boundary components are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let m = n + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let F be a closed, connected and orientable surface which is not a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Then we have a 1-connected-SGL map f : M → N on some m-dimensional closed and simply- connected manifold M into N satisfying the conditions of Theorem 2 with the no- tation being abused and ¯N and Wf being identified suitably as smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We review our main ingredient of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We consider a Morse function ˜fS2,l on a surface S2(l) obtained by removing the interiors of l > 1 copies of the unit disk D2 smoothly embedded in a 2-dimensional standard sphere enjoying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is a fundamental exercise on Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See [22, 23] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The image is denoted by [−a, a] ⊂ R for a real number a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ˜f −1 S2,l(a) and one of connected components of the boundary coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ˜f −1 S2,l(−a) and l − 1 of connected components of the boundary coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let {sj}l−2 j=1 denote the set of all singular points of ˜fS2,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For these points, we have ˜fS2,l(sj) = −a + 2a 1 l−1j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can consider a smooth deformation ˜fS2,l,[−1,1] : S2(l) × [−1, 1] → [−a, a] × [−1, 1] of the function enjoying the following properties by fundamental arguments on the theory of Morse functions, smooth functions, and differential topology of manifolds, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Around each singular point of the deformation, it is represented as the product map of a Morse function and the identity map on a line for suitable local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is a singular point of a so-called fold map and the class 12 NAOKI KITAZAWA of fold maps generalizes the class of Morse functions and that of special generic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See [7] for fold singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The restriction to the singular set of the deformation is a smooth embedding such that around the boundaries our condition on transversality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The image is [−a, a] × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ˜fS2,l,[−1,1]((p, −1)) = ˜fS2,l(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Outside the union of small neighborhoods of singular points sj, the values of ˜fS2,l,[−1,1] are constant in the interval [−1, 1] in S2(l) × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In the interval [−1, 1] in S2(l) × [−1, 1], the obtained functions are Morse functions whose singular sets are invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, at t ∈ [−1, 1], sj is mapped to ((−a + 2a 1 l−1j)(−t), t) ∈ [−a, a] × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We restrict ˜fS2,l,[−1,1] : S2(l) × [−1, 1] → [−a, a] × [−1, 1] to the preimage of a disk {x ∈ R2 | ||x|| ≤ 1 2} and compose this with the height function mapping x to ±|||x||2+c where c is some real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We adopt a manifold diffeomorphic to the 3- dimensional manifold of the domain as Ep and a function ˜fp,∂Wf as the obtained one: more rigorously a smooth function being C∞ equivalent to the obtained function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We construct our desired map as presented in Remark 1 with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We investigate ∂Ep and its fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' It is essentially same to investigate S2(l) × [−1, 1] and its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' They are diffeomorphic to the original 3-dimensional manifold Ep and the surface ∂Ep after the corners are eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ∂Ep is, by the original argument or fundamental ar- guments from Morse functions, a closed, connected and orientable surface of genus l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More precisely, we can consider a smooth isotopy from S2(l) × [−1, 1] to Ep by deforming the map in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can argue and have smooth maps, 1-dimensional manifolds diffeomorphic to circles or closed intervals, and other geo- metric objects as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In S2(l) × {−1}, we have exactly l − 1 smoothly embedded curves, each of which is denoted by cj : [−1, 1] → S2(l) × {−1} for an integer 1 ≤ j ≤ l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, we can have the family enjoying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Here the connected component ˜f −1 S2,l,[−1,1]((a, −1)) is denoted by C0 and a circle of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ˜f −1 S2,l,[−1,1]((−a, −1)) consists of exactly l−1 disjoint circles, each of which is denoted by Cj for an integer 1 ≤ j ≤ l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The disjoint union ⊔l−1 j=1Cj−1 and the boundary of the surface S2(l) × {−1} coincide of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – cj((−1, 1)) ⊂ Int S2(l) × {−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – cj(−1) ∈ Cj−1 and cj(1) ∈ Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – At boundaries, the condition on transversality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – The images of distinct curves are mutually disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – The images are sufficiently far from the singular set of the function ˜fS2,l,[−1,1]|S2(l)×{−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We consider the product of the image of each curve before and [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The boundary is regarded as a circle smoothly embedded in the boundary ∂ (S2(l) × [−1, 1]) and null-homotopic in S2(l) × [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The l − 1 circles in the boundary ∂ (S2(l) × [−1, 1]) are mutually disjoint and (by choosing the curves cj suitably first) we can obtain a situation such that after cutting ∂ (S2(l) × [−1, 1]) along the circles, we have a compact, connected and orientable surface whose Euler number is 4 − 2l and which SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 13 has exactly 2(l − 1) boundary connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In terms of [21], the obtained l − 1 circles form an independent and regular system of hyper- surfaces in the closed, connected and orientable surface ∂ (S2(l) × [−1, 1]) of genus l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let {Sj}l−1 j=1 denote the family of the smoothly embedded circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' (By considering suitable situations), we can have another family {Sj ′}l−1 j=1 of disjointly and smoothly embedded circles in ∂ (S2(l) × [−1, 1]) enjoying the following properties due to fundamental topological theory of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – Sj1 and Sj2 ′ do not intersect if j1 ̸= j2 and intersect at some one-point set if j1 = j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – In the one-point set, these distinct circles intersect satisfying a condi- tion on the transversality: the sum of the images of the differentials there is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' They give a so-called normal-crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – There exists a diffeomorphism which preserves the orientation and maps the disjoint union ⊔jSj onto ⊔jSj ′ and the disjoint union ⊔jSj ′ onto ⊔jSj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – We can choose an arbitrary point in the surface outside the union of these l−1+l−1 = 2(l−1) circles and connect the point and some point in each of the 2(l−1) circles by a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For each circle, we can have an element of π1(∂ (S2(l) × [−1, 1])) and the set of the resulting 2l − 2 elements generates the fundamental group of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Related to these arguments, [21] applies methods closely related to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This preprint is independent of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' However methods presented there must be important for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' There, to study homomorphisms from the fundamental group of a compact and connected manifold into free groups, closed intervals or circles disjointly embedded into surfaces, or more generally, such systems of hypersurfaces are studied, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We go back to our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Remember that S2(l) × [−1, 1] and Ep are essentially same in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We apply methods in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' By the obtained property on the surface, its fundamental group and nice systems of circles, for distinct two connected components of the boundary ∂ ¯N, we can choose suitable distinct diffeo- morphisms on ∂ ¯N for the two connected components and use the product map of the identity map on the base space and the diffeomorphism on the fiber for each connected components of the boundary ∂ ¯N to have a situation for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In the case F := S1 × S1 in Theorem 3, our construction gives simply generalized special genric maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For example, we consider the situation in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A smooth immersion ¯f : ¯N → N of an n-dimensional compact and simply- connected manifold ¯N is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' ¯N := Sk × D1 and N := Rk+1 with k ≥ 2 and n = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We easily have a smooth embedding of ¯N into N here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For our desired map here, first consider a smooth function on S3 whose image is a closed interval in R and whose singular set is mapped onto the boundary of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, the function is chosen as a Morse-Bott function such that the preimage of a point in the interior of the image is diffeomorphic to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We consider the product map of 14 NAOKI KITAZAWA this function and the identity map on Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' After that, we embed the image via the embedding ¯f to have our desired map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We present another phenomenon on SGL maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We explain about simply gen- eralized special generic maps in Example 2 (2) and Remark 1 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Hereafter, as the coefficient ring A, we take a principal ideal domain having the unique iden- tity element different from the zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' According to our main work of [18], as Remark 1 presents shortly, we can easily construct a natural simply generalized special generic map on some suitable m-dimensional closed and connected manifold M such that in Example 2 (2), we can choose jp as an arbitrary integer 1 ≤ j ≤ l+1 for some point p ∈ ∂ ¯N (if the boundary of the manifold ¯N has sufficiently many connected components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' According to our construction, the resulting reference ho- momorphisms along the curves between the homology groups of degrees at least 1 are always the zero homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This is also due to the structure of the (co)homology group of the product of spheres, understood by so-called K¨unneth theorem for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See [8] again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' K¨unneth theorem also shows the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can choose the cohomology duals to the elements of the homology groups being also the generators of the kernels of the presented reference homomorphisms along the curves and represented by the spheres in the product of the spheres, identified with F, in Example 2 (2), respecting some suitable bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' For this, universal coefficient theorem is important for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See [8] again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The set of these cohomology duals respecting the bases generates the co- homology ring of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In other words, we may expect the following phenomenon: these two conditions on the reference homomorphisms along the curves and the structure of the cohomol- ogy ring of F induce the fact that the reference homomorphisms along the curves between the homology groups of degrees at least 1 are always the zero homomor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Are phenomena as presented here observed in general?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Maps in Theorem 3 support Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We give a counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Main Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The answer to Problem 3 is false in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let k > 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Let A := Z/kZ, which is the naturally defined quotient ring of the ring Z and also a cyclic group of order k under the products for the quotient ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We need the notions of the Euler number and, more generally, the Euler class of a linear bundle and for them, see [24] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' This explains about classical theory of linear bundles systematically and essentially same notions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In our case or the case of linear bundles over closed, connected and orientable surfaces whose fibers are S1 or D2, Euler numbers and Euler classes are essentially same by fundamental algebraic topology such as Poincar´e duality for the surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can consider a linear bundle ˜ M 4,k over S2 whose fiber is the unit disk D2 and whose subbundle ∂ ˜ M 4,k obtained by restricting the fiber to ∂D2 ⊂ D2 is of Euler number k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' By some fundamental theory on construction of special genric maps of [27], we have a smooth map ˜f4,k : ˜ M 4,k → S2 × R enjoying the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' SMOOTH MAPS LIKE SPECIAL GENERIC MAPS 15 The image is denoted by S2 × [a, b] ⊂ S2 × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The singular set is mapped onto S2 × {b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The preimage of S2 × {a} and the boundary coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Around each point of S2 × {b}, it is locally represented as the product map of a height function on a copy of the unit disk D2 and the identity map on a 2-dimensional disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' More rigorously, on an open neighborhood U ⊂ S2 diffeomorphic to the interior of the unit disk D2, the restriction of ˜f4,k to the preimage of U × [a, b], we have some diffeomorphism ΦU : ˜f −1 4,k(U × [a, b]) → U × D2 and the relation (idU × ˜h2,[a,b]) ◦ ΦU = ˜f4,k : ˜f −1 4,k(U × [a, b]) → U × [a, b] where the notation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – idU is the identity map on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' – ˜h2,[a,b] is a (suitably scaled) height function on (a copy of) D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' An argument essentially same as this is also important in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content='7 (3) of our preprint [20] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' By composing the resulting map into S2 × R with the projection to R gives a function suitable for ˜f,∂Wf in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In Definition 3, Ep = ˜ M 4,k and F is diffeomorphic to ∂ ˜ M 4,k of course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can construct an SGL map as in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We have the homology exact sequence for the pair ( ˜ M 4,k, ∂ ˜ M 4,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' See [8] again for the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' By using some theory such as homology groups of closed, con- nected and orientable manifolds here and Poincar´e duality, we can find isomor- phisms represented via ”∼=”in the sequence → H3(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= A → H3( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= {0} → H3( ˜ M 4,k, ∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= H1( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= {0} → H2(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= A → H2( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= A → H2( ˜ M 4,k, ∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= H2( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= A → H1(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= A → H1( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) ∼= {0} and the inclusion induces an isomorphism between H2(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) and H2( ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' The cohomology ring H∗(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) of ∂ ˜ M 4,k is known to be generated by an el- ement of H1(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Furthermore, Hj(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) and Hj(∂ ˜ M 4,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' A) are known to be free for j = 0, 1, 2, 3 and of rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' We can see that this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' In our maps in our proof of Main Theorem 3, π1(F) is also known to be isomorphic to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' So we easily have cases for Theorem 2 with k = 1 there if other objects are suitably given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Acknowledgement The author would like to thank Takahiro Yamamoto for related discussions on our previous study [18], especially, on the terminology ”generalized special generic maps” and meanings of our studies in the singularity theory of differentiable maps and applications to geometry of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' These discussions have motivated the author to study further and contributed to our present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' 16 NAOKI KITAZAWA References [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFLT4oBgHgl3EQfnC-z/content/2301.12126v1.pdf'} +page_content=' Bott, Nondegenerate critical manifolds, Ann.' 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